1. science
2. physics

Advances in
ATOMIC, MOLECULAR, AND OPTICAL PHYSICS
V O L U M E 53
Editors
PAUL R. B ERMAN
University of Michigan
Ann Arbor, Michigan
C HUN C. L IN
University of Wisconsin
Madison, Wisconsin
E NNIO A RIMONDO
University of Pisa
Pisa, Italy
Editorial Board
C. J OACHAIN
Université Libre de Bruxelles
Brussels, Belgium
M. G AVRILA
F.O.M. Insituut voor Atoom- en Molecuulfysica
Amsterdam, The Netherlands
M. I NOKUTI
Argonne National Laboratory
Argonne, Illinois
Founding Editor
S IR DAVID BATES
Supplements
1. Atoms in Intense Laser Fields, Mihai Gavrila, Ed.
2. Cavity Quantum Electrodynamics, Paul R. Berman
3. Cross Section Data, Mitio Inokuti, Ed.
ADVANCES IN
ATOMIC,
MOLECULAR,
AND OPTICAL
PHYSICS
Edited by
G. Rempe
MAX - PLANCK INSTITUTE
FOR QUANTUM OPTICS
GARCHING , GERMANY
and
M.O. Scully
TEXAS A & M UNIVERSITY
AND
PRINCETON UNIVERSITY
Volume 53
AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK
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Contents
C ONTRIBUTORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
P REFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
xvii
Non-Classical Light from Artificial Atoms
Thomas Aichele, Matthias Scholz, Sven Ramelow and Oliver Benson
1.
2.
3.
4.
5.
6.
7.
8.
9.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Single Quantum Dots . . . . . . . . . . . . . . . . . . . . . . . .
Single-Photon Generation . . . . . . . . . . . . . . . . . . . . . .
A Single Photon as Particle and Wave . . . . . . . . . . . . . . .
A Multi-Color Photon Source . . . . . . . . . . . . . . . . . . .
Multiplexed Quantum Cryptography on the Single-Photon Level
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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30
30
Quantum Chaos, Transport, and Control—in Quantum Optics
Javier Madroñero, Alexey Ponomarev, André R.R. Carvalho, Sandro Wimberger,
Carlos Viviescas, Andrey Kolovsky, Klaus Hornberger, Peter Schlagheck,
Andreas Krug and Andreas Buchleitner
1.
2.
3.
4.
5.
6.
Introduction . . . . . . .
Spectral Properties . . .
Dynamics and Transport
Control through Chaos .
Conclusion . . . . . . . .
References . . . . . . . .
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34
34
41
59
67
68
Introduction . . . . . . . . . . . . . . . .
Single Atoms in a MOT . . . . . . . . . .
Preparing Single Atoms in a Dipole Trap
Quantum State Preparation and Detection
Superposition States of Single Atoms . .
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76
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Manipulating Single Atoms
Dieter Meschede and Arno Rauschenbeutel
1.
2.
3.
4.
5.
vii
viii
6.
7.
8.
9.
10.
11.
12.
Contents
Loading Multiple Atoms into the Dipole Trap . . . . . .
Realization of a Quantum Register . . . . . . . . . . . .
Controlling the Atoms’ Absolute and Relative Positions
Towards Entanglement of Neutral Atoms . . . . . . . .
Conclusions . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgements . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . .
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89
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106
107
120
136
136
137
1. General Linear Input–Output Transformation for a Linear Optical Device . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2. The Phase-Insensitive Amplifier . . . . . . . . . . . . . . . . . . . . .
3. The Multimode Phase Insensitive Amplifier . . . . . . . . . . . . . . .
4. The Nature of the Ancilla Modes . . . . . . . . . . . . . . . . . . . . .
5. An Optical Amplifier Working at the Quantum Limit . . . . . . . . . .
6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
140
141
143
144
147
148
148
Spatial Imaging with Wavefront Coding and Optical Coherence
Tomography
Thomas Hellmuth
1.
2.
3.
4.
5.
6.
Introduction . . . . . . . . . . . . . . . . . . . . . . .
Enhanced Depth of Focus with Wavefront Coding . .
Spatial Imaging with Optical Coherence Tomography
Conclusion . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgements . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . .
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The Quantum Properties of Multimode Optical Amplifiers Revisited
G. Leuchs, U.L. Andersen and C. Fabre
Quantum Optics of Ultra-Cold Molecules
D. Meiser, T. Miyakawa, H. Uys and P. Meystre
1.
2.
3.
4.
5.
6.
7.
8.
Introduction . . . . . . . . . . . . . . . . . . .
Molecular Micromaser . . . . . . . . . . . . .
Passage Time Statistics of Molecule Formation
Counting Statistics of Molecular Fields . . . .
Molecules as Probes of Spatial Correlations . .
Conclusion . . . . . . . . . . . . . . . . . . . .
Acknowledgements . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . .
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152
153
163
168
173
181
182
182
Contents
ix
Atom Manipulation in Optical Lattices
Georg Raithel and Natalya Morrow
1.
2.
3.
4.
5.
6.
7.
8.
9.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Theoretical Considerations . . . . . . . . . . . . . . . . . . . . . .
Review of One-Dimensional Lattice Configurations for Rubidium
Periodic Well-to-Well Tunneling in Gray Lattices . . . . . . . . .
Influence of Magnetic Fields on Tunneling . . . . . . . . . . . . .
Sloshing-Type Wave-Packet Motion . . . . . . . . . . . . . . . . .
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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187
190
196
208
213
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222
223
223
Femtosecond Laser Interaction with Solid Surfaces: Explosive Ablation and
Self-Assembly of Ordered Nanostructures
Juergen Reif and Florenta Costache
1.
2.
3.
4.
5.
6.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
Energy Coupling . . . . . . . . . . . . . . . . . . . . . . . .
Secondary Processes: Dissipation and Desorption/Ablation
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgements . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . .
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228
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233
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254
256
260
270
277
286
287
288
Characterization of Single Photons Using Two-Photon Interference
T. Legero, T. Wilk, A. Kuhn and G. Rempe
1.
2.
3.
4.
5.
6.
7.
8.
Introduction . . . . . . . .
Single-Photon Light Fields
Two-Photon Interference .
Jitter . . . . . . . . . . . .
Experiment and Results . .
Conclusion . . . . . . . . .
Acknowledgements . . . .
References . . . . . . . . .
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Fluctuations in Ideal and Interacting Bose–Einstein Condensates: From
the Laser Phase Transition Analogy to Squeezed States and Bogoliubov
Quasiparticles
Vitaly V. Kocharovsky, Vladimir V. Kocharovsky, Martin Holthaus,
C.H. Raymond Ooi, Anatoly Svidzinsky, Wolfgang Ketterle and Marlan O. Scully
x
Contents
1.
2.
3.
4.
5.
6.
7.
8.
9.
A.
B.
C.
D.
E.
F.
10.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
History of the Bose–Einstein Distribution . . . . . . . . . . . . . . . .
Grand Canonical versus Canonical Statistics of BEC Fluctuations . .
Dynamical Master Equation Approach and Laser Phase-Transition
Analogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Quasiparticle Approach and Maxwell’s Demon Ensemble . . . . . . .
Why Condensate Fluctuations in the Interacting Bose Gas are Anomalously Large, Non-Gaussian, and Governed by Universal Infrared Singularities? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Bose’s and Einstein’s Way of Counting Microstates . . . . . . . . . .
Analytical Expression for the Mean Number of Condensed Atoms . .
Formulas for the Central Moments of Condensate Fluctuations . . . .
Analytical Expression for the Variance of Condensate Fluctuations . .
Single Mode Coupled to a Reservoir of Oscillators . . . . . . . . . . .
The Saddle-Point Method for Condensed Bose Gases . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
293
298
315
328
357
372
390
394
395
395
397
399
401
402
404
408
LIDAR-Monitoring of the Air with Femtosecond Plasma Channels
Ludger Wöste, Steffen Frey and Jean-Pierre Wolf
1.
2.
3.
4.
5.
6.
7.
8.
9.
Introduction . . . . . . . . . . . . . . . . . . . . . .
Conventional LIDAR Measurements . . . . . . . . .
The Femtosecond-LIDAR Experiment . . . . . . . .
Nonlinear Propagation of Ultra-Intense Laser Pulses
White Light Femtosecond LIDAR Measurements .
Nonlinear Interactions with Aerosols . . . . . . . .
Conclusion . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgements . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . .
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413
415
419
421
427
433
437
438
439
I NDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C ONTENTS OF VOLUMES IN T HIS S ERIAL . . . . . . . . . . . . . . .
443
453
CONTRIBUTORS
Numbers in parentheses indicate the pages on which the author’s contributions begin.
T HOMAS A ICHELE (1), Nano Optics, Physics Department, Humboldt-Universität zu
Berlin, 10117 Berlin, Germany
M ATTHIAS S CHOLZ (1), Nano Optics, Physics Department, Humboldt-Universität zu
Berlin, 10117 Berlin, Germany
S VEN R AMELOW (1), Nano Optics, Physics Department, Humboldt-Universität zu Berlin,
10117 Berlin, Germany
O LIVER B ENSON (1), Nano Optics, Physics Department, Humboldt-Universität zu Berlin,
10117 Berlin, Germany
JAVIER M ADROÑERO (33), Physik Department, Technische Universität München, JamesFranck-Straße, D-85747 Garching, Germany; Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Str. 38, D-01187 Dresden, Germany
A LEXEY P ONOMAREV (33), Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Str. 38, D-01187 Dresden, Germany
A NDRÉ R.R. C ARVALHO (33), Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Str. 38, D-01187 Dresden, Germany
S ANDRO W IMBERGER (33), Dipartimento di Fisica Enrico Fermi and CNR-INFM, Università di Pisa, Largo Pontecorvo 3, I-56127 Pisa, Italy
C ARLOS V IVIESCAS (33), Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer
Str. 38, D-01187 Dresden, Germany
A NDREY KOLOVSKY (33), Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Str. 38, D-01187 Dresden, Germany
K LAUS H ORNBERGER (33), Arnold-Sommerfeld-Zentrum für Theoretische Physik,
Ludwig-Maximilians-Universität München, Theresienstr. 37, D-80333 München, Germany
P ETER S CHLAGHECK (33), Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany
xi
xii
Contributors
A NDREAS K RUG (33), Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer
Str. 38, D-01187 Dresden, Germany
A NDREAS B UCHLEITNER (33), Max-Planck-Institut für Physik komplexer Systeme,
Nöthnitzer Str. 38, D-01187 Dresden, Germany
D IETER M ESCHEDE (75), Institut für Angewandte Physik, Universität Bonn, Wegelerstr.
8, D-53115 Bonn, Germany
A RNO R AUSCHENBEUTEL (75), Institut für Angewandte Physik, Universität Bonn,
Wegelerstr. 8, D-53115 Bonn, Germany
T HOMAS H ELLMUTH (105), Department of Optoelectronics, Aalen University of Applied
Sciences, Germany
G. L EUCHS (139), Max Planck Research Group of Optics, Information and Photonics,
University of Erlangen-Nürnberg, Erlangen, Germany
U.L. A NDERSEN (139), Max Planck Research Group of Optics, Information and Photonics, University of Erlangen-Nürnberg, Erlangen, Germany
C. FABRE (139), Laboratoire Kastler-Brossel, Université Pierre et Marie Curie et Ecole
Normale Supérieure, Place Jussieu, cc74, 75252 Paris cedex 05, France
D. M EISER (151), Department of Physics, The University of Arizona, 1118 E. 4th Street,
Tucson, AZ 85705, USA
T. M IYAKAWA (151), Department of Physics, The University of Arizona, 1118 E. 4th
Street, Tucson, AZ 85705, USA
H. U YS (151), Department of Physics, The University of Arizona, 1118 E. 4th Street,
Tucson, AZ 85705, USA
P. M EYSTRE (151), Department of Physics, The University of Arizona, 1118 E. 4th Street,
Tucson, AZ 85705, USA
G EORG R AITHEL (187), FOCUS Center, Department of Physics University of Michigan,
Ann Arbor, MI 48109, USA
NATALYA M ORROW (187), FOCUS Center, Department of Physics University of Michigan, Ann Arbor, MI 48109, USA
J UERGEN R EIF (227), Brandenburgische Technische Universität Cottbus, Konrad-Wachsmann-Allee 1, 03046 Cottbus, Germany BTU/IHP JointLab, Erich-Weinert-Strasse 1,
03046 Cottbus, Germany
F LORENTA C OSTACHE (227), Brandenburgische Technische Universität Cottbus, KonradWachsmann-Allee 1, 03046 Cottbus, Germany BTU/IHP JointLab, Erich-WeinertStrasse 1, 03046 Cottbus, Germany
T. L EGERO (253), Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, 85748
Garching, Germany
Contributors
xiii
T. W ILK (253), Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, 85748
Garching, Germany
A. K UHN (253), Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, 85748
Garching, Germany
G. R EMPE (253), Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, 85748
Garching, Germany
V ITALY V. KOCHAROVSKY (291), Institute for Quantum Studies and Department of
Physics, Texas A&M University, TX 77843-4242, USA; Institute of Applied Physics,
Russian Academy of Science, 600950 Nizhny Novgorod, Russia
V LADIMIR V. KOCHAROVSKY (291), Institute of Applied Physics, Russian Academy of
Science, 600950 Nizhny Novgorod, Russia
M ARTIN H OLTHAUS (291), Institut für Physik, Carl von Ossietzky Universitat, D-2611
Oldenburg, Germany
C.H. R AYMOND O OI (291), Institute for Quantum Studies and Department of Physics,
Texas A&M University, TX 77843-4242, USA
A NATOLY S VIDZINSKY (291), Institute for Quantum Studies and Department of Physics,
Texas A&M University, TX 77843-4242, USA
W OLFGANG K ETTERLE (291), MIT-Harvard Center for Ultracold Atoms, and Department
of Physics, MIT, Cambridge, MA 02139, USA
M ARLAN O. S CULLY (291), Institute for Quantum Studies and Department of Physics,
Texas A&M University, TX 77843-4242, USA; Princeton Institute for Material Science and Technology, Princeton University, NJ 08544-1009, USA
L UDGER W ÖSTE (413), Physics Department, Freie Universität Berlin, Arnimallee 14,
14195 Berlin, Germany
S TEFFEN F REY (413), MIT, Department of Earth, Atmospheric, and Planetary Sciences,
77 Massachusetts Avenue, Cambridge, MA 02139, USA
J EAN -P IERRE W OLF (413), GAP-Biophotonics, University of Geneva, 20, rue de l’Ecole
de Médecine, 1211 Geneva 4, Switzerland
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HERBERT WALTHER
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PREFACE
Prof. Herbert Walther is a quantum optics star of galactic magnitude! Experimental physicists admire his ability to conduct experiments previously considered
impossible. Theoretical physicists eagerly look forward to the stunning results
that come out of his laboratory. His discoveries have brought increasingly new
life to both the theoretical and experimental quantum optical physicists. The scientific methods developed in his laboratory have become a mainstay to quantum
optics laboratories all over the world.
Three qualities of Herbert Walther stand out most clearly: His enormous energy, his unique dedication to science and his special eye for scientific quality.
He obviously must subscribe to the German motto: “Die Probleme existieren, um
überwunden zu werden” (problems exist to be overcome). This statement holds
true not only for scientific matters but also for science policy. Three examples
illustrating his qualities offer themselves:
In the early 1980s the Max-Planck-Society inherited the Ringberg castle located in the picturesque Bavarian mountains next to Lake Tegernsee. The MaxPlanck Institute for Quantum Optics was one of the first institutes that started
using this facility as a retreat to review its progress in the various groups and initiate novel research directions. It was during one of these early meetings when
Herbert Walther’s group was discussing the new possibilities in cavity quantum
electrodynamics offered by the unique combination of Rydberg atoms and high-Q
microwave resonators. Herbert Walther proposed to build a new type of maser
driven by a single atom. However, fresh ideas are rarely received with enthusiasm, especially by those who have to transfer the Gedanken experiments into
real experiments. It was argued that too many novel techniques, such as, atomic
beams and cryogenic equipment, which had only worked separately before, now
had to be combined into one single experiment: it was considered impossible to
make all these experimental tricks work at the same time. Herbert Walther tried
to convince the nay sayers about the feasibility of the experiment—without success. Finally he decided to follow a different route and attract students to do the
work. Indeed, several students, starting with Dieter Meschede, Gerhard Rempe,
Ferdinand Schmidt-Kaler, Georg Raithel, Oliver Benson, and Ben Varcoe, now all
faculty members at different scientific institutions, together with other students,
postdocs and visitors, planned and implemented today’s famous research line of
the micromaser.
Second, Herbert Walther is a great institution builder. He was a main driving
force responsible for building up the Max-Planck Institute for Quantum Optics to
xvii
xviii
PREFACE
one of the top institutions in the field worldwide and in fact a Mecca for many
international scientists visiting it religiously. Today it is hard to believe that in the
late 1970s the Institute was an institute on probation: The Max-Planck Society
had installed a research group in the newly emerged field of laser physics, the
so called “Projektgruppe für Laserforschung” (project group for laser research).
Herbert Walther was hired as one of the directors. In no time he was able to attract
many bright students to his group and bring the high society of laser physics to
the project group. Clearly enough, the Max-Planck Society then was given little
choice but to found a full fledged Max-Planck Institute. Many years later, Herbert
Walther had the unique opportunity to repeat this story of success on a much larger
scale: As Vice President of the Max-Planck Society, he was a leading authority
while setting up the new institutes in East Germany after reunification.
A third example illustrating Herbert Walther’s lack of fear was the hiring of
Prof. Theodor Hänsch who had previously turned down several offers from German universities. It seemed to be a hopeless task to lure Hänsch away from Stanford. Nevertheless, Herbert Walther was not afraid to compete. He first arranged
a Humboldt prize for Hänsch to get him used again to German life. He then managed to arrange an offer which could not be refused—a chair at the University
of Munich together with a directorship at the MPQ. In this way, Herbert Walther
achieved the impossible.
It is difficult to describe the impact of his works in just a few words. Those of
us who have been associated with him consider ourselves very fortunate, having
benefited from the relationship in many different ways. Needless to say, Herbert
Walther has trained a large number of students and other researchers, many of
whom have become authorities in the field. His students and colleagues have won
Nobel prizes; few are able to boast this line. But his humbleness and generosity
have no bounds and the optical community knows and appreciates all that he has
done for it.
We hope that some of the many shining aspects of his scientific life are reflected
in the present volume. All articles have been written by Herbert Walther’s former
students and collaborators, now grown up and dedicated to their own research.
But clearly enough, the nucleus of their work lies in Herbert Walther’s laboratory.
To our minds, when history is written, then one would find that many of the discoveries made in Herbert Walther’s laboratory will stand out as some of the most
fundamental discoveries in the discipline of quantum optics.
So with the preceding in mind, appreciation and admiration in our hearts, and a
special applause to his endearing spouse Margot, we dedicate this volume to you
Herbert. Vielen Dank!
Girish Agarwal
Gerhard Rempe
Wolfgang Schleich
Marlan Scully
ADVANCES IN ATOMIC, MOLECULAR AND OPTICAL PHYSICS, VOL. 53
NON-CLASSICAL LIGHT FROM
ARTIFICIAL ATOMS*
THOMAS AICHELE† , MATTHIAS SCHOLZ, SVEN RAMELOW
and OLIVER BENSON
Nano Optics, Physics Department, Humboldt-Universität zu Berlin, 10117 Berlin, Germany
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2. Single Quantum Dots . . . . . . . . . . . . . . . . . . . . . . . .
3. Single-Photon Generation . . . . . . . . . . . . . . . . . . . . . .
3.1. Correlation Measurements . . . . . . . . . . . . . . . . . . .
3.2. Micro-Photoluminescence . . . . . . . . . . . . . . . . . . .
3.3. InP Quantum Dots . . . . . . . . . . . . . . . . . . . . . . .
4. A Single Photon as Particle and Wave . . . . . . . . . . . . . . .
5. A Multi-Color Photon Source . . . . . . . . . . . . . . . . . . . .
6. Multiplexed Quantum Cryptography on the Single-Photon Level
6.1. A Single-Photon Add/Drop Filter . . . . . . . . . . . . . . .
6.2. Application to Quantum Key Distribution . . . . . . . . . .
7. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . .
9. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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30
1. Introduction
A photon is the fundamental excitation of the quantized electro-magnetic field.
Its introduction helped to get a deeper, yet more intuitive understanding of the
phenomenon light. The year 2005 celebrates the 100th anniversary of Einstein’s
ingenious explanation of the photoelectric effect using the concept of the photon.
Until today, the photon is a workhorse to test the foundation of quantum physics
against recurring efforts of a purely classical interpretation of nature [1,2]. More
* We would like to dedicate this article to Prof. Herbert Walther on behalf of his 70th birthday. He
pioneered quantum optics with single quantum systems and drew our attention to the beauty of the
single photon. The experiment we report in Section 4 of our article was motivated by his wonderful
experiments with single ions.
† Present address: CEA/Université J. Fourier, Laboratoire Spectrométrie, Grenoble, France.
1
© 2006 Elsevier Inc. All rights reserved
ISSN 1049-250X
DOI 10.1016/S1049-250X(06)53001-0
2
T. Aichele et al.
[1
F IG . 1. Photon number distributions of (a) thermal light, (b) a coherent state, and (c) a single-photon source (with 25% efficiency).
recently, single photons entered the stage to play an important role in the field of
quantum information processing. Bennett and Brassard [3] suggested, that data
can be transmitted without the possibility of eavesdropping, if information is encoded in the quantum state of single particles (for a review see [4] and references
therein). For transmission over large distances, the photon is currently the only
reasonable carrier of quantum information. Knill et al. [5] proposed an implementation of all-optical quantum gates for quantum computation using solely linear
optics and single-photons which is based on non-deterministically prepared entangled states and quantum teleportation [6]. Single photons have also been discussed
as transmitters of quantum information [7] between different knots of stationary,
matter-based qubits, such as ions [8–11], atoms [12,13], quantum dots [14,15],
and Josephson qubits [16–19].
In spite of their fundamental character, single photons cannot be generated easily by a classical light source. As photons obey Bose–Einstein statistics, classical
sources tend to emit photons in bunches. Figure 1 shows probability distributions
for various classical and non-classical states of light. Thermal light fields (a),
such as light from a bulb, have a smeared distribution with significant probabilities for larger photon numbers. Even laser light (b), which possesses the
narrowest classically obtainable photon number distribution, shows Poissonian
statistics pn = exp(−μ)μn /n! with average photon number μ. However, for applications in quantum information processing, single-photon operation of the light
source is crucial: For photonic quantum gates [5], but also quantum repeaters [20]
and quantum teleportation [21], multi-photon states may lead to wrong detection
events that cause wrong interpretations of the outcome of a quantum operation. In
quantum cryptography, an eavesdropper may split off additional photons to gain
partial information.
In contrast, an ideal single-photon source has a probability of one to measure
exactly one photon at a time (pn = δ1,n ). Such sub-Poissonian distributions—
with a width narrower than a Poissonian of the same average photon number—
are known to be non-classical and have to be described by means of quantum
mechanics. Additionally to the single-particle character, high purity of the spatial
1]
NON-CLASSICAL LIGHT FROM ARTIFICIAL ATOMS
3
and temporal mode is often required in many applications, for example, Hong–
Ou–Mandel-type [22] two-photon interference plays an important role in linear
optical quantum gates [5]. For useful operation, highly efficient photon generation
and the ability to trigger the emission time are also desirable. Real single-photon
sources show various loss mechanisms, like emission into uncontrolled optical
modes or absorption, so that a more realistic photon number distribution has a
certain zero-photon probability, as the one in Fig. 1(c) [23].
There are many ways to realize single-photon sources. The easiest is to approximate single-photon states by highly attenuated laser pulses: Due to their
Poissonian photon number distribution, the multi-photon probability scales linear
with the mean photon number, p2 ≈ μp1 /2, which approximates a singlephoton state for μ 1. However, the single-photon probability scales in the same
way, p1 ≈ μ for μ 1, which makes this method highly inefficient. Another
widely used method to generate single-photon states is spontaneous parametric down-conversion in non-linear crystals. Presently, these sources are the most
practical and brightest sources for non-classical light, such as entangled photon
pairs [24]. However, due to the stochastic nature of this process, only a limited
overall efficiency is offered while an increase of pump power to improve the photon rate leads to an increased probability to generate two-photon pairs. These facts
limit the potential of this method for future commercial quantum applications that
need high single-photon rates.
Another method is to use spontaneous emission from a single quantum emitter. Suppressing non-radiative decay mechanisms, these emitters represent, in
principle, single-photon sources with 100% efficiency. To make use of this high
efficiency, a strong control of the spatial emission mode is required which sets
a technical but no fundamental limit to today’s maximally achievable photon
efficiency. The variety of possible quantum emitters offered by nature allows a
multitude of realizations.
Discrete electronic transitions in atoms were the first to be investigated in 1977
by Kimble et al. [25]. Recent experiments used single atoms [26,27] and single
ions [28] coupled to microcavities to exploit effects of cavity quantum electrodynamics. In this way, not only the emission time, but also the spatial and temporal
mode of single photons can be controlled. The emission of an isolated single atom
is free from additional broadening due to coupling to the environment. Additionally, identical atoms emit identical photons which is a requirement for possible
applications in quantum information. Radiative cascades in atoms have also been
used for entangled-photon generation [1]. Moreover, the generation of stationary,
single- or few-photon Fock-states was demonstrated using Rydberg atoms and superconducting ultra high-Q cavities [29,30]. One drawback of atomic systems is,
however, the complexity of today’s atom traps or atomic beam experiments.
Transitions in single molecules and single nanocrystals also produce single
photons [31–33]. Nanocrystals are semiconductor crystals in the size of a few
4
T. Aichele et al.
[2
nanometers which are chemically produced as colloids [34]. Similar to quantum
dots (see below), nanocrystals show discrete energy levels, in contrast to bulk
crystals, leading to single-photon transitions. Molecules and nanocrystals have
similar properties with respect to single-photon emission. Both systems can be
operated even at room temperature which makes them cheap and easy to handle.
Their drawback is their susceptibility for photo-bleaching and blinking [34]. The
latter describes the effect of interrupted emission even on large timescales due to
the presence of long-lived dark states. However, this problem may be reduced by
improved synthesis.
In experiments where the single-photon character is the only important property, nitrogen–vacancy defect centers in diamonds are advantageous. These structures show room temperature single-photon emission without optical instabilities,
like blinking and bleaching [35,36], but have a broad optical spectrum at room
temperature together with comparably long lifetimes (12 ns).
This article focuses on single-photon generation using self-assembled single
quantum dots. Quantum dots are few-nanometer sized semiconductor structures
showing discrete electronic energy levels, in contrast to energy bands in bulk
semiconductors.1 Many properties of quantum dots (emission spectrum, electronic structure, etc.) resemble features known from atoms. For this reason quantum dots are also referred to as artificial atoms. To suppress electron–phonon
interaction and thermal ionization, quantum dots mostly need to be operated at
cryogenic temperatures, but experiments at increasingly higher temperature have
also been reported [37,38]. The emission from a quantum dot combines nearly
lifetime-limited narrow spectral lines and short transition lifetimes. In contrast
to its nanocrystal counterpart, quantum dots are optically very stable. They also
offer the possibility of electric excitation [39] and the implementation in integrated photonic structures [40]. Due to the variety of possible materials, quantum
dots have shown single-photon emission throughout the ultraviolet, visible, and
infrared spectrum. Moreover, it was proposed and demonstrated to use quantum
dot multi-photon cascades for the generation of entangled photon pairs [41,42].
2. Single Quantum Dots
For the calculation of electronic states in quantum dots (or artificial atoms), several schemes have been used at different levels of sophistication [43]. Figure 2
illustrates the simplest approach which assumes a spherical potential trap for electrons and holes. When the quantum dot is occupied by several quasi-free charge
1 Although colloidal nanocrystals are also quantum dots by this definition, to avoid confusion, here
the terminology of quantum dots is used for quantum dot structures grown on semiconductor substrates.
2]
NON-CLASSICAL LIGHT FROM ARTIFICIAL ATOMS
5
F IG . 2. Excitations in a quantum dot: (a) Exciton formed by an electron–hole pair, (b) biexciton
containing two electron–hole pairs, generally with a different energy than the exciton. (c) Schematic
term scheme for the exciton and biexciton decay cascade. The two dark excitons are indicated by gray
lines. Numbers indicate the electron, hole, and total spin.
F IG . 3. (a) Micro-photoluminescence image of InP quantum dots in GaInP. (b) Spectrum of a
single InP/GaInP quantum dot with the spectral lines of exciton and biexciton decay.
carriers (electrons or holes), Coulomb interaction has to be taken into account, as
well. While equally charged carriers repel each other, the energy of the system is
lowered for an electron–hole pair, and an exciton is formed (Fig. 2(a)). The recombination of the exciton leads to the emission of a single photon. Correspondingly,
two electron–hole pairs form a biexciton, but generally with a different energy
due to Coulomb interaction (Fig. 2(b)). When decaying, first one electron–hole
pair recombines, leading to the emission of a first photon. The remaining exciton
in the quantum dot emits a second photon with a different wavelength (Fig. 2(c)).
The quantum dot fine structure (see, for example, [44]) reveals a single biexciton ground state and four exciton ground states. Two of them are dark states and
participate in neither the biexciton nor the exciton decay. Figure 3 shows a photoluminescence image of an ensemble of InP quantum dots. The image was taken
through a bandpass filter to suppress excitation stray light from the optical excitation. The spectrum of a single InP quantum dot with two dominant spectral lines,
originating from the exciton and biexciton decay, is displayed in Fig. 3.
6
T. Aichele et al.
[2
Quantum dot samples can be fabricated by a variety of methods starting
from higher dimensional semiconductor heterostructures, like etching pillars in
quantum well systems or forming intersections of quantum wells or quantum
wires [43]. The growth of nanostructures on patterned substrates, such as grooves
or pyramids, led to successful quantum dot formation [45] and single-photon
emission [46], as well. These fabrication methods allow a high degree of position control which is advantageous for coupling the quantum dot to microcavities
and photonic devices.
So-called natural quantum dots [47] are formed by thickness fluctuations
mainly of quantum wells, but also in nanotube systems. In this environment, excitons are trapped in broader regions of the quantum well, where the confinement
energy is lowered, so that a potential minimum is formed. Such excitons exhibit
large oscillator strengths leading to short radiative lifetimes [48] as the lateral size
of natural quantum dots is usually much larger than the exciton Bohr radius.
The experiments described in this article are performed on self-assembled
quantum dots. These quantum dots are fabricated by epitaxial growth of one
crystal type on top of another. If the lattice constants differ noticeably, dislocations due to strain are created, and material islands are formed to minimize
the strain. A thin layer, which is known as the wetting layer, will remain, covering the substrate completely. This growth mode is called Stranski–Krastanov
growth. The wetting layer forms a quantum well which usually shows photoluminescence at energies above the quantum dot emission. There are different epitaxial
techniques like Molecular Beam Epitaxy (MBE) or Metal–Organic Vapor Phase
Epitaxy (MOVPE).
The InAs/GaAs material system is by far the most studied of all quantum dot
systems. Work on single-photon emission reported to date has predominately
been done on InAs dots emitting in the 900–950 nm region [39,49], but also at
1250 nm [50] and 1300 nm [51,52]. Photon correlation measurements at these
wavelengths require good infrared single-photon detectors. Single-photon generation on demand at 1300 nm will be very useful for quantum cryptography via
optical fibers. Nitride quantum dots (GaN in AlN) emit single photons in the ultraviolet region [53].
II–VI-type quantum dots have the advantage of short lifetimes (∼100 ps compared to ∼1 ns for the previously described III–V systems) which reduces the
probability of decoherence during the emission process and enables the generation
of single photons on demand with a small time uncertainty [37,54]. This suggests
a much higher maximum single-photon emission rate than for III–V dots. This
system also shows a larger energy splitting between the exciton and the biexciton
than the InAs/GaAs material system, which is useful to achieve a better filtering
of the exciton emission, enabling operation at higher temperatures. The refractive
index of ZnSe is lower than of GaAs which reduces photon losses due to total
internal reflection at the sample surface.
3]
NON-CLASSICAL LIGHT FROM ARTIFICIAL ATOMS
7
In the experiments reported here, InP dots in a GaInP matrix are used to generate single photons in the 640–690 nm range as well as photon pairs and triplets.
In principle, this material system can be used to generate single photons between
620 nm and 750 nm which fits to the maximum efficiency of silicon avalanche
photo diodes (over 70% at around 700 nm).
3. Single-Photon Generation
3.1. C ORRELATION M EASUREMENTS
The measurement of intensity correlations is a standard method for testing singlephoton emission: The intensity correlation of a light field is detected at two points
in time, resulting in the second-order coherence function g (2) (t1 , t2 ). In the case
of stationary fields, it has the form
g (2) (τ = t1 − t2 ) =
: Iˆ(0)Iˆ(τ ) : ,
Iˆ(0)2
(1)
where : : denotes normal ordering of the operators. This function is proportional
to the joint probability of detecting one photon at time t = 0 and another at t = τ .
This function has several characteristic properties: As each random process is assumed to become uncorrelated after a sufficiently long timescale, the normalized
correlation function tends to a value of unity for large times. It can further be
shown [55] that for all classical fields g (2) (0) 1 and g (2) (0) g (2) (τ ) hold. For
classical light fields, this prohibits values smaller than unity.
The case g (2) (0) > 1 is characteristic for thermal light sources. In this case, the
photons are bunched, which means that there is an increased probability to detect
a second photon soon after a first one (Fig. 4(a)). For coherent light fields, such as
continuous laser light, g (2) (τ ) = 1 for all τ which indicates a Poissonian photon
number distribution and photons arriving randomly (Fig. 4(b)).
If, however, the probability to detect a second photon soon after a first detection
event is reduced compared to an independent process, g (2) (0) < 1 (Fig. 4(c)).
This effect is called anti-bunching. As mentioned before, this case is reserved
to non-classical states with sub-Poissonian photon statistics. For photon number
states |n, with exactly n photons, g (2) (0) = 1 − 1/n and in the special case of a
single-photon state (n = 1), g (2) (0) = 0. For statistical mixtures of one- and twophoton states (or more), intermediate values can also be obtained. In the case of a
pulsed source, the second-order coherence function possesses a peaked structure.
Here, a missing peak at τ = 0 indicates the generation of one and only one photon
per pulse (Fig. 4(d)).
A straightforward method to measure the second-order coherence function
would be to simply note the times of detector clicks and to compute the correlation
8
T. Aichele et al.
[3
F IG . 4. Top: illustrative distribution of the photon arrival time, bottom: second-order coherence
function g (2) (τ ) of (a) a thermal light source (for example, a light bulb), (b) coherent light (laser
light), (c) a continuously driven single-photon source, and (d) a pulsed single-photon source.
F IG . 5. (a) Scheme of the Hanbury Brown–Twiss setup. (b) Correlation measurement of a spectral
line of a single quantum dot over a timescale that is large compared to the average time between
detection events (7.7 µs for the black and 20 µs for the gray curve). The logarithmic scale emphasizes
the exponential behavior. The dip at delay time τ = 0 (see zoom in the inset) indicates single-photon
emission.
function according to Eq. (1). However, this approach prevents the measurement of timescales shorter than the detector’s dead time (≈50 ns for avalanche
photo detector modules [56]). To overcome this problem, a Hanbury Brown–
Twiss arrangement is chosen [57] as depicted in Fig. 5(a), consisting of two photo
detectors monitoring the two outputs of a 50:50 beam splitter. With this setup,
the second detector can be armed right after the detection event of the first. For
small count rates, one can neglect the case, where the first detector is already
armed while the second one is still dead. Losses, like photons leaving the wrong
beam splitter output or undetected photons, simply lead to a global decrease of
the measured, non-normalized correlation function.
3]
NON-CLASSICAL LIGHT FROM ARTIFICIAL ATOMS
9
Technically, it is very difficult to acquire absolute detection times with a resolution in the nanosecond regime. Additionally, the sheer amount of detection events
needed to reach a reasonable statistics (107 events are typical for a count rate of
105 s−1 ) makes the computation of the correlation function very time-consuming.
Instead, only the time differences between detection events are usually registered
and binned together in a histogram. An electronic delay shifts the time origin and
enables the observation of asymmetric cross-correlation functions (see Section 5).
Using a time-to-amplitude converter, time differences can be measured very precisely. This method has the additional advantage that the measurement can be
tracked online. However, the function measured in this way is the waiting time
distribution [58] d(t) rather than the second-order correlation function g (2) (t).
The function d(t) is defined as:
d(t) = (Prob. density to measure a stop event at t
after a start event at time 0)
× (Prob. that no stop det. occurred before)
t
(2)
= T g (t) + rD 1 − d t dt ,
(2)
0
where the transmission T was introduced to account for possible photon losses
and rD describes the detector dark count rate. For large t it follows:
t
(2)
(2) d(t) = T g (t) + rD exp −
T g t + rD dt
0
≈ const × e
−(rc +rD )t
.
(3)
The second line indicates that for long time differences, the measured histogram
decays exponentially on a timescale given by the detector count rate. Figure 5(b)
shows such a large-time measurement of a single-photon source (see also the
inset). Only if the average arrival time of the photons t = rc−1 (rc : photon count
rate) is much larger than the observed time t between start and stop event, the
probability, that no stop detection has occurred before, is approximately 1 and
g (2) (t) ≈ d(t).
3.2. M ICRO -P HOTOLUMINESCENCE
In order to perform experiments with single quantum dots, several requirements concerning the setup have to be fulfilled: As self-assembled quantum dot
samples—even on a so-called low-density sample—have quantum dot densities
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F IG . 6. Basic scheme of a micro-PL setup (FMs: mirrors on flip mounts, DM: dichroic mirror,
PH: pinhole, BP: narrow bandpass filter, APDs: avalanche photo detectors).
of 108 . . . 1011 cm−2 , a sufficient spatial resolution is required to select a single quantum dot or at least only as few as possible. At the same time, a high
collection efficiency is preferred to gain a maximum amount of photons. These
two requirements can be combined by choosing a micro-photoluminescence (PL)
setup which is a well-established setup in single molecule spectroscopy.
Figure 6 shows the experimental setup. The sample is mounted inside a
continuous-flow liquid Helium cryostat which can be cooled down to 4 K. Optical access for the excitation of the sample and collection of the emitted light is
provided through a thin glass window. The sample is excited by either a pulsed
(Ti:Sapphire, pulse width 400 fs, repetition rate 76 MHz, frequency-doubled to
400 nm) or a continuous wave (Nd:YVO4 , 532 nm) laser. Thus, the excitation is
off-resonant and creates charge carriers in the continuum which are subsequently
captured by the quantum dot. The laser light is sent into the microscope objective via a dichroic mirror. The microscope system with a numerical aperture of
NA = 0.75 has a lateral resolution of 0.5 µm which allows the resolution of individual quantum dots on the sample. The collected PL light is filtered spatially
by imaging onto a pinhole in order to block stray and PL light from neighboring sites. Spectral filtering can be performed with a narrow bandpass interference
filter. The light transmitted through these filters is directed onto a CCD camera
for imaging or to a grating spectrograph for spectral analysis (Fig. 3 shows a
corresponding image and spectrum). Finally, a Hanbury Brown–Twiss correlation setup is used to measure the second-order coherence function. It consists of
two avalanche photo diodes (APDs) and correlation electronics that collects the
time differences between start and stop detector in a histogram. The overall time
resolution of the correlation setup is 800 ps.
3]
NON-CLASSICAL LIGHT FROM ARTIFICIAL ATOMS
11
3.3. I N P Q UANTUM D OTS
InP quantum dots grown in a GaInP matrix are a particularly interesting system for generating single photons for free-space experiments as their emission
wavelength around 690 nm allows the highest possible detection efficiency of
commercial Si APDs. However, they show disadvantages in fiber-coupled applications because their losses in glass fibers are much higher than at infrared
wavelengths. On the other hand, infrared photo detection suffers from low efficiency and bad signal-to-noise ratio. Thus, single photons from InP quantum dots
are particularly important for free-beam experiments.
The sample used in this section was grown by Metal–Organic Vapor Phase
Epitaxy (MOVPE).2 Figure 7(a) shows the structure of the sample. On a GaAs
wafer, a 300 nm thick GaInP layer was deposited, followed by 1.9 mono-layers of
InP which form the quantum dots and another 100 nm layer of GaInP. The density
of dots emitting around 690 nm was estimated to be about 108 cm−2 by imaging
through a narrow bandpass filter. In order to increase light extraction efficiency,
a 200 nm thick Al layer was deposited on top of the sample to form a mirror. The
sample was then glued upside down with epoxy onto a Si substrate, and the GaAs
substrate was removed using a selective wet etch. For the purpose of increasing the
light extraction efficiency, the use of a metallic mirror is preferable to a distributed
Bragg reflector (DBR), as metal mirrors reflect strongly at all angles, resulting in
a larger integrated reflectivity if a point-like emitter is assumed.
Figure 7(b) shows a spectrum of a single InP quantum dot at 10 K. The lower
graph is an unfiltered spectrum. The excitation power density was adjusted to have
only one dominant emission line. By measuring the intensity of this spectral line,
F IG . 7. (a) Structure of the InP/GaInP sample. (b) PL spectra taken on a single InP quantum dot
at cw excitation and a temperature of 10 K. The bottom spectrum was taken without filtering, the top
spectrum through a narrow bandpass filter. An offset was added for separating the graphs. The black
line in the inset is a spectrum over a larger wavelength range, the gray line shows the efficiency of the
single-photon detectors.
2 This sample was provided by the group of Prof. W. Seifert from Lund University (Sweden).
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F IG . 8. Measurement of the g (2) -function at continuous excitation. The gray curve is the expected
correlation function for an ideal single-photon source, but limited time resolution. The right graph is
a magnification of the dip in the left plot.
a linear dependency on the excitation power was observed, indicating an exciton
transition. Additional emission lines appear with increasing laser power. A spectrum taken over a wider wavelength range is displayed in the inset of Fig. 7(b) and
shows that all the emission within a very broad wavelength range originates from
the dot under study. The inset also shows the detection efficiency of the APDs
with its maximum right at the emission wavelength of the quantum dot. When
placing a narrow 1 nm bandpass filter into the beam path, only light from a single
transition of this quantum dot is transmitted (upper curve of Fig. 7(b)).
Figure 8 shows a correlation function (see Eq. (1)) measured on the exciton
spectral line of this dot performed at continuous excitation [59]. The total count
rate was 1.1 × 105 counts per second. The dashed gray line in this figure is the
calculated correlation function obtained by taking into account the limited time
resolution of the Hanbury Brown–Twiss setup. This function is modelled as a
convolution of the expected shape of the ideal correlation function g (2) (τ ) = 1 −
exp(−γ τ ) and a Gaussian distribution with a width of the system’s time resolution
of 800 ps. 1/γ is the timescale of the anti-bunching dip and depends on both
the transition lifetime and the excitation timescale. This timescale is used as a fit
parameter here. A zoom into the region around the origin of the left graph is given
in the right graph of Fig. 8. The excellent agreement between calculations and
measurement indicates that the quantum dot device generates single photons and
that the minimum dip value of 5% (relative to the value at large time differences)
is mostly due to the limited time resolution. The characteristic timescale 1/γ of
the anti-bunching dip is fitted to 2.3 ns. The measurement in Fig. 8 is the quantum
dot counterpart of first measurements performed with single trapped ions [60].
In Fig. 9, correlation measurements at pulsed excitation are displayed [59]. The
total count rate was 4.4 × 104 counts per second for the measurement in Fig. 9(a).
It is observed that the peak at zero delay time is vanishing almost completely. This
amounts to single-photon generation on demand: Upon each laser pulse creating
an exciton, one and only one photon is produced.
4]
NON-CLASSICAL LIGHT FROM ARTIFICIAL ATOMS
13
F IG . 9. Second-order coherence function measured on a single InP quantum dot at pulsed excitation: (a) at 8 K and (b) between 20 and 50 K.
This measurement is a prerequisite for succeeding single-photon experiments
because it proofs single-photon emission and excludes the possibility to observe
light from several quantum dots, leading to ensemble averaging of the results.
Additionally, for quantum cryptography experiments, only a normalized area of
the peak zero delay time clearly below 0.5 ensures secure transmission of the
encryption key.
The graphs in Fig. 9(b) show measurements at higher temperatures. When increasing the temperature, the emission intensity of the quantum dot decreases
which can be attributed to thermal carrier escape. Moreover, broadening of the
spectral lines due to phonon interactions [61–63] leads to an increased incoherent
background when other spectral lines start to overlap with the filter transmission
window. Both of these effects deteriorate the quality of single-photon generation.
However, up to 27 K, the peak zero delay time at is still almost completely suppressed. With increasing temperature, this peak slowly starts to grow, but it has a
relative area still below 0.5 even at 50 K, indicating that a single quantum dot’s
transition still dominates the emission.
4. A Single Photon as Particle and Wave
The wave–particle duality lies at the heart of quantum mechanics. With respect to
light, the wave-like behavior is perceived as being classical and the particle aspect
as being non-classical while for massive microscopic objects, like neutrons and
atoms, the opposite holds. The occurrence of an interference pattern is a manifestation of the wave-nature of matter.
Already in 1909, soon after the introduction of the concept of light ‘quanta’,
Taylor observed experimentally that there is no deviation from the classically predicted interference pattern if a double-slit interference experiment is performed
with very weak light, even if the intensity is so small that on average only a
single photon is present inside the apparatus [64]. Later, this observation was
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F IG . 10. Experimental setup for the simultaneous Michelson and Hanbury Brown–Twiss experiment.
accounted for by quantum mechanics and was confirmed by more precise experiments [65,66]. There exists an exact correspondence between the interference of
the quantum probability amplitudes for each single photon to travel along either
path of an interferometer, on the one hand, and the interference of the classical
field strengths on the different paths, on the other hand. Therefore, the outcome
of any first-order interference experiment can be obtained by describing light as
a classical electromagnetic wave, independent of the statistical distribution of the
incident photons. In more recent experiments, Grangier et al. [67] performed a
series of experiments with single photons from atoms. In a first step, they showed
the single-photon character of the atomic emission by observing the corresponding anti-bunched behavior of the intensity correlation function. In a second step,
they inserted the photons into a Mach–Zehnder interferometer and observed an
interference pattern with varying path difference, a feature that displays the wave
nature of light. Braig et al. [68] implemented a similar experiment using a diamond defect center as the emitter and observed single-photon statistics after
detecting interference in a Michelson interferometer.
In this section, an experiment is described that combines these two experimental techniques in a single step for simultaneous observation of interference and
anti-bunching of the quantum dot fluorescence [69]. Therefore, a Michelson interferometer and the Hanbury Brown–Twiss setup were set in series as displayed
in Fig. 10.
The clicks of the detectors (APDs) can be evaluated in two ways: (I) When
looking for coincidences between clicks from start and stop APD, anti-bunching
is observed, revealing the particle nature of light. The only effect from the change
between constructive and destructive interference on this measurement is an over-
4]
NON-CLASSICAL LIGHT FROM ARTIFICIAL ATOMS
15
F IG . 11. Measured correlation function and interference pattern, respectively, for a single quantum
dot at pulsed excitation ((a) and (b)) and at cw excitation ((c) and (d)).
all change of the coincidence rate, independent of the delay time between start and
stop events. Since the latter is short compared to the timescale of the arm length
variation in the Michelson interferometer, the non-normalized second-order coherence function changes just by a constant factor. (II) On the other hand, one
can count the clicks of either APD while the arm length of the Michelson interferometer is changed. A modulation of the single detector count rate represents
interference which directly demonstrates a wave-feature of light. Since the detector produces a classical electrical pulse that can, after detection of a photon, be
easily split into two parts, it is also possible to perform these two measurements
simultaneously.
Figure 11 displays the results of such a combined measurement when exciting
a quantum dot with a pulsed ((a) and (b)) and a cw laser ((c) and (d)), respectively.
In Figs. 11(a) and (c), the autocorrelation functions of the two measurements
are plotted, expressed by the number of coincidences. The non-classical antibunching effect is clearly visible since the number of coincidences exhibits a
pronounced minimum at zero delay time. In contrast, Figs. 11(b) and (c) depict the
single-detector count rate at an integration time of 10 ms, dependent on the path
difference on the interferometer. It shows the expected first-order interference pattern that reveals the wave-like nature of the emitted single-photon radiation.
The described combination of the two experimental techniques, Michelson interferometer and Hanbury Brown–Twiss correlation setup, forms an extension to
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[5
the experiments of Grangier et al. [67], as one and the same photon contributes
to both the measured interference pattern and the anti-bunched correlation function. In this sense, the described experiment is similar to the classic experiment
of Taylor [64] and to the experiments described in Refs. [65,66], but gives an unequivocal evidence of the particle nature of light: Instead of using weak light
fields with classical photon number statistics (with a super-Poissonian photon
number distribution), the anti-bunching effect shows that the quantum dot photoluminescence represents number states that can only be described within the
frame of quantum mechanics. In a similar work, Höffges et al. [70] simultaneously
performed heterodyne and photon correlation measurements in the resonance fluorescence of a single ion.
5. A Multi-Color Photon Source
The potential of single semiconductor quantum dots as emitters in photonic devices is not only the generation of single photons on demand. Quantum dots are
promising candidates for the generation of entangled photon pairs. It was proposed [41] and demonstrated [42] to make use of polarization correlations in the
biexciton–exciton cascade. But as will be demonstrated in the next section, even
without entanglement formation, multi-photon cascades find applications in quantum communication experiments.
Here, intensity cross-correlations between several different quantum dot transitions on the InP quantum dot sample are measured. Similar exciton–biexciton
cross-correlation measurements have also been reported on InAs quantum
dots [23,71,72] and II–VI quantum dots [73,74]. Such experiments answer several purposes: First, they are an important tool for identifying the nature of the
investigated spectral lines, such as resulting from an exciton, biexciton, triexciton,
or emerging from the same or different quantum dots. Second, they give information about the different decay and excitation timescales in multi-photon cascades.
Finally, polarization resolved cross-correlations form a first step towards the observation of entangled photon pairs.
In order to detect correlations between different transitions, a variant of the
second-order coherence function is considered. The cross-correlation function is
defined in a similar way as in Eq. (1), but with the intensity operators assigned to
different field modes α and β:
(2)
gαβ (τ ) =
: Iˆα (t)Iˆβ (t + τ ) :
.
Iα (t)Iβ (t)
(4)
In the experiments described here, the two modes represent spectral lines of
two quantum dot transitions. In order to distinguish this cross-correlation from
5]
NON-CLASSICAL LIGHT FROM ARTIFICIAL ATOMS
17
F IG . 12. Modified Hanbury Brown–Twiss setup for cross-correlation experiments. The different
colors of the two beams represent two spectral lines selected by the bandpass filters.
the second-order coherence function, the latter is also referred to as the autocorrelation function.
The experiment is performed by filtering of the two spectral lines for each photo
detector in the Hanbury Brown–Twiss system individually. Here, this was realized
by placing narrow bandpass filters directly in front of each APD, as sketched in
Fig. 12. In this way, the resulting correlation function will show an asymmetry
with respect to the time origin, as start and stop detection events now arise from
different processes and a change in sign of the time axis accords with an effective
exchange of start and stop detector.
To get a first idea about the origin of the distinct spectral lines, their different
scaling with the excitation intensity was investigated. Figure 13 shows PL spectra
of a quantum dot, taken at various excitation intensities at 8 K. In the following, the reference excitation power density P0 was kept constant at 1 nW/µm2 .
The spectral behavior in Fig. 13 is typical in terms of line spacing and power dependence, for an InP dot emitting in this energy range. At low excitation power
density, a single sharp emission line at 686.3 nm (1.8155 eV) is present in the
spectrum (X1 ). As the excitation power is increased, a second line (X2 ) appears
about 0.6 nm (1.5 meV) beside the exciton emission. When further increasing the
excitation power, additional lines appear. The integrated photoluminescence intensity of X1 increases linearly with the excitation intensity whereas X2 shows a
quadratic dependence. This behavior is a good indication of excitonic and biexcitonic emission, respectively. The lines appearing at high excitation power density,
such as X3 , are attributed to a multi-exciton of higher complexity. Especially,
X3 is assumed to originate from a triexciton, as will be proven later.
For such a complex excitation as the triexciton, it is necessary to invoke additional states to the single-particle ground states of the quantum dot. Figure 14(a)
shows two of the possible triexciton decays together with a simplified decay chain
of a triexciton in (b). The triexciton X3 recombines to an excited biexciton X2∗ that
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F IG . 13. Power dependent spectroscopy of a single InP quantum dot showing the lines used in
the correlation measurements. The excitation intensity is given as a multiple of P0 = 1 nW/µm2 .
Lines X1 , X2 , and X3 are assigned to different excitations, as described in the text.
rapidly relaxes to the biexciton ground state X2 which in turn recombines via the
exciton X1 to the empty ground state G of the quantum dot [44].
After the different spectral lines have been characterized and pre-identified,
additional information can be gained by performing cross-correlation measurements between these emission lines. Figure 15(a) shows the cross-correlations of
the exciton and biexciton line of that dot at different cw excitation power densities [75]. A strong asymmetric behavior is observed: At positive times, when
the detection of a biexciton photon starts the correlation measurement and the
detection of an exciton photon stops it, photon bunching occurs, as here the detection of the starting biexciton photon projects the quantum dot into the exciton
state which has now an increased probability of recombining shortly after. On the
other hand, if the correlation measurement is started by the exciton photon, which
prepares the dot in the ground state, and stopped by the biexciton photon (negative times in Fig. 15(a)), a certain time is needed until the dot is re-excited. In
this measurement, effectively the recycling time of the quantum dot is observed
which explains the strong anti-bunching for negative times. The population of the
biexciton state is dependent on the laser power, and the excitation time decreases
when the laser power increases. Similar measurements have also been performed
5]
NON-CLASSICAL LIGHT FROM ARTIFICIAL ATOMS
19
F IG . 14. Illustration of the multi-exciton cascade in quantum dots. (a) Occupation of electron and
hole states in the decay of the triexciton state X3 to the biexciton ground state X2 and to an excited
state X2∗ . (b) Decay cascade model used in the discussions and the rate equation approach. Dashed
arrows indicate excitation, solid arrows radiative decay, and the open arrow a non-radiative relaxation.
The dashed state C symbolizes an effective cut-off state as explained in the text.
20
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[5
F IG . 15. Measured cross-correlation functions (a) between the exciton and biexciton line and
(b) between the biexciton and triexciton line of the quantum dot that was also used for Fig. 13 at
different excitation intensities (P0 = 1 nW/µm2 ). (c) Exciton–biexciton cross-correlations of a second quantum dot.
on another quantum dot (depicted in Fig. 15(c)). While its timescales are similar,
it shows a more pronounced bunching peak.
In the same way, the cross-correlation of the biexciton emission with the triexciton emission was measured. This is shown in Fig. 15(b). Its behavior is similar
to the exciton–biexciton case, but with different timescales apparent.
The presence of the combined bunching/anti-bunching shape is a unique hint
for observing a decay cascade of two adjacent states. In contrast, the crosscorrelation function of spectral lines of two independent transitions (for example,
from two quantum dots) would show no (anti-)correlations, at all. It can be concluded that there is a three-photon cascaded emission from the triexciton via
biexciton and exciton to the quantum dot ground state. Together with the infor-
5]
NON-CLASSICAL LIGHT FROM ARTIFICIAL ATOMS
21
mation of the different scaling of the spectral lines with excitation power, this
justifies the previous assignments to these lines.
In order to support the interpretation of the obtained correlation data, the photon
cascade was analyzed using a common rate model [23,71]. The rate equations
correspond to the scheme shown in Fig. 14(b) where only two transition types
account for the dynamics of the excitonic states: spontaneous radiative decay and
re-excitation at a rate proportional to the excitation power. As the excitation is
performed above the quantum dot continuum, the relaxation of the charge carriers
into the multi-exciton states, as well as the relaxation of the excited biexciton
after the triexciton decay, should also be taken into account. But as this process
happens on a much faster timescale (several 10 ps [76]) than the state lifetimes
(≈ns), it is neglected in this consideration. The according rate equation ansatz
then reads:
⎞
⎛
γ1
0
0
0
−γE
γ2
0
0 ⎟
⎜ γE −γE − γ1
d
⎟
⎜
n(t) = ⎜ 0
−γE − γ2
γ3
0 ⎟ n(t) (5)
γE
⎠
⎝
dt
−γE − γ3 γC
0
0
γE
0
0
0
γE
−γC
with n(t) = (nG (t), n1 (t), n2 (t), n3 (t), nC (t)) and γi = τi−1 . Here n1 , n2 , and
n3 represent the populations of the exciton, biexciton, and triexciton, respectively,
with corresponding decay times τ1 , τ2 , and τ3 . nG is the population of the empty
ground state, and τE−1 is the excitation rate. In order to truncate the ladder of states
connected by rates in this model, an effective cut-off state with population nC and
lifetime τC was introduced. This accounts for population and depopulation of all
higher excited states via excitation and radiative decay, respectively.
This rate equation can be solved analytically [77]. The general solution is a
sum of decaying exponentials with different time constants. The initial conditions
are defined by the transition that forms the start event in the Hanbury Brown–
Twiss measurement which prepares the quantum dot in the next lower state α, so
that nα (0) = 1 and nγ =α (0) = 0. On the other hand, the detection of a photon
from the stop transition dictates the shape of the cross-correlation function, as
(2)
gαβ (t) ∝ nβ (t). Therefore it is clear that the cross-correlation function on the
positive and negative side is described by two completely different functions with
a possible discontinuity at τ = 0. In the experiment (Fig. 15), this discontinuity
is washed out, due to the finite time resolution of the detectors. Because of this
smoothing of the experimental data, the minima in the graphs of Fig. 15 are shifted
towards the anti-bunching side, as well.
The model was used to describe the auto- and cross-correlation data in Figs. 14
and 15. The results are shown as gray lines in these graphs. The lifetimes of exciton, biexciton, and triexciton were taken from independent measurements. The
excitation rate was chosen to optimally fit the correlation functions in these figures, but was kept linear to the experimental excitation power P throughout the
22
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[6
graphs. In this way, apart from the one-time initialization of the experimentally
inaccessible values τ3 , τC , and τE , the normalization was the only real fit parameter in all graphs. No vertical offset was used to compensate the lift of the
anti-bunching dips. Apparently, the model describes the experimental data very
well. Minor deviations can be explained by the long-term variation of the excitation power due to a spatial drift of the sample while taking data or by the presence
of additional states neglected in this model.
6. Multiplexed Quantum Cryptography on the Single-Photon
Level
6.1. A S INGLE -P HOTON A DD /D ROP F ILTER
Among the requirements for single-photon sources, high efficiencies and high
emission rates are a major priority in order to raise the statistical significance of
experimental outcomes or to enhance the bandwidth for quantum communication protocols. The overall efficiency can be improved by using passive optical
elements such as integrated mirrors (compare the InP sample Fig. 7(a)), solid immersion lenses to enhance the optical collection efficiency [78], or by resonant
techniques that embed the quantum emitters in microcavities [79,80]. The latter
method exploits the Purcell effect [81] in order to enhance the emission rate in
a certain well-defined resonant cavity mode. The Purcell effect can also substantially modify the overall spontaneous emission rate. For a single-photon source,
which relies on the decay of an excited state, the (modified) spontaneous lifetime
determines the maximum photon generation rate.
In classical communications, multiplexing is a well-established technique to
increase the transmission bandwidth. It is the transmission and retrieval of more
than one signal through the same communication link (sketched in Fig. 16). This
is usually accomplished by marking each signal with a physical label, such as the
wavelength. At the receiver, the signals are identified by using filters tuned to the
carrier frequencies [82]. Losses when merging and separating the signals can be
compensated by amplification of the classical signal. For single-photon channels,
F IG . 16. Transmission of N optical signals distinguished by their wavelengths through the same
fiber using multiplexing.
6]
NON-CLASSICAL LIGHT FROM ARTIFICIAL ATOMS
23
the no-cloning theorem [83] prevents the amplification of qubit information so
that losses have to be kept minimal and effective separation of photons with different wavelengths is required. Moreover, a multi-color single-photon source is
needed to provide distinguishable photons.
In this section, an interferometric technique is described to perform multiplexing on a single-photon level (see also Ref. [84]). The biexciton–exciton cascade
in quantum dots provides an excellent source for the required photon pairs with
well-separated energies and strong correlation in the emission time. As a proofof-principle, a quantum key distribution experiment using the BB84 protocol [3]
was performed.
In order to use several independent qubits in a single communication channel
simultaneously, they have to be distinguishable by at least one physical property,
and a method is needed to merge and divide them at the sender and receiver side,
respectively. For photons, a reasonable choice would be to distinguish them by
their wavelength and to use their polarization to encode quantum information.
A common way to separate light with different wavelengths uses diffractive or
refractive optics. However, these techniques are unfavorable, especially in inhomogeneously broadened systems, like a sample of self-organized quantum dots.
Here, the wavelength of the two photons as well as their wavelength difference
may vary from dot to dot. When using diffractive and refractive optics, a complete
realignment of the beam paths for each individual quantum dot under consideration would be required. Moreover, diffractive optics suffer from losses due to
diffraction into different orders.
A superior method is to use interferometric techniques, like the one sketched
in Fig. 17(a). Two photons with different wavelengths λ and λ + λ enter a
Michelson interferometer with variable arm lengths. Retro-reflector prisms are
used to obtain a lateral shift between input and output beam. Due to the difference in wavelength, the two photons undergo different interference conditions.
As long as the path difference s between the two interferometer arms is significantly smaller than the coherence length scoh , the probability to find a photon
(with wavelength λ) at one of the two interferometer output ports is:
1 1
± cos(2πs/λ).
(6)
2 2
The signs + and − correspond to the interferometer output ports labeled as 1
and 2 in Fig. 17, respectively.
In order to illustrate how this can be used to separate the two photons, the interference pattern p1 (s) for two different wavelengths is plotted in Fig. 17(b). For
s ≈ 0, each wavelength shows the same interference pattern. But for increasing path difference, they run out of phase and at a certain position (indicated
by the dotted line in Fig. 17(b)) the two interference patterns are in opposite phase, i.e. each photon interferes constructively at a different output port.
p1,2 (s, λ) =
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F IG . 17. Sketch of the experimental setup: (a) Two photons with different energies enter a Michelson interferometer that consists of a 50:50 beam splitter and two retro-reflectors. (b) Scheme of the
intensity interference pattern at one interferometer output for two distinct wavelengths versus the path
difference between the two interferometer arms. (c) Combining the separated photons: The two interferometer output ports are coupled to an optical fiber each, one of them delayed by half of the
excitation laser repetition time, and recombined at a beam splitter, again.
The smallest path difference for which such a wavelength separation occurs is
s0 = λ(λ + λ)/(2λ). As long as s0 scoh , such a situation can always be
achieved. Note that this condition simply reflects the spectral distinguishability
of the two photons which is a general limit for wavelength separation. When s0
is in the order of or bigger than the coherence length, the interference visibility
decreases and none or only poor photon separation is performed. In this case, the
setup would basically act as a 50:50 beam splitter.
It can be seen that such an interferometric technique is highly insensitive to
changes in wavelength difference. For example, when changing the quantum
emitter or in case of spectral drifts, these effects can be compensated for by correcting the interferometer arm lengths which causes no change in the exiting, final
beam direction. Moreover, as long as s0 scoh , the main losses that occur in such
a system are caused by partial back-reflections at the interfaces of optical components which can be strongly suppressed by appropriate anti-reflection coatings.
Figure 18 shows a set of spectra taken from one interferometer output port.
For Fig. 18(a), white light illumination was used, and the path difference was
set to s = 40 µm. According to Eq. (6), a sine-like modulation with the period
λ = 5.5 nm is observed in the spectrum. Switching to spectra with discrete lines,
the full power of this method becomes evident. In the top graph of Fig. 18(b),
6]
NON-CLASSICAL LIGHT FROM ARTIFICIAL ATOMS
25
F IG . 18. (a) Spectrum of a white light source observed through the Michelson interferometer.
(b) Spectrum of a few InP quantum dots through the interferometer. In the top graph, one arm was
blocked resulting in the original spectrum. In all five graphs, the intensity axes are equally scaled, for
comparison. The numbers indicate three arbitrarily chosen spectral lines whose brightness depends on
the path difference in the interferometer.
an unfiltered few-quantum dot spectrum is displayed with several well-separated
spectral lines. It was obtained by blocking one interferometer arm. But when unblocking and exposing the lines to interference, it was possible to align the path
difference for selectively switching on and off individual lines. This is the case in
the lower four graphs, indicated by three arbitrarily selected spectral lines. In the
other interferometer output, the opposite picture would be visible. In this way, an
inventive spectral switch can also be achieved.
In a further step, the setup was expanded by the part sketched in Fig. 17(c). By
adjusting the arm length difference in the unblocked Michelson interferometer,
a situation was achieved where the exciton and biexciton lines show constructive
interference in either output port and destructive in the other. After filtering, these
two output beams were coupled into two multi-mode fibers. The different fiber
lengths provided a relative delay time of 6.6 ns (half the repetition time of the
pulsed Ti:Sa laser). Behind the fibers, the two beams were merged and detected
by the Hanbury Brown–Twiss detectors. In this way, the train of exciton photons
was shifted in between the biexciton photon train. This enables the simultaneous
observation of the two photon sources and leads to a stream of single photons
with a doubled repetition rate. Ideally, a second, inverted Michelson arrangement
will be used to merge the two beams leaving the fibers. However, for simplicity,
a 50:50 beam splitter was used instead.
Figure 19(b) shows correlation measurements of the light merged behind the
fibers. For comparison, an exciton correlation function is displayed in Fig. 19(a).
Both figures exhibit the characteristics of a pulsed single-photon source. While
the impinging photons in Fig. 19(a) have a time separation of 13.2 ns determined
by the excitation laser repetition rate of 76 MHz, the photon stream in Fig. 19(b)
possesses only half the repetition time. Still, clear anti-bunching is visible.
26
T. Aichele et al.
[6
F IG . 19. Intensity correlation of (a) the exciton spectral line and (b) the multiplexed signal.
The last graph demonstrates how the maximum emission rate of a single-photon
source based on spontaneous emission is limited. As the photo detectors cannot
distinguish between the energies of the two photons, a similar correlation measurement would have been obtained if photons from two excitonic transitions had
been recorded, but at a doubled excitation rate. In both situations, the time period between excitation events approaches the spontaneous lifetime which is in
the order of 1 ns for the quantum dot transitions as reflected by the peak widths.
Thus, the peaks start to overlap, and the photons cannot be assigned to individual
excitation pulses any more which is vital for their use in quantum communications. The result in Fig. 19(b) is already on the onset of this process. However, in
the presented kind of experiment, adjacent photons remain distinguishable with
respect to their wavelength, and an assignment of each photon to a certain pulse
can be preserved.
6.2. A PPLICATION TO Q UANTUM K EY D ISTRIBUTION
An important application of single-photon multiplexing is quantum key distribution. In this techniques, single photons (as required by the BB84 protocol [3]) or
entangled photon pairs (as used by the Ekert protocol [85]) were used to secretly
distribute cryptography keys among distant parties. Eavesdropping is prevented
as the no-cloning theorem [83] forbids to copy the quantum states of the distributed photons. A review and detailed discussion is given by Gisin et al. [4].
Long distance experiments have been successfully realized with weak coherent
laser pulses [4,86] and down-converted entangled photon pairs [87]. Realizations
of the BB84 protocol with single-photon states were performed using diamond
defect centers [88] and single quantum dots [89].
6]
NON-CLASSICAL LIGHT FROM ARTIFICIAL ATOMS
27
F IG . 20. Possible implementations of the multiplexer using the BB84 protocol. In both schemes,
the polarization is modulated between rectilinear and circular polarization using the polarizer P with
an electro-optic modulator (EOM). Bob’s detection side is realized by another EOM, a polarizing
beam splitter (PBS), and detectors (D). ω1 and ω2 indicate the two energies of the photons, and M is
the Michelson arrangement. In (a), the two photons are recombined without delay time, in (b), a delay
time is introduced, and the two beams are then recombined with a beam splitter BS.
For most protocols, information is stored in the photon’s polarization whereas
the exact wavelength is unimportant. Thus, multiplexing—as described previously—can provide an increased communication bandwidth without loss of security. Figure 20(a) shows a possible implementation of interferometric multiplexing in the BB84 protocol. On Alice’s side, a cascaded photon source, such as
a single quantum dot, provides two closely emitted single photons with different
energies upon each excitation pulse. In a first Michelson interferometer, the two
photons are separated. In the same way as in the conventional protocol, polarizers define a fixed polarization3 and electro-optic modulators (EOMs) randomly
modulate between (H, V , L, R) polarization for each photon. This destroys any
polarization correlation between the two photons, thus providing independent
qubits. An inverted Michelson interferometer recombines the two photons in a
single channel for transmission. On the other side, Bob uses the same arrangement to separate the photons. A second set of EOMs is used to randomly change
the bases. Polarizing beam splitters in combination with two APDs detect the polarization state of each photon. In this way, the transmission rate will be doubled
compared to a protocol using only one photon per pulse.
To demonstrate this application, a simplified proof-of-principle experiment as
sketched in Fig. 20(b) was set up: The excitonic and biexcitonic photons from the
3 For already polarized photons, the polarizers would be ideally aligned for optimum transmission.
28
T. Aichele et al.
[6
quantum dot single-photon source were separated by a Michelson interferometer,
fiber-coupled, and delayed (see also Fig. 17). The photon pulse rate was doubled,
but the average photon number per pulse was halved. Deviating from the proposed
scheme, Bob’s detection consisted of a second EOM, an analyzing polarizer, and
an APD, with the EOM randomly switched between the two bases. Since Bob
measures the same state as Alice only in one fourth of the cases, a reduction of
the effective count rate of 50% follows, compared to a scheme with two detectors.
In this configuration, two-photon events can create a possible insecurity, but in our
setup, the collection efficiency is estimated to be p ≈ 10−3 [77]. The probability
to collect two adjacent photons is p11 = p 2 ≈ 10−6 and thus much smaller than
the probability to collect one photon and loose the next, p10 = p(1 − p) ≈ 10−3 .
At higher collection efficiency, the setup in Fig. 20(a) is favored.
The transmission distance was 1 m. For exciting the quantum dots, a pulsed
diode laser (λ = 635 nm, pulse width 125 ps) with a repetition rate of 10 kHz was
used which was adapted to the modulation rate of the EOM drivers. These drivers
consist of a digital-to-analog converter steered by a computer card and a subsequent high-voltage amplifier to supply the EOMs with half- and quarter-wave
voltages. A rectangular voltage signal acts as a trigger for the laser pulses, the
EOM switching, and the detection gate for acquiring Bob’s detection events. The
trigger and the detection gate were shifted towards the end of the EOM switching
period in order not to affect photon polarization by initial voltage spikes of the
EOM driver. The presence of these spikes dictated the maximum modulation rate.
The choice of the random bases and data acquisition were controlled by a Labview program. An improved software-based random number generator provided
the randomness of the bases.
In the images of Fig. 21, the results of a quantum key distribution are visualized. In a first step, Alice and Bob exchanged quantum information resulting in
a common sequence of random bits. A series of random number tests checked
and confirmed the randomness of the key. This key then encrypted Fig. 21(a) by
applying an exclusive-OR (XOR) operation between every bit of image and key.
The result is shown in Fig. 21(b) into which the randomness of the key was trans-
F IG . 21. Visualization of the quantum key distribution. After exchanging the key, Alice encrypts
image (a), a photography of Berlin’s skyline taken out of our lab window, and sends the encrypted
image (b) to Bob. After decryption with his key, he obtains image (c).
7]
NON-CLASSICAL LIGHT FROM ARTIFICIAL ATOMS
29
ferred. Then, Fig. 21(b) was classically submitted to Bob who decrypted it by
applying another XOR operation with his received key, yielding Fig. 21(c).
Altogether, the experiment was run with the following parameters: After the
electronic gating of Bob’s detector signals, the rate of usefully exchanged photons is found to be 30 s−1 whereas the dark count rate is reduced to 0.75 s−1 . The
probability to transmit photons through the two EOMs with crossed polarizations
was measured to be 6.8%. After comparing Alice’s and Bob’s keys, an error rate
of 5.5% was found. The presence of transmission errors leads to the necessity of
error correction. This requires the exchange of redundant data which opens an
eavesdropping loophole for gaining partial information of the message. With the
experimental parameters, the number of secure bits per pulse is 5 × 10−4 (following Lütkenhaus [90]) which is a typical value for current single-photon quantum
cryptography experiments (≈1 × 10−3 secure bits per pulse, see Refs. [88,89]).
The Michelson add/drop filter might also find applications in linear optical
quantum computation (LOQC). Since gates in LOQC have only limited success
probabilities, parallel processing may increase the efficiency of gates or at least
improve the statistical significance of a computational result. The method, which
was demonstrated here, also allows the spatial separation of two polarization
entangled photons (produced, for example, according to the proposal described
in [41]) without destroying their entanglement. Thus, they can be subsequently
used in a multitude of experiments and applications.
7. Summary
In this article, we have described single-photon generation with single InP quantum dots which emit in the visible spectrum around 690 nm. At this wavelength,
highest detection efficiencies with Si-based photo detectors are currently available which makes InP quantum dots preferable for free-space quantum optical
applications. The single-photon character of this source enables the performance
of fundamental quantum optics experiments, where the wave- and particle-aspect
of light can be observed simultaneously. We demonstrated single-photon statistics
and cross-correlations of various transitions from multi-excitonic states including
biexciton and triexciton decays. Multi-photon generation from single quantum
dots may find applications in quantum cryptography devices since a higher rate
of photons also enhances the maximum transmission rate of the quantum information. Therefore, a method to perform multiplexing was presented, similar to
the classical technique, but on the single-photon level. A typical application—
the BB84 quantum key distribution protocol—was performed to demonstrate this
method.
Quantum dot single-photon sources have reached a state where they can be implemented as ready-to-use non-classical light sources in a number of experiments.
30
T. Aichele et al.
[9
The on-demand character of the emission together with the potential for entangled pair generation will be extremely useful in all-solid-state implementations of
quantum information devices. Quantum cryptography and quantum computing,
but also interfacing of (small-scale) quantum information systems will be future
tasks of ‘quantum photonics’.
8. Acknowledgements
We acknowledge W. Seifert for providing the quantum dot sample. We thank
V. Zwiller, G. Reinaudi, and J. Persson for valuable assistance. This work was
supported by Deutsche Forschungsgemeinschaft (SFB 296) and European Union
(EFRE).
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ADVANCES IN ATOMIC, MOLECULAR AND OPTICAL PHYSICS, VOL. 53
QUANTUM CHAOS, TRANSPORT,
AND CONTROL—IN QUANTUM OPTICS*
JAVIER MADROÑERO1,2 , ALEXEY PONOMAREV2 ,
ANDRÉ R.R. CARVALHO2 , SANDRO WIMBERGER3 , CARLOS VIVIESCAS2 ,
ANDREY KOLOVSKY2 , KLAUS HORNBERGER4 , PETER SCHLAGHECK5 ,
ANDREAS KRUG2,† and ANDREAS BUCHLEITNER2
1 Physik Department, Technische Universität München, James-Franck-Straße, D-85747 Garching,
Germany
2 Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Str. 38, D-01187 Dresden, Germany
3 Dipartimento di Fisica Enrico Fermi and CNR-INFM, Università di Pisa, Largo Pontecorvo 3,
I-56127 Pisa, Italy
4 Arnold-Sommerfeld-Zentrum für Theoretische Physik, Ludwig-Maximilians-Universität München,
Theresienstr. 37, D-80333 München, Germany
5 Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2. Spectral Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1. Parametric Level Dynamics and Universal Statistics . . . . . . . . . . .
2.2. Spectral Signatures of Mixed, Regular-Chaotic Phase Space Structure .
3. Dynamics and Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1. Atomic Conductance Fluctuations . . . . . . . . . . . . . . . . . . . . .
3.2. Web-Assisted Transport in the Kicked Harmonic Oscillator . . . . . . .
3.3. Ericson Fluctuations in Atomic Photo Cross Sections . . . . . . . . . .
3.4. Photonic Transport in Chaotic Cavities and Disordered Media . . . . .
3.5. Directed Atomic Transport Due to Interaction-Induced Quantum Chaos
4. Control through Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1. Nondispersive Wave Packets in One Particle Dynamics . . . . . . . . .
4.2. Nondispersive Wave Packets in the Three Body Coulomb Problem . . .
4.3. Quantum Resonances in the Dynamics of Kicked Cold Atoms . . . . .
5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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66
67
68
* We dedicate this paper to Herbert Walther, at the occasion of his 70th anniversary, in reverence
to his contributions to the foundations of quantum optics, as well as to identifying the “quantum
signatures of chaos” in the lab. Happy birthday!
† Present address: Siemens Medical Solutions, Erlangen, Germany.
33
© 2006 Elsevier Inc. All rights reserved
ISSN 1049-250X
DOI 10.1016/S1049-250X(06)53002-2
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J. Madroñero et al.
[2
Abstract
Chaos implies unpredictability, fluctuations, and the need for statistical modelling.
Quantum optics has developed into one of the most advanced subdisciplines of
modern physics in terms of the control of matter on a microscopic scale, and, in
particular, of isolated, single quantum objects. Prima facie, both fields therefore appear rather distant in philosophy and outset. However, as we shall discuss in the
present review, chaos, and, more specifically, quantum chaos opens up novel perspectives for our understanding of the dynamics of increasingly complex quantum
systems, and of ultimate quantum control by tailoring complexity.
1. Introduction
Quantum optics has nowadays largely accomplished its strictly reductionist program of preparing, isolating and manipulating single quantum objects—atoms,
ions, molecules, or photons—such as to access the very fundaments of quantum theory, from quantum jumps [2,3] over the measurement process [4] and
decoherence [5], to quantum nonlocality and entanglement [6], in the laboratory.
The field turns “complex” now, by building up—or “engineering”—complexity
from the bottom, with nonlinear Hamiltonian dynamics [7], particle–particle interactions [8,9], disorder [10] or noise [11,12] as essential ingredients. Somewhat
unexpectedly, quantum optics therefore makes contact with quantum chaos—the
theory of finite size, strongly coupled quantum systems.
While for a long time under the suspicion of rather mathematical interest, coming up with “large fluctuations and hazardous speculations”, quantum chaos [1]
now finds an ever expanding realm of experimental applications [7,13–30]. In addition, it provides novel tools for the understanding and the robust control [14,
28,29,31,32] of the dynamics of increasingly “complex” quantum systems. In the
present review, we recollect some of the generic features encountered within such
“chaotic” quantum systems, and spell out their potential for the control of quantum dynamics in light-matter interaction.
2. Spectral Properties
There are different ways to approach quantum chaos. Possibly the most suggestive one proceeds along the semiclassical line, juxtaposing classical phase space
structures or dynamics on the one side, and the quantum spectral density or wave
function evolution in phase space, on the other [33–35]. The specific motivation
of this program lies in the intricate nature of the semiclassical limit (“h¯ → 0”,
2]
QUANTUM CHAOS, TRANSPORT, AND CONTROL
35
meaning the vanishing of Planck’s quantum when compared to typical classical
actions on macroscopic scales), and, hence, of the emergence of classical from
quantum dynamics at sufficiently large actions. This is an extremely attractive
approach, with a beautiful mathematical and theoretical machinery, leading to
important practical consequences, such as the rather recent semiclassical elucidation of the helium spectrum [36–41]. However, it is—by construction—bound to
quantum systems with a well-defined classical counterpart, since it derives quantum features from the backbone of the underlying classical dynamics.
While we shall adopt the semiclassical perspective for the motivation or interpretation of some of the results to be discussed in this paper, we will often deal
with systems which lack a well-defined classical analog. Therefore, most of our
observations will be derived directly from the quantum spectrum of the specific
systems under study.
2.1. PARAMETRIC L EVEL DYNAMICS AND U NIVERSAL S TATISTICS
On the spectral level, quantum chaos is tantamount to the destruction of good
quantum numbers [42,43]. Since the latter express symmetries, or dynamical invariants, of the specific system under study, quantum chaos occurs when these
symmetries are destroyed, e.g., by the nonperturbative coupling of initially separable degrees of freedom. If a well-defined classical Hamiltonian dynamics underlies the quantum dynamics, good quantum numbers are inherited from the
classical constants of the motion, and their destruction is paralleled by the invasion of classical phase space by chaotic motion.
Good quantum numbers can be considered, in a bounded system with a discrete
spectrum, as the labels attributed to individual eigenvalues of the Hamiltonian.
Symbolically, we may write for a system with three degrees of freedom:
H (λ) |n m(λ) = En(λ) m |n m(λ) .
(1)
These labels are good labels in the sense that, if H (λ) depends parametrically on a
real scalar λ, the eigenvectors |n m(λ) do not (ex)change their specific character
over a finite interval of λ.
The corresponding good quantum numbers loose their significance for the identification of individual eigenstates as soon as different eigenstates of H (λ) are
strongly mixed by a perturbation which couples at least two of the degrees of
freedom represented by the quantum numbers n, , and m, on arbitrarily small
intervals of λ—they are “destroyed” by the perturbation-induced coupling.
In the jargon of quantum chaos, the parametric evolution of the eigenvalues E (λ) of some Hamiltonian H (λ) parametrized by the real scalar λ is called
“regular level dynamics” if completely classifiable by good quantum numbers.
“Chaotic” or “irregular level dynamics” (also “level spaghetti”) is encountered
36
J. Madroñero et al.
[2
when all good quantum numbers are destroyed. Such irregular level dynamics
alone is one possible indicator of quantum chaos, without any recourse to some
analogous classical dynamics.1
A nice illustration of the transition from regular to irregular level dynamics is
provided by the Floquet–Bloch spectrum generated by the Bose–Hubbard Hamiltonian under static tilt,
L
L
L
J †
W HB = −
aˆ l+1 aˆ l + h.c. + F
dl nˆ l +
nˆ l (nˆ l − 1).
2
2
l=1
l=1
(2)
l=1
The Hamiltonian is formulated in terms of the creation and annihilation operators aˆ l† and aˆ l of a bosonic atom at the lattice site l, with the associated number operators nˆ l . It describes the dynamics of N ultracold bosonic atoms in a
one-dimensional optical lattice of length L and lattice constant d. The implicit
single band approximation assumes that no excitations to the first conduction
band of the lattice can be mediated by the tilt, F d Egap , nor by thermal activation, kT Egap , with Egap the band gap. J and W quantify the
strength of the nearest neighbor tunneling coupling J , and of the on-site interaction strength W between the atoms, respectively, which compete with a static
forcing of strength F . A suitable gauge transform reestablishes the translational
invariance in space apparently broken by the static field term in (2), and additionally introduces an explicit, periodic time dependence with the Bloch period
TB = 1/F [44]. The time evolution operator for one Bloch cycle in this time
dependent coordinate frame is the Floquet–Bloch operator associated with HB .
Figure 1 displays the level dynamics of the one cycle propagator, parametrized
by F , for different values of the ratio of tunneling coupling to interaction strength.
Clearly, when J and W become comparable, the eigenstates of the Floquet–Bloch
operator interact strongly for any value of F , while in the limit W J (and
equally so for J W ) individual eigenstates are clearly identifiable over large
intervals of F . In this specific model—which is actually realized in laboratory
experiments which load Bose Einstein condensates (BEC’s) into periodic optical lattices [8,45]—the transition from regular dynamics to quantum chaos is
apparent and unambiguous. Yet, this interacting multiparticle system has no welldefined classical counterpart! Further down in this review (see Section 3.5), we
will analyze the dynamical (and experimentally highly relevant) consequences of
this transition. At present, it is enough to state that the qualitative transition observed in Fig. 1 is actually qualitatively underpinned by the cumulative spacing
1 The term “dynamics” is motivated by considering the parameter λ as some generalized time, with
the eigenvalues E (λ) some generalized particle position evolving under variations of λ.
2]
QUANTUM CHAOS, TRANSPORT, AND CONTROL
37
F IG . 1. Spectrum of the Floquet–Bloch operator generated by HB as defined in (2), as a function
of 1/F , for N = 4 particles distributed over a lattice with L = 7 wells (periodic boundary conditions). Only states with quasimomentum κ = 0 are shown, in order to separate different symmetry
classes [44]. The particle–particle interaction strength and the tunneling coupling are set equal to
W = 0.032, and J = 0.00076 (top) and J = 0.038 (bottom), respectively. As we tune the tunneling
coupling to a value comparable to the interaction strength, the “individuality” of the energy levels
drowns in an irregular pattern: isolated avoided crossings between different energy levels which can
be labeled by the interaction energy between the different particles of a given multiparticle eigenstate
in the lattice [44] (for weak tunneling coupling, the distribution of the particles over the lattice characterizes a given eigenstate very well, except for resonant tunneling enhancements at isolated values
of F ) are replaced by strongly interacting levels, for arbitrary values of F .
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J. Madroñero et al.
[2
F IG . 2. Cumulative level spacing distribution of the Floquet–Bloch operator generated by HB
(Eq. (2)), for N = 7 bosonic atoms distributed over a lattice of length L = 9 (periodic boundary
conditions), static tilt F = 0.01, tunneling strength J = 0.038, interaction strength W = 0.032
(full line). The statistics is obtained from the unfolded spectrum [43] with the symmetry class defined
by quasimomentum κ = 0 [44]. The dashed and dash-dotted line indicate the RMT prediction for
Poissonian and Wigner–Dyson statistics, respectively.
distribution,
s
I (s) =
P s ds ,
(3)
0
with P (s) the probability distribution of the (normalized and unfolded, see,
e.g., [43]) spacings s between adjacent eigenphases of the Floquet–Bloch operator [44]. Inspection of Fig. 2 clearly shows that I (s) (and equally so P (s),
but the comparison of I (s) with the random matrix prediction is known to be
more reliable, in particular in the vicinity of s = 0) exhibits Poissonian statistics, P (s) = exp(−s), in the regular limit, and Wigner–Dyson statistics,
P (s) = πs exp(− π4 s 2 )/2, in the chaotic limit (more precisely, the level spacings
faithfully reproduce the COE statistics of random matrices of the circular (C) orthogonal (O) ensemble (E) [46]). Hence, by simply tuning the ratio of J and W , in
the perfectly deterministic Hamiltonian (2), we induce a spectral structure which
enforces a statistical description if we seek for a robust, quantitative description
of the system dynamics.
Another example of chaotic level dynamics is shown in Fig. 3, where we display the parametric evolution of the eigenphases of the Floquet operator of the
kicked harmonic
τ oscillator. The Floquet operator—or one cycle propagator—
U = exp(−i 0 H (t ) dt /h¯ ), with τ the kicking period, is generated by the
2]
QUANTUM CHAOS, TRANSPORT, AND CONTROL
39
F IG . 3. Spectrum of the Floquet operator generated by Hkho in (4), as a function of the
Lamb–Dicke parameter η, for a fixed phase space structure indicated by the single trajectory runs over
40,000 kicks in the insets (u and v are suitably defined, canonical phase space variables, see [32]).
Only eigenphases with an overlap larger than 10−3 with the initial state |ψ0 are represented. Filled
circles represent |ψ0 = |0, while dots refer to a displaced vacuum centered at (1.3, 3.0) (top) and
(1.2, 2.0) (bottom).
Hamiltonian
†
Hkho = hν
¯ aˆ aˆ + K
∞
mν †
cos
η
a
ˆ
+
a
ˆ
δ(t − nτ ).
k2
n=0
(4)
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J. Madroñero et al.
[2
This is a paradigmatic example of a quantum chaotic system which, on the classical level, does not obey the Kolmogorov–Arnold–Moser (KAM) theorem (which
guarantees stability with respect to small perturbations) [47], due to the degeneracy of the unperturbed spectrum of the harmonic oscillator. In (4), aˆ and aˆ † represent the annihilation and creation operators of the harmonic oscillator modes of
the translational degree of freedom (for a particle of mass m), and K measures
the strength of the kicking mediated √
by the periodically flashed standing wave
potential with wave vector k. η = k h¯ /2mν is the experimentally easily tunable Lamb–Dicke parameter, which essentially measures the ratio of the width
of the harmonic oscillator ground state in units of the wave length of the kicking
potential.
Hkho can be realized in semiconductor heterostructures [48] as well as with
cold, harmonically trapped ions, and allows for unlimited, superdiffusive energy
growth (i.e., for trapped ions, unlimited heating) under rather precisely defined
conditions, as we will see further down in this review. This specific dynamical
behavior has once again its root in the largely irregular level dynamics shown in
Fig. 3, which is here illustrated for two different ratios q = 2π/τ ν = 5 (top)
and q = 6 (bottom) of kicking period τ and oscillator period 1/ν, under variation
of η. These two choices correspond to a crystalline and quasicrystal [49] symmetry of the classical phase space structure, as indicated by the classical sample
trajectories shown in the corresponding insets. The crystal case still bears some
remnants of regularity, with regularly aligned avoided crossings coexisting with
apparently randomly distributed anticrossings of variable size. The quasicrystal
case, in contrast, exhibits an extremely complicated level structure, with no apparent regularity left. The details and structure of the level dynamics remain to
be understood, but part of its peculiarities can already be exploited for novel perspectives of quantum control, as we shall see further down in Section 3.2.
2.2. S PECTRAL S IGNATURES OF M IXED , R EGULAR -C HAOTIC P HASE S PACE
S TRUCTURE
In quantum systems with a well-defined classical analog which exhibits mixed
regular chaotic phase space structure [21,31,36,50–62], the parametric evolution of the eigenenergies does not exhibit an unambiguously “chaotic” structure.
Eigenenergies associated with eigenstates that are localized in phase space domains of regular motion are only weakly affected by the adjacent chaotic phase
space component and evolve, in general, smoothly under variations of some control parameter λ. Since regular domains of phase space are associated with local
dynamical invariants, these states can actually be labeled with good quantum
numbers, and undergo, in general, only locally avoided crossings with states living
on the chaotic phase space component. Consequently, such states “go straight” in
3]
QUANTUM CHAOS, TRANSPORT, AND CONTROL
41
F IG . 4. Parametric evolution of the spectrum of a microwave driven hydrogen atom, in suitably
rescaled energy units, under variation of the driving field amplitude F0 (measured in units of the
Coulomb field experienced by the Rydberg electron propagating along an unperturbed Kepler orbit
with principal quantum number n0 ) [31]. Two energy levels, which anticross at F0 0.036, clearly
“go straight” in this plot, and only weakly interact with the “level spaghetti” background: They represent eigenstates of the atom in the field which are localized on elliptic regions in the classically mixed
regular-chaotic phase space, and are therefore shielded against strong interaction with states living in
the chaotic phase space component.
the energy level dynamics, with almost constant slope, as displayed in Fig. 4 for
the (quasi)energy level associated with a wave packet eigenstate of a microwavedriven Rydberg state of atomic hydrogen (see also Section 4.1 below). In a
rather abstract sense, such states can therefore sometimes be attributed solitonic
character [63]—they anticross with “chaotic” eigenstates without changing their
characteristic features like localization properties, dipole moments, or the like.
Conversely, the soliton-like motion under variations of λ can serve as an identifier
for eigenstates which are shielded from the irregular part of the spectrum, even in
the absence of an unambiguous classical dynamics—examples are found, e.g., in
microwave driven Rydberg states of alkali atoms [64], with their nonhydrogenic
multielectron core which induces quantum mechanical diffraction effects on top
of the semiclassical Rydberg dynamics [57,65].
3. Dynamics and Transport
The specific spectral structure of a given quantum system fully determines the
associated time evolution. If we initially prepare our system in the state |ψ0 , the
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J. Madroñero et al.
action of the time evolution operator is given by
exp(−iEn t/h¯ )|En En |ψ0 ,
U (t)|ψ0 =
[3
(5)
n
where we assume, for simplicity, a discrete spectrum {En } of H . Alternatively,
the energies En may be thought of as complex eigenvalues En − iΓn /2 of some
effective Hamiltonian, with the decay rates Γn representing, for instance, the
nonvanishing coupling to a continuous part of the spectrum [66–69]. In most experiments, some sort of (auto)correlation signal like
ψ0 |φn 2 exp(−iEn t/h¯ )
C(t) = ψ0 |U (t)|ψ0 =
n
2
Γn
→
ψ0 |φn exp(−iEn t/h¯ ) exp − t
2
n
(6)
is measured [70,71], which, besides the purely spectral ingredients En and Γn
also includes a local “probe” |ψ0 |φn |2 of the spectrum, in the vicinity of the
state |ψ0 with which the time evolved wave function is to be correlated. Also
ionization or survival probabilities which are often encountered in atomic ionization experiments or in model systems which probe quantum mechanical phase
space transport are closely related to such correlation functions, possibly amended
by an additional summation over a (discrete or continuous) set of “test functions”
|ψ0 [53,72–75].
3.1. ATOMIC C ONDUCTANCE F LUCTUATIONS
It is immediately clear from the form of (6) that the dynamics of a chaotic quantum system in the sense of chaotic level dynamics as illustrated in Section 2 will
exhibit a sensitive parameter dependence, reflecting the parametric evolution of
the spectrum. A nice example is provided by the ionization yield of one electron
Rydberg states under microwave driving—which probes the asymptotic electron
transport induced by the external perturbation. In such type of experiments [21,
50,76–88], one electron Rydberg states (with excitations to principal quantum
numbers around n0 70) are exposed to a microwave field of frequency ω and
amplitude F , for an adjustable interaction time t. The experimentally easily accessible ionization yield Pion is formally given [73] by
ψ0 |φj 2 exp(−Γj t).
Pion = 1 −
(7)
j
The sum extends over the complete spectrum of the atom dressed by the field,
though weighted by the overlap of the (field free) initial state with the atomic
3]
QUANTUM CHAOS, TRANSPORT, AND CONTROL
43
F IG . 5. Typical distribution of the ionization rates Γj and local weights Wj = |ψ0 |φj |2 entering
the expression (7) for the ionization yield Pion of an atomic Rydberg state under electromagnetic
driving. In the upper plot, 500 spectra of a one-dimensional model atom initially prepared in the
Rydberg state |n0 = 100 are accumulated, for driving field frequencies ω/2π = 13.16 . . . 16.45 GHz,
at fixed photonic localization length = 1 (see Eq. (9)). In the lower plot, one single spectrum of
the three-dimensional hydrogen atom initially prepared in the state |n0 = 70 0 = 0 m0 = 0, at
ω/2π = 35.6 GHz and = 1 is shown. There is no apparent correlation between ionization rates and
local weights—which also manifests in the parameter dependence of Pion itself, see Fig. 6.
dressed states for the specific choice of ω and F . Typically, several hundreds to
thousands dressed states contribute to the representation of |ψ0 [89,90].
Under changes of ω or F , not only the decay rates Γj of the individual dressed
states will fluctuate, but, equally important, the local weights |ψ0 |φj |2 —as a
corollary of the destruction of good quantum numbers in the realm of quantum chaos: The characteristic properties of the system eigenstates vary rapidly
with the control parameter (here ω or F ), and so does the decomposition of the
(parameter-independent) initial state |ψ0 . In general, the fluctuations of decay
rates and overlaps are uncorrelated, as illustrated in Fig. 5, for typical driving frequencies and amplitudes, and for a one-dimensional model of the driven atom,
as well as for the real, three-dimensional system. While one might believe that
these fluctuations average out under the summation in (7), this is actually not
the case—Fig. 6 shows the ionization yield of atomic hydrogen, initially prepared in the unperturbed n0 = 100 Rydberg state, under microwave driving
with variable frequency. Indeed, Pion fluctuates rapidly with the scaled frequency
ω0 = ω × n30 [93] in this plot, at fixed n0 . This is the dynamical manifestation
of the sensitive ω0 -dependence of the quantities which determine Pion , according
to (7). While this sensitive dependence shows that the mere ionization yield for
44
J. Madroñero et al.
[3
F IG . 6. Ionization yield Pion , Eq. (7), of a one-dimensional Rydberg atom launched in the Rydberg
state |n0 = 100, as a function of the scaled driving field frequency ω0 = ω × n30 , at fixed localization
length = 1 (see Eq. (9)). The strong fluctuations of the signal under variations of ω0 are characteristic
of a strongly localized (in the sense of Anderson [91]) transport process (here on the energy scale, and
induced by the external driving) in disordered media [92].
given ω and F does not provide a robust characterization of the electronic transport process induced by the external drive, a statistical analysis allows for some
insight: The atomic conductance [94]
2
1 gatom =
(8)
ψ0 |φj Γj ,
j
formally equivalent to the time derivative of the ionization yield at t = 0 (with the average spacing between adjacent energy levels), exhibits a log-normal distribution, i.e., ln gatom is normally distributed, when sampled for a fixed photonic
localization length [95]
6.66F02 n0
n20 −1
E
.
=
(9)
=
1
−
7/3
ω
n2c
ω0
The latter is a measure of the typical decay length of the electronic population
distribution over the near resonantly coupled Rydberg states away from the atomic
initial state |ψ0 , and determines the asymptotic continuum transport on average,
according to [93]:
ln gatom ∼ 1/.
(10)
In particular, this proportionality relation together with the lognormal distribution for fixed localization length, which are established in Figs. 7 and 8 for a
one-dimensional hydrogen atom (which is a reliable model for the description of
real 3D hydrogen under external microwave driving, when initially prepared in
3]
QUANTUM CHAOS, TRANSPORT, AND CONTROL
45
F IG . 7. Average value of the natural logarithm of the atomic conductance g vs. the inverse photonic localization length 1/, for a one-dimensional Rydberg atom initially prepared in the state |n0 with principal quantum number n0 = 40, 60, 70, 90, 100 (from top to bottom). Clearly, the direct proportionality (10) predicted by the Anderson picture is very well satisfied for sufficiently large values
of n0 [93].
F IG . 8. Distribution (histograms) of the atomic conductance g of a one-dimensional Rydberg
atom [93,94,97], sampled over 500 different spectra with photonic localization length = 0.2, in
the frequency range ω0 = 2.0 . . . 2.5, for initial principal quantum number n0 = 40 (left) and
n0 = 100 (right). The log-normal fit is excellent for n0 = 100, in perfect quantitative agreement
with the Anderson picture. Finite size effects lead to discrepancies between the numerical distribution
of ln g and the lognormal fit at lower excitations around n0 = 40.
an extremal parabolic state [96,97]), provide strong quantitative support for the
analogy between electronic transport along the energy axis in periodically driven
atomic Rydberg states and electronic transport across one-dimensional disordered
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J. Madroñero et al.
[3
wires [92,94,98–100]: Destructive quantum interference of the many transition
amplitudes connecting the initial atomic state to the atomic continuum, in the
atomic problem, and the left and the right edge of the disordered wire, in the
mesoscopic problem, leads to an exponential suppression of the quantum transport, as opposed to diffusive transport in a classical description. This phenomenon
is known as Anderson localization [91,101–104] (also strong localization), and
was baptized dynamical localization [17,19,105–115] in the realm of quantum
chaos, where dynamical chaos substitutes for disorder.
3.2. W EB -A SSISTED T RANSPORT IN THE K ICKED H ARMONIC O SCILLATOR
An alternative scenario for the detection of chaos-induced fluctuations on the level
of quantum transport properties is provided by cold, harmonically trapped ions
under periodic kicking. We already have seen in Section 2.1 that the energy level
dynamics of the kicked harmonic oscillator which is realized in such a setting
exhibits many avoided crossings of variable size. Indeed, if we launch a wave
packet in the harmonic oscillator ground state and monitor its mean energy as
time evolves, the energy growth rate is found to depend sensitively on the precise value of the Lamb–Dicke parameter η, which is easily tuned in state of the
art ion trap experiments. Figure 9 shows such behavior,
√ for three different values
of η, at fixed classical phase space structure (η ∼ h¯ determines the effective
size of h¯ with respect to the typical classical action of the harmonic oscillator;
also see Fig. 3). Correspondingly, the mean energy extracted by the atoms from
the kicking field, after a fixed interaction time, exhibits strong, apparently random
fluctuations with the Lamb–Dicke parameter, as illustrated in Fig. 10. Once again,
this can be directly associated with the avoided crossings in the energy level diagram in Fig. 3, and is strongly reminiscent of the atomic conductance fluctuations
encountered in Fig. 6. Note, however, that the classical phase space structure of
the kicked harmonic oscillator is different from the phase space structure of the
harmonically driven Rydberg atom, since we are here dealing with a non-KAM
system. The signature of this non-KAM structure in the spectral statistics is hitherto unexplored, and represents a formidable challenge, both for random matrix
theory, as well as for computational physics.
We can nonetheless precisely identify the universal cause of the locally enhanced energy absorption of the trapped ions from the kicking field, by inspection
of the eigenstates which undergo the specific avoided crossing, at a given value
of η: Fig. 11 shows the Husimi phase space projections [69] of those eigenfunctions which account for the dominant part in the decomposition of the ionic initial
state |ψ0 = |0 in the vicinity of η = 0.464 (the associated level anticrossing
is shown by the inset in Fig. 9), i.e., at a value where strongly enhanced heating
of the ions is observed. While for Lamb–Dicke parameters slightly below and
3]
QUANTUM CHAOS, TRANSPORT, AND CONTROL
47
F IG . 9. Mean energy of the kicked harmonic oscillator, Eq. (4), for crystal symmetry, q = 6,
kicking strength K = 2.0, and initial state |ψ0 = |0. Tiny changes of the Lamb–Dicke parameter
from η = 0.459 (a) over η = 0.464 (b) to η = 0.469 (c) lead to a locally dramatic enhancement of the
energy absorption by the trapped particle from the kicking field, with respect to the classical heating
process. This local enhancement can actually be traced back to an avoided crossing of the continuation
of the eigenphase associated with |ψ0 in the level dynamics (inset) with a “web-state” (see Fig. 11)
reaching far out to high energies in the harmonic oscillator phase space. The above values of η are
indicated by the corresponding labels, in the inset. Filled black circles indicate an overlap of more
than 1% of the associated eigenstate with the initial state |ψ0 .
F IG . 10. Mean energy (left vertical axis) after 600 (full line) kicks vs. the Lamb–Dicke parameter η. The classical phase space structure is fixed by K = 2.0 and q = 6. Locally strongly enhanced
energy absorption can always be traced back to avoided crossings of the initial state with web states,
as apparent from the underlaid energy level dynamics (right vertical axis).
48
J. Madroñero et al.
[3
F IG . 11. Husimi representations of the eigenstates associated with the labels a (left column) and
c (right column) in the inset of Fig. 9, in the rescaled phase space coordinates v/2η = −60 . . . +60
and u/2η = −60 . . . +60 of the insets of Fig. 3. The top left and bottom right plot represent web
states associated with the top left and bottom right branch of the avoided crossing shown in the inset
of Fig. 9. At η = 0.464, i.e., at the center of that avoided crossing, they strongly mix with the continuation (bottom left and top right branch of the avoided crossing, and bottom left and top right Husimi
representation in the present figure) of |ψ0 , thus giving rise to efficient transport from the trap center
to high energy states of the harmonic oscillator, along the stochastic web of the underlying classical
phase space flow. Since the avoided crossing of the web state with the localized state occurs at fixed
phase space structure, this is a pure quantum tunneling effect, without classical analog.
slightly above this critical value the eigenstate which is strongly localized in the
vicinity of the origin of phase space has the largest weight in the initial state decomposition, an eigenstate localized on the stochastic web has equal weight right
at η = 0.464. The existence of such web states is a peculiarity of non-KAM systems and is at the very origin of the observed enhanced energy growth, simply
since the stochastic web reaches out to infinity, and therefore provides an efficient transport channel to high energy states of the oscillator. Since the avoided
crossing which mediates the coupling of the initial state to the web state occurs
under changes of the effective value of h¯ (via η), at fixed phase space structure,
we have here—much as in the above case of strong localization in the ionization
process of periodically driven atoms—a pure quantum effect without classical
analog, leading now to a dramatic enhancement of the asymptotic transport, as
3]
QUANTUM CHAOS, TRANSPORT, AND CONTROL
49
compared to the classical dynamics. A closely related phenomenon has been observed in the conductance across semiconductor superlattices, in the presence of
a tunable magnetic field [48]. Since there the magnetic field allowed to switch
between localized and delocalized (i.e., web-) states, web-states mediate, in some
sense, metal-insulator like transitions.
3.3. E RICSON F LUCTUATIONS IN ATOMIC P HOTO C ROSS S ECTIONS
In the preceding two subsections, we encountered examples of a sensitive dependence of asymptotic transport on some control parameter, typical of quantum
chaotic systems, in explicitly time dependent transport processes. As a third example, we now consider the continuum decay of Rydberg electrons induced by static
external fields, which can be probed through the photoabsorption cross section
for a probe laser beam from the atomic ground state into the Rydberg spectrum.
Indeed, an atomic one electron Rydberg system exposed to perpendicularly oriented, static electric and magnetic fields, allows us to realize such a situation: The
Hamiltonian reads
p2
B
B2 2
(11)
+ Vatom (r) + Lz +
x + y 2 + F x,
2
2
8
in atomic units, with F and B the strength of the electric and magnetic field, respectively, and Lz the angular momentum projection on the magnetic field axis.
If Vatom (r) is given by the hydrogenic Coulomb potential, the diamagnetic term
in (11) is known to induce chaotic motion in the bound space dynamics of the
Rydberg electron. For B = 0 the electric field, while leaving the dynamics completely integrable, induces a Stark saddle and, hence, strong coupling of the bound
eigenenergies with the continuum part of the spectrum. If both external fields are
present, all symmetries of the unperturbed Coulomb problem are destroyed, and
one faces a truly three-dimensional problem which exhibits dynamical chaos. In
the case of alkali atoms, the additional presence of a multielectron core is not expected to suppress the signature of the classically chaotic Coulomb dynamics, on
the spectral level [57,116,117].
Due to the suppression of the ionization threshold by the electric field, the high
lying Rydberg states can acquire relatively large autoionization rates Γj , with
an average value Γ¯ which can become larger than the mean level spacing of
the (quasi)discrete energy levels Ej , i.e., Γ¯ > . In this regime of overlapping
resonances, Ericson fluctuations [118–122] are expected in the photoabsorption
cross section
HExB =
σ (E) =
4π(E − E0 ) |g|T |Ej |2
Im
ch¯
Ej − iΓj /2 − E
j
(12)
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[3
F IG . 12. Distribution of the resonance widths Γj which contribute to the photo cross section σ (E),
Eq. (12), in an energy interval which covers the experimentally [27,123] scanned region. The dashed
line indicates the average (local) spacing of the resonance states on the energy axis. Approx. 65%
of them exhibit overlapping widths, Γj > .
from the atomic ground state |g into the Rydberg regime at energy E: Boundcontinuum transition amplitudes which mediate the decay of individual resonances couple to overlapping intervals of continuum states, and thus may interfere. Consequently, one expects interference structures in the cross section which
can no more be attributed to individual resonance eigenstates with a specific
width Γj , but are rather due to the interference of several decay channels, and
exhibit typical widths smaller than Γ¯ . If a classical analog dynamics is available,
these structures are predicted to be correlated on an energy scale which is determined by the dominant Lyapunov exponent of the classically chaotic dynamics,
i.e., by the shortest decorrelation time scale of the classical dynamics [120].
Indeed, the transition into the Ericson regime has recently been observed in
the photoionization cross section of rubidium Rydberg states in the presence of
crossed fields [27,123]. A detailed theoretical analysis of the experimental situation shows that the laboratory results indeed entered the regime of overlapping
resonances, and approx. 65% of all resonance eigenstates contributing to the photoabsorption signal have widths which are larger than the mean level spacing .
Figure 12 shows the numerically calculated distribution of resonance widths over
the energy range probed by the experiment, under precisely equivalent conditions
as in the experiment (fixed by the strength of the magnetic and electric fields).
Besides the strongly fluctuating background signal, the cross section σ (E) displayed in Fig. 13 also shows some narrow resonances on top, which stem from
isolated resonances with Γj < . However, many of the structures with a width
smaller than Γ¯ can no more be associated with single isolated resonances, and
thus indicate the interference of different decay amplitudes.
Thus, we observe the coexistence of individually resolved resonances with
Ericson fluctuations. This it is not too surprising, since the original Ericson sce-
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51
F IG . 13. Numerically obtained photo cross section (12) of rubidium Rydberg states in crossed
electric and magnetic fields [124], deduced from a parameter free diagonalization of the Hamiltonian (11), using exactly the experimental parameters [27], B = 2.0045 T, F = 22.4 kV/m.
nario was inspired by highly excited compound nuclei with a large number of
essentially equally weighted decay channels, while we are here dealing with a
low-dimensional atomic decay problem, where different decay channels (e.g.,
through different angular momentum channels) have certainly different weights
and equally different effective bound-continuum coupling constants.
Once again, due to the underlying chaotic level structure—here additionally
complicated by resonance overlap—the experimentally accessible cross section
shows erratic fluctuations, essentially uncorrelated on energy scales which are
larger than the inverse of the characteristic life time of the ion–electron compound (which, in a classical picture, is determined by the largest Lyapunov exponent).
3.4. P HOTONIC T RANSPORT IN C HAOTIC C AVITIES AND
D ISORDERED M EDIA
In the previous section, we showed how the fine interplay between overlapping
and isolated resonances determines the nature of the fluctuations in the transport
properties of chaotic systems. In this section, we shall consider a novel kind of
systems for which this interplay has also a determinative role: random lasers.
In contrast to standard lasers, random lasers do not possess mirrors. They are
a class of nonlinear amplifiers realized in disordered dielectrics with a fluctuating
dielectric constant that varies randomly in space. Light amplification is provided
by an active optical medium, while the multiple chaotic scattering of photons in
the random medium constitutes the feedback mechanism. Due to multiple scattering, the time spent by the light inside the active medium is enhanced. This, in
turn, increases the probability of stimulated emission, making the field amplification efficient. Laser oscillations emerge when the radiation losses are overcome
by the light amplification.
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J. Madroñero et al.
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In recent years, several experiments on random lasers (see Ref. [125] for a
review) as well as on lasers in chaotic resonators [126,127] have attracted considerable interest in the characterization of the properties of light emitted by these
devices. Most striking are the generic signatures of the underlying disorder of
the random media in the emission spectra: In samples with a low density of scatterers [128], light is only weakly confined and we expect the resonant modes to
overlap. Once the pump energy exceeds the laser threshold, the onset of lasing is
signaled by a collapse of the thermal emission spectrum into a single broad peak
with a width of a few nanometers at the center of the amplification bandwidth.
For samples with a high density of scatterers [128], on the other hand, some wellresolved resonant modes exist. As soon as the laser enters the operation regime
above threshold, several very sharp peaks appear, the frequencies (within the amplification bandwidth) and strengths of which fluctuate strongly from sample to
sample.
The above-mentioned features of the emission spectra cannot be explained by
standard laser theory [129–131]. The reasons are twofold: First, in random lasers
the spatial structure of the resonant modes as well as their frequencies depend
on the statistical properties of the disordered medium. Random lasers, therefore,
must be analyzed in an statistical fashion. Second, due to the absence of mirrors, light in random lasers is only weakly confined, giving rise to spectrally
overlapping resonances. Recently, based on a field quantization method for open
systems with large outcoupling losses [132–134], a quantum theory of random
lasing incorporating both effects, random scattering of light and mode overlap,
was proposed [135].
For a random laser with an active medium composed of two-levels atoms, the
quantum Langevin equations of motion for the field variables are
∗
Hλλ aλ (t) +
gλp
σ−p (t) + Fλ (t).
a˙ λ (t) = −i
(13)
λ
p
Here, aλ is the annihilation operator of the field mode λ, and σ−p is the dipole
operator of the pth atom. The coupling amplitudes gλp between field and atoms
are proportional to the atomic dipole d and to the field amplitude u(r) at the position of the atom, gλp ∝ duλ (r p ). Equation (13) should be complemented with
the equations of motion for the atomic operators, which we have omitted as they
remain the same as those found in standard laser theory [129]. There are drastic
differences between Eq. (13) and the independent-oscillator equations of standard
laser theory. They arise from the fact that in order to account for the strong coupling of the field with the outside, all internal modes must now leak into the same
external channels, i.e., they are coupled to the same bath. Hence, the internal dynamics of the field is determined by the non-Hermitian operator H, accounting for
the system’s losses due to the coupling with the exterior, and coupling the different
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QUANTUM CHAOS, TRANSPORT, AND CONTROL
53
modes aλ . Additionally, and consistently with the fluctuation–dissipation theorem, the noise operators Fλ of the different modes are correlated, Fλ† Fλ = δλλ
(the expectation value is defined with respect to the state of the bath).
The emission properties of random lasers are determined by the complex eigenvalues ωk − iΓk /2, and the nonorthogonal eigenfunctions R(r) of H. Due to the
strong correlation among modes, the relation between the mean frequency separation of the real frequencies ωk , and the average decay rate Γ¯ of the modes
is of crucial relevance for the emission spectra. In the regime of overlapping resonances, Γ¯ > , typically many broad modes will contribute to the emitted
radiation. The resulting spectrum is then a smooth function of the frequency. On
the contrary, in the regime of isolated resonances, Γ¯ < , the spectrum consists
of a set of sharp peaks located at the resonant frequencies of the system. More
striking, however, is the effect of the mode correlations on the coherence time
of the random laser emission. For single mode lasing the coherence time δτ is
inversely proportional to the laser line width δω. The latter was first calculated
for standard lasers by Schawlow and Townes [136], by taking into account the
spontaneous emission noise, and was found to decrease for increasing output intensities, δωST ∼ 1/I . In random lasers, however, the noise correlation between
different modes leads to an enhancement of the line width. One then has [132]
δω = KδωST ,
(14)
where K 1 is the so called Petermann factor [137–139]. K can be related to
the self-overlap of the nonorthogonal laser mode R(r), and is a measure of the
correlations in the system. Hence, the coherence time of a random laser is smaller
than the coherence time of a standard laser with the same output intensity.
The signatures of the underlying disorder in random lasers are also present in
the photon statistics of the emitted light. Though for light propagating in a disordered material the photon statistics below threshold is well understood [140,141],
only recently the nonlinear optical regime above threshold has been investigated [142–145]. As an example, we evaluate the mean photocount of the emitted
field from a chaotic laser resonator in the regime of single-mode lasing [142]. We
consider the coupling of the cavity to the outside to be weak, so that all resonances
in the cavity are well defined. In this perturbative limit, the non-Hermitian operator H in Eq. (13) becomes diagonal, and the laser mode a decouples from all
other modes. Moreover, since the cavity opening is small, we can replace R(r) by
the orthogonal close cavity modes u(r). In chaotic resonators the amplitude u(r)
at a point r behaves like a Gaussian random variable, and is uncorrelated with the
amplitude at any other point, provided it lies further apart than an optical wave
length λ [146,147]. As we shall show, these spatial fluctuations induce strong
mode-to-mode fluctuations in the laser emission.
In its steady-state, the laser is characterized by three parameters comprising
the effects of the active medium on the field: The linear gain A, the nonlinear
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J. Madroñero et al.
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saturation B, and the total loss rate C. The photon number distribution giving the
probability to find the laser field at a time t with n photons is [130]
Pn = N
(Ans /C)n+ns
,
(n + ns )!
(15)
where the symbol N stands for a normalization constant, and the nonlinear saturation B enters through the so called saturation photon number ns = A/B. When
the number of atoms in the active medium is large, A and B are shown to acquire
sharp values. C = Γ +κ, on the other hand, is the sum of the photon escape rate Γ
due to the cavity opening, and the absorption rate κ accounting for all other loss
mechanisms of the radiation inside the resonator. While here κ may be considered
fixed, the photon escape rate depends on the resonator mode u. Thus, inasmuch as
the resonator mode represents wave chaos, Γ , and therefore C, become random
numbers. The distribution P (Γ ) over an ensemble of modes in time-reversal invariant cavities is a well-know result from random-matrix theory [148,149], and is
given by the χν2 distribution. Here, ν is an integer, counting the number of escape
channels at the opening of the resonator. For the case ν = 1, the corresponding
distribution is known as the Porter–Thomas distribution.
For a single-mode laser, the mean output intensity is given by I = Γ n, where
n is the mean photon number inside the cavity. Over an ensemble of chaotic
cavity modes the mean output intensity fluctuates from one mode to the other. Its
distribution is given by
P (I ) = dΓ P (Γ )δ I − Γ n .
(16)
Note that the right-hand side involves a twofold average, the quantum optical
average with the distribution Pn (represented by the brackets . . .) and the ensemble average over the cavity modes with distribution P (Γ ). We evaluate numerically P (I ). The results for an ensemble of chaotic cavities with one escape
channel are plotted in Fig. 14, for two different sets of parameters. In both cases
¯ i.e., they correspond to lasers above threshold in the ensemble averA > C,
age. We note that all distributions are strongly non-Gaussian. They are all peaked
as I −1/2 at small intensities, and present a second peak for maximal intensity.
Furthermore, for one of the parameter sets (dashed lines) the distribution P (I )
displays a shoulder for submaximal I . This last feature is seen to be a signature of
spontaneous emission [142]. Thus, for lasers in resonators with irregular shape the
chaotic nature of the cavity modes gives rise to fluctuations of the photocount on
top of the quantum optical fluctuations known from laser theory. Chaos-induced
fluctuations are found when a single-mode photodetection is performed over an
ensemble of modes.
In recent years, in the light of nonlinear optical effects, the investigation on multiple scattering of photons has received new impetus. A fresh and fertile field for
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QUANTUM CHAOS, TRANSPORT, AND CONTROL
55
F IG . 14. Distribution P (I /Imax ) as a function of the dimensionless mean intensity I /Imax , for
one escape channel and two sets of parameters. Rates are given in units of A ≡ 1, the nonlinearity is
B = 0.005. The solid line corresponds to κ = 0.7, Γ¯ = 0.02; the dashed line to κ = 0.7, Γ¯ = 0.2.
interesting physics is found in this region where nonlinear optics and wave chaos
intersect. Random lasers are just one example of the kind of problems encountered there. Other relevant examples constitute studies of coherent backscattering
of light by a cloud of cold atoms. In these system, for sufficiently high intensities
of the incident light, nonlinearities becomes relevant and a new class of coherent
effects are seen to arise [150,151]. In the near future, new questions concerning
the consequences of nonlinear effects for the strong localization of light are likely
to move into focus.
3.5. D IRECTED ATOMIC T RANSPORT D UE TO I NTERACTION -I NDUCED
Q UANTUM C HAOS
All the above examples of transport in quantum chaotic systems stem from the
realm of one (active) particle dynamics—where we also include the phenomena
observed with alkaline atoms, since the multielectron atomic core only induces
additional quantum diffraction effects, which can be accounted for on the one
particle level. In our last example, we consider now an interacting many-particle
problem, which is motivated by recent progress in the manipulation of ultracold
atoms loaded into optical lattices, and which establishes, in some sense, the experimentally “controlled” version of multiparticle quantum chaos originally thought
of by Bohr [152] and Wigner [153] when they modelled compound nuclear reactions.
One of the prominent models to describe the dynamics of matter waves in optical potentials is defined by the Bose–Hubbard Hamiltonian (2) which we already
encountered above. Indeed, it can be shown that (2) exhibits Wigner–Dyson statistics in a broad interval of tunneling coupling J and interaction strength W , for
56
J. Madroñero et al.
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filling factors n¯ = N/L, N the particle number and L the lattice length, even in
the absence of any static forcing, i.e., for F = 0 [154]. Surprisingly, this was realized only recently, despite the fact that (2) is a standard “working horse” for quite
a big community—which, however, is mostly interested in ground state properties
rather than dynamics. Only recent experiments in quantum optics laboratories [8,
155–160] have triggered enhanced interest in dynamics, and hence in the excitation spectrum of the many-body Hamiltonian.
On the dynamical level, the chaotic character of the Bose–Hubbard spectrum induces the rapid decay of single particle Bloch oscillations across a onedimensional lattice, for not too large static forcing (such that the static term in (2)
does not dominate the symmetry of the problem) [161,162]. The single particle
dynamics can be defined equally well by the reduced single particle wave function of the bosonic ensemble, or by a second, spin-polarized fermionic component
loaded into the lattice [163]. We shall here adopt the latter scenario, where noninteracting fermionic atoms interact with a bosonic “bath”. The corresponding
two-component Hamiltonian writes
HFB = HF + HB + Hint ,
(17)
and decomposes into the (single particle) fermionic part
L
L
JF |l + 1l| + h.c. + F d
|lll|,
HF = −
2
l=1
the (many particle) bosonic part
L
L
JB †
WB aˆ l+1 aˆ l + h.c. +
nˆ l (nˆ l − 1),
HB = −
2
2
l=1
(18)
l=1
(19)
l=1
and a term which mediates the collisional interaction between fermions and
bosons,
Hint = WFB
L
nˆ l |ll|.
(20)
l=1
Here we built in the assumption that only the fermions experience the external
static force—this can be arranged by preparing the fermionic and bosonic component in appropriate internal electronic states, which couple differently to external
fields.
Since in (17) there is a clear separation between “system” (the fermions) and
“bath” (the bosons), we can derive a master equation for the time evolution in
the fermionic degree of freedom, in Markovian approximation [163]. A crucial
ingredient for this derivation is the chaotic level dynamics of the bath degree of
freedom, what ensures a broad distribution of frequencies of the bath modes, such
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QUANTUM CHAOS, TRANSPORT, AND CONTROL
57
as to act as a Markovian environment, with a rapid decay of the bath correlations,
on the relevant time scales of the system dynamics [164]. One ends up with
(F )
∂ρl,m
∂t
=−
i
(F )
HF (t), ρ (F ) l,m − γ (1 − δl,m )ρl,m ,
h¯
(21)
where ρ is the fermionic one particle density matrix, and the relaxation rate γ is
completely determined by the parameters of our original Hamiltonian (17):
γ =
2
τ n¯ 2 WFB
2
3n¯ 2 WFB
.
h¯ JB
(22)
h¯ 2
In other words, we can “engineer” incoherent Markovian dynamics in a perfectly
Hamiltonian system, (17), by exploiting the chaotic dynamics of one system component. The resulting decay of the fermionic Bloch oscillations is illustrated in
Fig. 15, where perfect agreement of the actual decay rate (resulting from an exact numerical propagation of the dynamics generated by (17)) with the analytical
expression (22) is observed.
The collisional interaction of the fermions with the bosonic bath provides a
relaxation mechanism which, in the theory of electronic conductance across a periodic potential, is the necessary ingredient for observing a net current across the
lattice [165]. Yet, in Fig. 15 we do not observe any net drift of the electrons. This
F IG . 15. Bloch oscillations of the fermionic mean velocity in the optical lattice, under a static
tilt F d = 0.57 × JF , with JF = JB , and WFB = 0.101 × JF , 0.143 × JF , 0.202 × JF (from top to
bottom). The bosonic bath, which is the source of the collisionally induced damping of the oscillations,
is composed of N = 7 particles, distributed over a lattice of length L = 9. v0 = JF d/h¯ . The typical
time scale of the interaction induced decay fits the time scale predicted by Eq. (22) (dash-dotted lines)
very well [163].
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J. Madroñero et al.
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is due to the fact that we are here dealing with a perfectly closed system, without
attaching any leads—in particular, we are dealing with a finite size bath, which,
consequently, has a finite heat capacity. Therefore, the initial state of the bath
plays a crucial role for the effective fermionic transport across the optical lattice:
If prepared in the thermalized state (as in Fig. 15), with equal population of all
energy levels of the bath, no net energy flux can occur from the fermionic into the
bosonic degree of freedom, and, hence, no net drift velocity of the fermions can
emerge. In contrast, if we prepare the bath in a low temperature state, with only the
ground state and few excited states initially populated, the bath can absorb energy
from the fermions, via collisions, and the fermionic component acquires a nonvanishing drift—which lasts until the bath is fully thermalized. This is illustrated
in Fig. 16, together with the corresponding energy increase of the bath. Figure 17
shows the resulting current (fermionic drift velocity v)
¯ voltage (static tilt F experienced by the fermionic component) characteristics under variations of F , which
displays a marked transition from Ohmic behavior (small F ) to negative differential conductance (large F )! Note that such behavior was earlier predicted for
semiconductor superlattices [166], on the basis of a semiclassical theory with a
phenomenologically determined relaxation rate γ , whereas the present scenario
allows for the experimental tuning of the relaxation rate, on the basis of our microscopic theory (with crucial input from the theory of quantum chaos).
F IG . 16. Mean velocity v(t) of the fermionic component (top, solid line) for a low temperature
(kB T 2.86 × JB ) bath, under static tilt F d = 0.143 × JF , with WFB = 0.143 × JF , WB /JB = 3/7,
JB = JF , N = 7, L = 9. The solid line in the bottom plot shows the associated time evolution of
the mean energy EB of the bath. Dashed lines in both plots indicate the result for a thermalized bath
(kB T 150 × JB ), when no net energy exchange between the fermions and the bosons is possible.
Clearly, only for the low temperature bath do we observe a nonvanishing drift velocity (i.e., a directed
current) of the fermions across the lattice.
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QUANTUM CHAOS, TRANSPORT, AND CONTROL
59
F IG . 17. Current–voltage (expressed as drift velocity v¯ vs. tilt F d) characteristics for the directed
fermionic current across the optical lattice (stars) [163], for the same parameters as in Fig. 16. The
continuous line shows the prediction of a phenomenological model of charge transport in semiconductor superlattices [166], with the relaxation rate γ extracted from Eq. (22). A clear transition from
Ohmic to negative differential conductance at large tilt potentials is observed.
4. Control through Chaos
We have seen in the preceding sections that quantum chaos is tantamount to strong
coupling of the various degrees of freedom of a given quantum system, of the
destruction of good quantum numbers, and that all this usually leads to large fluctuations of various observables under slight changes of some control parameter,
or to decoherence-like reduced dynamics. Though, does quantum chaos provide
us with any means not only to describe, but also to control complex quantum
systems in a robust way?
Indeed, there is a positive response to this question, at least for periodically
driven quantum systems with a mixed regular-chaotic structure of the underlying
classical dynamics. The phase space of such systems decomposes into domains
of regular and of chaotic motion, see Fig. 18, which are associated with elliptic (i.e., stable) and hyperbolic (i.e., unstable) periodic orbits. Elliptic periodic
orbits are surrounded by elliptic islands in phase space, which define regions
of regular, i.e., integrable classical motion. A classical particle launched within
such an island cannot leave it (or, in higher dimensions, only on rather long time
scales [167,168]), and the only way for a quantum particle to leave the island is by
tunneling. It is rather obvious on semiclassical grounds [169], and has also been
realized by approximating the quantum dynamics in elliptic islands by a quantum
pendulum [170], that such regular regions in classical phase space lend support
for quantum eigenstates localized on top of them, provided the island’s volume
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F IG . 18. Example for the surface of section of the classically mixed regular-chaotic phase space of
a periodically driven system in a one-dimensional configuration space—here derived from the equations of motion of a one-dimensional hydrogen atom under periodic electromagnetic driving (in dipole
coupling) [31]. The phase space—spanned by the classical action I (measured in units of some reference action n0 ) and the conjugate angle θ —decomposes into essentially three main components:
a near-integrable (weakly perturbed) part (for actions below approx. 0.9), a prominent resonance island structure centered around (θ = π, I /n0 = 1.0), and a chaotic region—the complement of
near-integrable and island domain.
is large enough to accommodate the typical phase space volume hf (with f the
number of degrees of freedom) of a quantum state. Later on it was realized that, in
periodically driven systems, these quantum eigenstates faithfully follow the time
evolution of the elliptic trajectory they are anchored to [20,171–173], and that
their localization properties are preserved by the elliptic island—i.e., by the underlying nonlinearity of the classical dynamics—thus protecting them against the
usual dispersion of quantum wave packets in unharmonic systems. Hence, elliptic
islands in classical phase space give rise to the emergence of nondispersive wave
packets on the quantum level [31]. The only mechanism which limits their life
time (as long as incoherent processes can be excluded [31,174,175]) is tunneling
from the island into the surrounding chaotic sea, which, however, is strongly suppressed in the semiclassical limit of large classical actions as compared to h¯ [176].
Since elliptic structures in mixed regular chaotic classical dynamics are ubiquitous, so are nondispersive wave packets in the microscopic world. And the
classical nonlinear dynamics bears yet another blessing: The KAM theorem guarantees that elliptic islands in classical phase space are extremely robust against
perturbations—i.e., for sufficiently small perturbations, an elliptic island is possibly slightly distorted in phase space, though preserves its topology. While KAM
might appear of essentially mathematical interest on a first glance, this statement
has indeed very far-reaching consequences on the experimental level: Note that it
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QUANTUM CHAOS, TRANSPORT, AND CONTROL
61
is very hard to prevent conventional Rydberg wave packets, built, e.g., on a Stark
manifold (by exciting a coherent superposition of the Stark levels, with a laser
pulse, from the atomic ground state) from dispersion [71]—since any small (uncontrolled) perturbation shifts the Stark levels and thus induces an unharmonicity
in the spectrum, leading to dispersion of the wave packet. In contrast, a nondispersive wave packet anchored to an elliptic island in classical phase space is
essentially inert against any perturbation which is not strong enough to destroy
the island, as a consequence of KAM. In other words, the KAM theorem as one of
the fundamental theorems of classical nonlinear dynamics shields nondispersive
wave packets against technical noise (alike stray fields, etc.). It is this robustness
which allows the experimentalist to realize and manipulate nondispersive wave
packets in the laboratory [28], over time scales which exceed “traditional” wave
packet dynamics by orders of magnitude!
4.1. N ONDISPERSIVE WAVE PACKETS IN O NE PARTICLE DYNAMICS
The simplest realization of nondispersive wave packets is provided by an unharmonic, bounded, one-dimensional system under periodic driving, described by
the Hamiltonian
Hwp = H0 (z) + λV (z) cos(ωt).
(23)
Transformation to the action-angle variables (I, θ ) of H0 allows one to rewrite
this as
Hwp = H0 (I ) + λ
m=+∞
Vm (I ) cos(mθ − ωt),
(24)
m=−∞
where we assumed, for simplicity, that the Fourier amplitudes Vm (I ) are real [31].
Reminding ourselves of θ = Ωt, with Ω the classical roundtrip frequency along
the unperturbed trajectory with action I , we immediately realize that choices of
the driving frequency ω such that sθ − ωt 0, for some term m = s in the
above sum in (24), will lead to a separation of time scales in the time evolution
generated by Hwp . While all terms in (24) except the one with m = s will oscillate rapidly, a resonance will occur between the external drive at frequency ω
and the unperturbed motion along the trajectory with sΩ(I ) = ω. In other words,
proper choice of the driving frequency allows one to selectively address a specific trajectory of the unperturbed dynamics, via this resonance condition. For
s = 1, a suitable coordinate transformation, followed by a secular approximation (which averages over the rapidly oscillating terms in (24), at resonance), and
a final quadratic expansion around the action of the resonantly driven classical
orbit yields a pendulum Hamiltonian, which establishes the backbone of the typical phase space structure of an elliptic island at weak perturbation amplitudes,
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J. Madroñero et al.
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F IG . 19. Typical phase space structure in the vicinity of a resonantly driven trajectory of a
bounded, one-dimensional system, in action-angle coordinates I and θ . I is measured in units of
some reference action n0 . The external driving frequency is chosen such as to match the unperturbed
roundtrip frequency of the trajectory with action I /n0 = 1.0. The consequent separation of time
scales in (24) induces an onion-like, elliptic island structure centered around (θ = π, I /n0 = 1.0),
already at weak perturbation strengths λ. With increasing λ chaos invades phase space, at the expense
of the elliptic island and of near integrable regions at low actions. However, comparison with Fig. 18
also shows that the center of the elliptic island survives (actually to rather large values of λ [31,63]),
what is a consequence of the KAM theorem, and identifies elliptic islands as very robust topological
structures in classical phase space.
F IG . 20. Electronic density of a nondispersive electronic wave packet in a periodically driven,
one-dimensional Rydberg atom. The wave packet starts (at phase ωt = 0 of the driving field) at the
outer turning point of the classical eccentricity one orbit, is reflected from the Coulomb singularity
at ωt = π , and precisely refocuses at the outer turning point, without dispersion, after one complete
field cycle.
displayed in Fig. 19. The KAM theorem essentially guarantees that the core of
this structure survives even a considerable increase of λ, whilst all the remaining
phase space volume may undergo a dramatic metamorphosis, as evident from a
comparison of Figs. 18 and 19.
Figure 20 shows the configuration space representation of a nondispersive wave
packet launched along the Rydberg orbit with principal quantum number n0 = 60,
for the one-dimensional Coulomb problem [20]. This model describes the dynam-
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QUANTUM CHAOS, TRANSPORT, AND CONTROL
63
ics of quasi one-dimensional (i.e., extremal parabolic) Rydberg states of atomic
hydrogen in a near resonant field reasonably well [96,97]. Such a nondispersive
electronic wave packet propagating without dispersion along a highly excited Rydberg orbit has recently been excited and probed in laboratory experiments with
lithium atoms [28,29]. In particular, these experiments succeeded to demonstrate
the extremely long life time of these objects, by probing the electron’s position on
its Rydberg orbit after 15,000 cycles of the driving microwave field. This is equivalent to 15,000 Kepler orbits of the unperturbed Coulomb dynamics, and thus by
approximately three orders of magnitude longer than the life time of any Rydberg
wave packet so far generated in the laboratory. Furthermore, the experimentally
measured life time only gives a lower bound for the wave packet’s endurance,
since longer probing times were not possible due to the geometry of the experimental setup. Theory predicts life times of approx. 106 Kepler orbits, at these
excitations [31,176].
4.2. N ONDISPERSIVE WAVE PACKETS IN THE T HREE B ODY
C OULOMB P ROBLEM
The above scenario of nondispersive one particle wave packets can be generalized for the three body Coulomb problem, naturally realized in the helium atom.
A very nontrivial complication arises here from the fact that the electron–electron
interaction term in the helium Hamiltonian
HHe =
p1 2
2
1
p2 2
2
−
+
+
−
,
2
2
r1
r2
|r1 − r2 |
(25)
generates classically chaotic dynamics even in the absence of any external perturbation [39]. This is nowadays identified as the cause of the failure of the early
semiclassical quantum theory to come up with a quantitative description of the helium spectrum [41]. Furthermore, doubly excited states of helium have a finite autoionization probability, again due to the electron–electron interaction [177,178].
Hence, the helium atom itself has to be treated as an open system, and bears
some similarity with the crossed fields problem which we discussed in Section 3.3
above. Indeed, Ericson fluctuations are also expected in the photoabsorption cross
section of helium [179], for sufficiently high excitations, though the required energy range has not yet been reached in the lab [23].
Thus, since the classical phase space structure of the helium atom is globally
chaotic, our above motivation of the typical elliptic island structure on which
to build nondispersive wave packets is not straightforward, since there are no
global action-angle variables for irregular classical dynamics. However, we can
focus on specific regular domains in the classical phase space of the helium
64
J. Madroñero et al.
[4
F IG . 21. Characteristic frozen planet trajectory of the unperturbed three body Coulomb problem.
The inner electron precesses on highly eccentric ellipses, with a rapid Kepler oscillation between the
inner and the outer turning point. Upon average over the inner electron’s rapid motion, Coulomb attraction due to the screened Coulomb potential of the nucleus and electron–electron repulsion conspire
such as to create an adiabatic, shallow binding potential for the outer electron [181]. Consequently,
the outer electron is locked upon the precessing motion of the inner electron, leading to a strong
correlation of both electrons’ positions.
F IG . 22. Phase space structure for the outer electron of the (collinear) frozen planet configuration [182], in the absence (a) and in the presence (b) of an external, near resonant driving field. If the
external field frequency is chosen to match a resonance condition with the unperturbed outer electron’s
motion, secondary resonance islands emerge as in (b).
atom, which are elliptic islands themselves.2 These lend support for stable eigenstates of the unperturbed helium atom—the most prominent thereof being the
frozen planet configuration [36,180]. Figures 21 and 22 show a typical classical, highly correlated two-electron trajectory, and the phase space structure
of the frozen planet configuration, respectively. Given the regular phase space
structure with well-defined, stable periodic orbits as shown in Fig. 22, we are
back to our original setting: If we apply an external field with a frequency near
2 Indeed, by mapping an f degrees of freedom system on a periodically driven f − 1 degrees of
freedom system, where the periodic time dependence of the drive is provided by the periodic time
dependence of the remaining degree of freedom, these islands can be made formally equivalent to
those considered above [31,47].
4]
QUANTUM CHAOS, TRANSPORT, AND CONTROL
65
F IG . 23. Top: Husimi representation of a nondispersive two-electron wave packet propagating
along the collinear frozen planet orbit of the planar helium atom [184], in the phase space coordinates
of the outer electron along the quantization axis defined by the linear field polarization vector, for different phases ωt = 0 (left), π/2 (middle), and π (right). Very clearly, the electronic density faithfully
traces the resonantly driven frozen planet trajectory, as obvious from a comparison with the classical
phase space structure shown below (on identical scales).
resonant with one of the stable periodic orbits of the classical phase space of
the unperturbed system, we induce elliptic islands which propagate along the
unperturbed trajectory, phase-locked on the period of the drive. Consequently,
for sufficiently high excitations, we find nondispersive two-electron wave packets [182,183] propagating along the frozen planet trajectory, as illustrated in
Fig. 23 for an excitation to the fifth autoionization channel (in other words,
the inner electron is launched along an extremal parabolic orbit with principal
quantum number N = 6). Note that a quantum treatment of the planar three
body Coulomb problem (an accurate treatment of the fully three-dimensional
problem is hitherto out of reach, due to the size of Hilbert space when many
angular momenta are coupled by the driving field) predicts life times of approx. 1000 driving field periods (or, due to the resonance condition on drive
and unperturbed two-electron orbit, 1000 frozen planet periods) for these wave
packet eigenstates [184,185]. This prediction can be expected to be reliable, on
the basis of a comparison of typical He autoionization rates in 1D, 2D, and
3D configuration space [186]. The predicted two-electron wave packet’s life
times are considerably less than the life times predicted for the one electron
problem considered in the previous section, though still much longer than life
times of conventional Rydberg wave packets, and thus eligible for applications
in coherent control. Recently, the excitation of another type of nondispersive
two-electron wave packets has been suggested, with both electrons far from the
nucleus [187].
66
J. Madroñero et al.
[4
4.3. Q UANTUM R ESONANCES IN THE DYNAMICS OF K ICKED C OLD ATOMS
Nondispersive wave packets as those discussed above are ubiquitous, and can be
realized in any driven quantum system with an unharmonic spectrum (the unharmonicity guarantees the selectivity of the addressing of a specific classical
trajectory by the near resonant drive) and mixed regular-chaotic phase space [31].
Importantly though, their creation is not necessarily restricted to the realm of
semiclassical physics, where h¯ becomes small in comparison to the classical actions of the dynamics. This has been realized recently, in the treatment of quantum
resonances [188] and quantum accelerator modes [189] in the translational degree
of freedom of periodically kicked cold atoms loaded into one-dimensional optical
lattices which are flashed periodically. Such quantum resonances occur due to the
close similarity of the kicked atom Hamiltonian
HKA
+∞
p2
=
δ(t − mτ )
− K cos(kx)
2
m=−∞
(26)
with the kicked rotor, apart from the different boundary conditions (an infinite periodic lattice in the atomic problem, a circle in the case of the kicked rotor [188]).
They are excited by kicking periods τ = 2π, integer, since then the kicks are
synchronized with the exact revivals of the free evolution of the rotor dynamics
(we omit here the discussion of the specific value of the atomic quasimomentum,
which implies further restrictions, though is not indispensable for our present argument), leading to ballistic energy growth, for the appropriately prepared initial
quasimomentum state of the atoms [188].
If one considers the quantum dynamics close to the resonance condition, i.e., at
τ = 2π+, with a small detuning , it turns out [188,189] that the time evolution
generated by the Hamiltonian (26) can be obtained from the formal quantization
of some well-defined classical dynamics described by a map, with the detuning
taking the role of h¯ ≡ τ (which itself remains constant and can be arbitrarily
large!).
The quantum accelerator modes are created when an additional static potential
(such as provided by gravity) is added to the Hamiltonian of Eq. (26). For appropriate parameters, this Stark field allows the experimentalist to design classical
nonlinear-resonance islands (classical in the above sense of taking the role of h)
¯
embedded in a surrounding chaotic sea. These islands support ballistic transport,
which—in contrast to the ballistic motion at quantum resonance—is directed due
to the destruction of the translational invariance by the Stark field (see the accelerated tail of the atoms’ momentum distribution in Fig. 24).
In this generalized classical picture, both quantum resonances and quantum accelerator modes are nothing but quantum eigenstates anchored to elliptic islands
in the phase space of that classical map, i.e., a variant of our above nondispersive
5]
QUANTUM CHAOS, TRANSPORT, AND CONTROL
67
F IG . 24. (Courtesy of Gil Summy.) Time dependence (measured by the number of pulses or kicks)
of the atomic momentum distribution under periodic kicks along the gravitational field [15], in a
reference frame freely falling with the atoms. Besides the bulk of the atomic ensemble, which does
not acquire momentum, there is an atomic component which exhibits ballistic acceleration. This is the
experimental signature of a quantum accelerator mode.
wave packets. This mode-locking of the external drive to the intrinsic characteristic frequency of the system allows the experimentalist to efficiently transfer
large momenta to the atoms. Once again, these modes are robust against perturbations [190], are clearly identifiable in laboratory experiments [14–16], see Fig. 24,
and offer a variety of experimental applications, such as for high precision measurements of the gravitational constant [14].
5. Conclusion
As quantum optics addresses the dynamics of more and more complex quantum
systems, methods imported from quantum chaos provide useful tools for identifying statistically robust quantities for their description, and also to control their
time evolution. In this review, we have seen examples for characteristic universal
features of chaotic quantum systems on the spectral as well as on the dynamical
level, in such different settings like ultracold atoms in periodic optical potentials,
excitation and ionization processes of one and two-electron atoms subject to static or oscillating external fields, random laser theory, and cold atoms kicked by
standing light fields. The chosen examples are far from exploring all the diversity of current experimental and theoretical activities at the interface of quantum
68
J. Madroñero et al.
[6
optics and chaos—we did not discuss here the recently predicted and observed
universal ionization threshold of one electron Rydberg states under microwave
driving [84,117], weak and strong localization phenomena in the scattering of
photons off clouds of cold (or ultracold) trapped atoms (with close connections to
random lasing) [191], nor the complementary scenario of matter wave transport
in disordered optical or magnetic potentials [10,115], or the role of incoherent
processes which might compete with coherent quantum transport in complex dynamics [12,86,174,175]. Nonetheless, we hope that the examples treated already
give a flavor of the potential applications of quantum chaos, from the microscopic
modelling of an atomic current across a periodic potential, by using a chaotic
bosonic system as a bath which provides the necessary relaxation processes, to
nondispersive, one and two-electron wave packets which, due to their extraordinarily long life times and robustness against technical noise (inherited from the
KAM theorem), might find applications in robust quantum control schemes or as
quantum memory, in the context of quantum information processing. In particular, the analogies between quantum chaos and quantum transport in disordered
systems are currently coming into focus, and hold a panoply of intriguing challenging questions, to be tackled in the near future.
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ADVANCES IN ATOMIC, MOLECULAR AND OPTICAL PHYSICS, VOL. 53
MANIPULATING SINGLE ATOMS
DIETER MESCHEDE and ARNO RAUSCHENBEUTEL
Institut für Angewandte Physik, Universität Bonn, Wegelerstr. 8, D-53115 Bonn, Germany
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
2. Single Atoms in a MOT . . . . . . . . . . . . . . . . . . . .
2.1. Magneto-Optical Trap for Single Atoms . . . . . . . .
2.2. Dynamics of Single Atoms in a MOT . . . . . . . . . .
2.3. Beyond Poissonian Loading . . . . . . . . . . . . . . .
3. Preparing Single Atoms in a Dipole Trap . . . . . . . . . . .
4. Quantum State Preparation and Detection . . . . . . . . . .
5. Superposition States of Single Atoms . . . . . . . . . . . . .
6. Loading Multiple Atoms into the Dipole Trap . . . . . . . .
7. Realization of a Quantum Register . . . . . . . . . . . . . .
8. Controlling the Atoms’ Absolute and Relative Positions . . .
8.1. An Optical Conveyor Belt . . . . . . . . . . . . . . . .
8.2. Measuring and Controlling the Atoms’ Positions . . . .
8.3. Two-Dimensional Position Manipulation . . . . . . . .
9. Towards Entanglement of Neutral Atoms . . . . . . . . . . .
9.1. An Optical High-Finesse Resonator for Storing Photons
9.2. A Four-Photon Entanglement Scheme . . . . . . . . . .
9.3. Cold Collisions in Spin-Dependent Potentials . . . . .
10. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . .
11. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . .
12. References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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76
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Abstract
Neutral atoms are interesting candidates for experimentally investigating the transition from well-understood quantum objects to many particle and macroscopic
physics. Furthermore, the ability to control neutral atoms at the single atom level
opens new routes to applications such as quantum information processing and
metrology. We summarize experimental methods and findings in the preparation,
detection, and manipulation of trapped individual neutral atoms. The high efficiency
and the observed long coherence times of the presented methods are favorable for
future applications in quantum information processing.
75
© 2006 Elsevier Inc. All rights reserved
ISSN 1049-250X
DOI 10.1016/S1049-250X(06)53003-4
76
D. Meschede and A. Rauschenbeutel
[1
1. Introduction
Neutral atoms have played an outstanding role in our understanding of the microscopic world through quantum physics. Countless details of quantum mechanics
have been discovered and experimentally investigated with dilute gases of atoms.
With the advent of tunable, narrowband lasers around 1970, it became possible
to use laser light as an agent to control not only the internal quantum state of
atoms but also the motional degrees of freedom. The first observation of individual atomic particles was successful in 1978 by P. Toschek and collaborators [1].
The experimenters realized essential premises to observe individual Barium ions:
A strong electromagnetic radio frequency trap (Paul trap) to store ions in a small
volume and for extended periods of time, and an efficient optical detection by
resonance fluorescence from a narrowband tunable laser.
As a result of this breakthrough, trapped ions became prime objects for studying and illustrating light–matter interactions at the ultimate microscopic level, i.e.,
single particles interacting with well-controlled light fields. Interesting advances
in the 1980s include the observation of quantum jumps [2–4], anti-bunching in
resonance fluorescence [5], ion crystals [6,7], and more.
A similar degree of control was achieved for neutral atoms beginning in 1994
[8–10]. The origin for this delay with respect to ions is straightforwardly associated with the much weaker trapping forces available for a neutral atomic particle
in comparison with a charged particle. Neutral atoms can be localized in space by
exerting radiation pressure (magneto-optical trap, MOT), in the effective potential of an optical dipole trap (DT), or by magnetic traps (MT) if the atom carries
a permanent magnetic moment. A simple calculation shows that for typical laser
beam intensities trapping depths do not exceed 1 K for the MOT, 10 mK for DTs,
and 1 K for typical MT designs [11].
Experimental accomplishments in handling microscopic particles since 1980
have led to the demonstration of many quantum processes at an elementary level.
Perhaps even more importantly they have initiated new lines of research where
the control of atomic systems—and in particular atom–atom interactions—have
opened the route to study novel many particle systems. The celebrated realization
of Bose–Einstein condensation with neutral atoms in 1995 [12,13] has catapulted
experiments with neutral atoms into a central and unique role: they allow the
study of many particle systems with tailored interactions in a highly controlled
environment. It has already been shown with ultracold samples of atoms containing 10,000s of atoms, that novel quantum states, for instance, induced by quantum
phase transitions, can be realized and investigated [14]. A combination of these
methods with an experimental access to the atomic constituents at the single particle level promises deep insight into the physics of many particle systems and
their application, e.g., in quantum simulation and quantum information processing [15].
2]
MANIPULATING SINGLE ATOMS
77
It is the aim of this article to describe the state of art in the manipulation of
single neutral atoms. It is focused on well-known optical traps for neutral atoms,
usually employed for trapping much larger samples of atoms. In an alternative
approach, single neutral atoms can be prepared through the interaction with a
single mode of a low loss optical resonator which is of relevance for the field of
cavity-QED. For more information about this field we refer to [16].
2. Single Atoms in a MOT
2.1. M AGNETO -O PTICAL T RAP FOR S INGLE ATOMS
The magneto-optical trap, proposed by J. Dalibard and realized by D. Pritchard
and coworkers in 1987 [17], has revolutionized experimental work in atomic and
optical physics, because it allows to directly prepare and confine cold, i.e., low
velocity atoms from a background gas at room temperature. The MOT relies on
spatially modulated, velocity dependent radiation pressure forces exerted by red
detuned laser beams in combination with a magnetic quadrupole field. It remains
to this day the work horse of physics with cold atoms and serves in nearly all
experiments to initially prepare an ensemble of atoms at very low velocities.
The MOT capture rate is determined by the gradient of the magnetic quadrupole field, the diameter and the detuning of the trapping laser beams, as well as the
partial pressure of the atomic species to be trapped [18]. The loss rate, on the other
hand, is determined by collisions with the residual gas and exothermic intra-trap
collisions. In a conventional MOT with a quadrupole field gradient of 10 G/cm,
cm-wide beams, and a red detuning of the trapping laser beams of about −2γ ,
where γ is the natural linewidth of the atomic resonance line, typically 109 atoms
are captured with characteristic temperatures below 1/2 the Doppler temperature. For Caesium atoms, which are used in the experiments described here, the
Doppler temperature is TDopp = h¯ γ /2kB = 125 µK.
Single atom preparation and observation in a MOT is achieved by taking several
MOT parameters to the limits [8–10]: Since atom capture is mostly determined
by the time available for radiation pressure deceleration, the trapping rate is
dramatically reduced by small laser beam diameters (≈1 mm) and strong field
gradients (up to several 100 G/cm) [19], and of course, very low partial pressure (<10−14 mbar) of the trapped atomic species. Very low residual gas pressure
(≈10−11 mbar) also makes storage times of order 1 min and more possible. In our
experiment, the magnetic field gradient can be ramped up and down within typically 20–30 ms time scale which allows to actively control trap loading processes
(see Section 2.3).
Resonance fluorescence is collected from a 2.1% solid angle by a self-made
microscope objective with a diffraction limit below 2 µm [20], and recorded with
78
D. Meschede and A. Rauschenbeutel
[2
F IG . 1. Schematic of experimental setup of the magneto-optical trap. A diffraction limited microscope objective (working distance 36 mm, NA = 0.29) collects fluorescence from a 2.1% solid
angle and directs half of the signal towards an intensified CCD camera (ICCD, approx. 10% quantum
efficiency at 852 nm, one detected photon generates about 350 counts on the CCD chip). The other
half of the fluorescence signal is transmitted by the beamsplitter and focused onto an avalanche photodiode (APD, 50% quantum efficiency). Alternatively, the ICCD can be replaced by a second APD in
order to measure photon correlations (see below). The ICCD image shows the fluorescence of a single
Caesium atom trapped in the MOT. One pixel corresponds to approximately 1 µm, exposure time is
1 s. Interference and spatial filters (IF, SF) are used to suppress background.
either an intensified CCD camera or with avalanche photodiodes. Spectral as well
as spatial filtering helps to suppress stray light and reduces background to typically below 20,000 counts/s while the fluorescence of a single atom contributes
typically R = 60,000 counts/s to the fluorescence signal. The “portrait” of a single Caesium atom illuminated with trapping laser beams at the 852 nm D2 line is
shown in Fig. 1 for a 1 s exposure time.
The rate of photons recorded by the APDs reflects the time evolution of the
number of trapped atoms in Fig. 2: Prominent upward steps indicate loading,
downward steps disappearance of an individual atom from the trap. Neglecting background, the number of counts is proportional to the atom number N
through CT = N · f · T , where f is the fluorescence rate detected from individual atom and T is the integration time of the counter. The width CT of the
individual steps in Fig. 2 is dominated, to better than 99%, by shot noise, i.e.,
2]
MANIPULATING SINGLE ATOMS
79
F IG . 2. Left: Time chart clip of resonance fluorescence from neutral atoms trapped in a MOT.
Well-resolved equidistant fluorescence levels (step size f · T , see text) correspond to integer numbers
of atoms. Right: Distribution of count rates shows shot noise limited detection, here for an average of
about 2 atoms.
√
√
CT CT = Nf T . In order to distinguish N from N + 1 atoms with better
than 99% confidence, the step√size f T must be larger than the peak widths by
a factor of ≈5, i.e., f T /5 Nf T . Thus the minimal time to detect N atoms
with negligible background is T 25N/f , which for f = 6 · 104 results in
T N · 400 µs, many orders of magnitude shorter than the storage and hence the
processing time, see the next section.
For purely random loading and loss processes, the distribution of the occurrences for atom numbers N should exhibit a Poissonian distribution. In reality,
deviations are observed as a result of atom–atom interactions as discussed below
in more detail.
An interesting application of the single atom MOT has been developed by Z. Lu
and coworkers [21]: The ATTA method (Atom Trap Trace Analysis) makes use of
extreme selectivity of the magneto-optical trap with respect to atom species and
spatial detection. The sensitivity of the method for the detection of rare species is
essentially limited by the number of atoms that can be sent through the trapping
volume only.
2.2. DYNAMICS OF S INGLE ATOMS IN A MOT
In the MOT, trapped atoms continuously scatter near-resonant light. During these
excitation and de-excitation processes, the atoms are optically pumped from one
state to another in their multilevel structure. Furthermore, due to the random
transfer of momentum in each scattering event, they undergo diffusive motion
in the trap volume. Finally, the interaction between atoms in the presence of nearresonant light can induce inelastic collisions causing departure from the trap.
Substantial information about all relevant dynamical processes can be retrieved
from photon correlations in the resonance fluorescence which are imposed by the
80
D. Meschede and A. Rauschenbeutel
[2
atomic dynamics. We analyze photon correlations either by the classic configuration introduced by Hanbury Brown and Twiss [22], in order to overcome detector
dead times at the shortest nanosecond time scale, or by directly recording photon
arrival times with a computer and post-processing.
From this data, second order auto- or cross-correlation functions are derived.
In the photon language, g (2) (τ ) describes the conditional probability to observe a
second photon with a delay τ once a first photon was observed:
(2)
gAB (τ ) =
nA (t + τ )nB (t)
,
nA (t)nB (t)
where . . . denotes time averaging, and A and B symbolize the two quantities
correlated with each other.
The dynamics of a single (or a few) Caesium atoms trapped in the MOT can be
derived from these measurements at all relevant time scales [23]:
(a) Rabi-Oscillations. Excitation and de-excitation of electronic atomic transitions occurs at the nanosecond time scale. The corresponding measurement of the
auto-correlation function is shown in Fig. 3(a) and shows (after substraction of
the background) the famous phenomenon of anti-bunching, i.e., the second order
correlation function shows non-classical behavior at τ = 0, g (2) (0) = 0 [5,24].
Damping of the Rabi oscillations occurs at the 30 ns free space lifetime of the
excited Caesium 6P level. The data also show that with increasing number of
atoms the rate of stochastic coincidences rapidly increases: Anti-bunching can be
observed at the level of a single or very few atoms only.
(b) Optical Pumping. It is known that optical pumping of multi-level atoms
plays a central role for the realization of sub-Doppler temperatures in MOTs and
optical molasses [25,26]. The single atom MOT has allowed to directly observe
(2)
optical pumping by measuring, e.g., the cross-correlation glr (τ ) for left- and
right-hand circularly polarized fluorescent light, see Fig. 3(b): Observation of a
lefthanded photon projects the atom into a strongly oriented quantum state from
which the observation of right-handed photons is significantly reduced. Atomic
motion through the spatially varying polarization of the near-resonant trapping
light field induces optical pumping and causes this orientation to relax. From the
data one can estimate that it takes several microseconds for an atom to travel a
distance of λ/2, i.e., the length over which typical polarization variations occur.
(c) Diffusive dynamics. If one half of the image of the trapping volume is
blocked, the intensity measured at the detector indicates the presence of the atom
in the open or in the obstructed half of the trapping volume: If an atom is detected
in the visible part of the MOT, it will stay there and continue to radiate into the
detector until it vanishes into the oblique part by diffusion. Fig. 3(c) shows this
effect in the intensity autocorrelation measurement of a single atom moving about
in a MOT. A diffusion model agrees well with the observations, showing that the
so-called position relaxation time is of the order of 1 ms, as directly seen from the
2]
MANIPULATING SINGLE ATOMS
81
F IG . 3. Time domain measurements of atomic dynamics in a MOT by photon correlations (a)–(c)
and direct observation (d). See text for details.
experimental data. The average kinetic energy and hence the diffusion constant of
the atom is controlled by the detuning of the trapping laser beams.
(d) Cold collisions. The time chart of Fig. 3(d) shows the slow load and loss
dynamics at the seconds to minutes time scale similar to the one which has already been presented in Fig. 2. One of the most interesting properties is the
observation of two-atom losses (arrows), which occur much more frequently than
what can be expected if one assumes Poissonian-distributed, i.e., independent,
one-atom losses [27]. The analysis of the occurrence of such two-atom losses reveals that their rate is proportional to N (N − 1), where N is the total number of
atoms trapped in the MOT. Its origin thus clearly stems from a two-body process.
A detailed examination shows that inelastic collisions which are induced by the
trapping laser light, so-called radiative escape processes [28], are the dominant
mechanism for these two-atom losses. This experiment shows that atom–atom
interactions can be observed at the level of only two atoms.
2.3. B EYOND P OISSONIAN L OADING
Stochastic loading of the MOT is acceptable for applications with very small numbers of atoms. For instance, if MOT parameters are such that on average a single
82
D. Meschede and A. Rauschenbeutel
[3
atom populates the trap, Poissonian statistics predicts about 37% probability of
single atom events. For many experiments, implementation of control loops does
not offer a significant advantage in this case.
Some of the most interesting future routes of research with neutral atoms systems, however, will be directed towards small (“mesoscopic”) systems of neutral
atoms with controlled interactions. In experiments it will thus be essential to load
an exactly known number of, e.g., 5–20 atoms in a much shorter time than offered
by stochastic fluctuations of the atom number. In the MOT the random loading
process can be manipulated by controlling the magnetic field gradient, the trapping laser beam properties, or the flux of atoms entering the trap volume. Several
strategies for controlling the exact number of trapped atoms have already been
investigated or are currently studied:
In the experiment by Schlosser et al. [29] an optical trap providing very strong
confinement was superposed with the MOT (see also Section 3). Light assisted
atom–atom interaction prevents presence of more than one atom in the trap which
thus fluctuates between 0 and 1 atom occupation numbers only. Suppression of
two-atom occupation of a purely magnetic trap was also observed by Willems et
al. [30].
An active feedback scheme for a single Cr atom MOT has been introduced
by McClelland and coworkers [31]: If the trap is empty, rapid loading (≈5 ms)
is achieved by directing the flux from a source of Cr atoms through light forces
into the MOT volume. Using the MOT fluorescence as the indicator loading is
terminated when a single atom is detected in the trap, and it is dumped if the
trap contains more than one atom. An average single atom occupation probability
exceeding 98% has been demonstrated in this experiment. The authors estimate
that such a device may deliver individual atoms up to a rate of about 10 kHz.
In our laboratory, we have begun to explore a loading scheme, where we rapidly
load a preset mean number of atoms into our MOT by temporarily lowering its
magnetic field gradient. After this forced loading, the magnetic field gradient is
ramped up again and the actual number of trapped atoms is determined by analyzing the level of fluorescence with a software discriminator [32]. As a result
of this analysis, the atoms are either loaded into an optical dipole trap for further
experiments, see Section 6, or, in case the MOT does not store the desired atom
number, the atoms are discarded and the forced loading of the MOT is repeated.
3. Preparing Single Atoms in a Dipole Trap
While the MOT is an excellent device for the preparation of an exactly known
number of neutral atoms, it relies on spontaneous scattering of near-resonant laser
light which is highly dissipative and makes precise quantum state control of the
trapped atoms impossible. We have found in our experiments that preparation of
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MANIPULATING SINGLE ATOMS
83
F IG . 4. Scheme of the experimental set-up. See text for details.
a sample of an exactly known number (1–30) of atoms in a MOT and subsequent
transfer to an optical dipole trap (DT) makes a very efficient instrument for experiments investigating quantum control of small ensembles of neutral atoms. A very
tightly confining dipole trap for similar objectives was demonstrated by Schlosser
et al. [29].
In our experiment (Fig. 4), the DT is generated by a focused and far off resonant
Nd:YAG or Yb:YAG laser beam at λ = 1.06 µm and 1.03 µm, respectively. The
laser beam is split into two arms and can be used in a single beam configuration
(traveling wave), or in a configuration of two counterpropagating beams (standing wave). We routinely reach transfer efficiencies from the MOT into the DT and
vice versa in excess of 99% [33]. The dipole trap provides an approximately conservative, harmonic potential with bound oscillator quantum states for the neutral
atoms. Focusing of the trapping laser beam power of several Watts to a 10–30 µm
waist provides strong confinement of the atom in the transverse direction, and application of a standing wave with 0.5 µm modulation period exerts even stronger
forces in the longitudinal direction. The dipole trap provides a typical potential
depth of order UTrap /k ≈ 1 mK. After transfer from the MOT, we measure temperatures of 50–70 µK, significantly below the 125 µK Doppler temperature for
Caesium atoms [34]. Sub-Doppler cooling is enhanced during transfer from the
MOT into the dipole trap since the atomic transition frequencies are light shifted
towards higher frequencies and hence the cooling lasers are effectively further red
detuned.
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F IG . 5. Left: ICCD-image of atomic fluorescence in the optical dipole trap under continuous illumination with molasses beams, exposure time 0.5 s. In the horizontal direction, the width of the
fluorescent spot is determined by the resolution of our imaging system. In the vertical direction the
spot shows the extension of atomic trajectories corresponding to a temperature of about 50–70 µK in
the trap of depth 1 mK. Right: Characteristic parameters of the dipole trap. Shaded areas schematically
indicate MOT and molasses laser beams.
We have also realized a method to continuously illuminate an atom in the dipole
trap with an optical molasses and to observe its presence through fluorescence
detection. The laser cooling provided by the molasses in this case balances the
heating forces. In Fig. 5 we show an ICCD image of a trapped atom as well as
characteristic parameters of the dipole trap.
4. Quantum State Preparation and Detection
Neutral atoms are considered to be one of several interesting routes towards the
implementation of quantum information processing. Fundamental information
processing operations such as the famous quantum CNOT gate must be realized through physical interaction of the qubits [35]. For neutral atoms, several
concepts, including photon exchange mediated by cavity-QED [36–38], or cold
collisions [39,40] have been proposed. Each of these concepts relies on tight control of the quantum evolution of atomic qubits which already poses important
experimental challenges.
In our experiments, hyperfine ground states of the Caesium atom are employed
as qubits, the elementary units of quantum information storage. It is well known
from the Caesium atomic clock that the microwave transition operated at νhfs =
9.2 GHz between the long lived |F = 4 and |F = 3 hyperfine states provides
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MANIPULATING SINGLE ATOMS
85
efficient means of internal quantum state manipulation. It is thus expected that
specific hyperfine states of the Caesium atom are excellent candidates to serve as
qubit states with, e.g., |0 = |F = 4 and |1 = |F = 3. The first step in these
applications is to prepare and detect (“write” and “read”) arbitrary quantum state
into Caesium prepared in the DT.
During the transfer from the MOT into the dipole trap, an atom is normally
prepared in the |F = 4 state. This is achieved by switching off the MOT cooling
laser, near resonant with the |F = 4 → |F = 5 transition, a few milliseconds
before switching off the MOT repumping laser, resonant with the |F = 3 →
|F = 4 transition. After this transfer, we can populate the |F = 4, mF = 0
magnetic substate using resonant optical pumping on the |F = 4 → |F = 4
and |F = 3 → |F = 4 transition of the λ = 852 nm D2-line multiplet for about
5 ms with linear π-polarized light. In the mF = 0 states, the influence of ambient
magnetic field fluctuations is strongly suppressed, a favorable condition for the
observation of long dephasing times described in Section 5. On the other hand,
using circular σ − -polarized light, atoms can be pumped to the |F = 4, mF = −4
state. This state allows fine tuning of its energy level by external magnetic fields
which is essential for position selective addressing and the implementation of a
neutral atom quantum register (see Section 7). Finally, an initial pure |F = 3
quantum state can be prepared by switching off the MOT repumping laser about
10 ms before switching off the MOT cooling laser. In this way, the |F = 4 state
is depleted while transferring the atom from the MOT into the DT. In our trap,
residual light scattering of the DT lasers causes relaxation of the hyperfine state
populations of the |F = 3 and |F = 4 Caesium ground states at a time scale of
several seconds or more, depending on the trapping laser intensity.
For unambiguous detection of the hyperfine state of the trapped atoms, we currently use a destructive “push-out” method [41], which discriminates the F = 3
and F = 4 levels with excellent contrast of better than 1:200 (Fig. 6). Discrimination is realized by ejecting atoms from the trap if and only if they are in the F = 4
state and by monitoring the presence or absence of the atom after this procedure.
For this purpose, a saturating laser beam resonant with the F = 4 → F = 5
cycling transition is applied transversely to the dipole trap axis. When the trap
depth is lowered to approximately 0.12 mK, atoms in F = 4 are pushed out in
less than 1 ms by scattering on average 35 photons. Atoms in the |F = 3 state
are not affected by the push-out laser. In the last step, the remaining atoms are
either detected at a given dipole trap site by imaging with the ICCD camera, or
by observing their fluorescence after recapture in the MOT. A fluorescing site indicates projection to the F = 3 quantum state, an empty site that was occupied
before is equivalent to projection to the F = 4 quantum state.
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F IG . 6. Detecting the quantum state of a single neutral atom. Upper trace: An atom is prepared in
the MOT and transferred to the dipole trap in state |F = 4. A resonant push-out laser removes the
atom from the trap. When the MOT lasers are switched on again, stray light is observed only. Lower
trace: In the dipole trap, the atom is transferred to the dipole trap in state |F = 3. The push-out laser
is invisible for an atom in |F = 3. After switching on the MOT lasers the 1 atom fluorescence level
is recovered. See text for details on atom state preparation.
5. Superposition States of Single Atoms
The two hyperfine states form a pseudo spin-1/2 system, which can be manipulated by spin rotations, induced by shining in microwave radiation resonant
with the atomic clock transition. For instance, spin-flips are caused by so-called
π
π
π-pulses (|0 → |1, |1 → −|0), where for a given magnetic field amplitude B⊥ and transition moment μ the microwave pulse duration τ is defined by
Ωτ = (μB⊥ /h¯ )τ = π. We have found that in our geometrically complex apparatus, the power of our 33 dBm microwave source is most efficiently directed
at the experimental region with a simple open ended waveguide. We find a minimal pulse length of 16 µs for a π-pulse. Arbitrary quantum state superpositions
cos(Ωτ/2)|0 + eiφ sin(Ωτ/2)|1 can be generated by varying the pulse area Ωτ
and phase φ, and a π/2-pulse generates superpositions with even contributions of
the two quantum eigenstates.
Future applications of the trapped atom quantum states as qubits depend crucially on the question whether coupling to the environment (“decoherence”) or
to technical imperfections and noise (“dephasing”) can be suppressed to such a
degree that coherent quantum evolution is preserved at all relevant time scales.
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87
Promisingly long coherence time in dipole traps have been first observed by
Davidson et al. [42].
In the Bloch vector model, the longitudinal and transversal relaxation time constants T1 and T2 , are introduced phenomenologically. T1 describes the relaxation
of the population difference of the two quantum states to their thermal equilibrium, T2 the relaxation of the phase coherence between the two spin states. While
spontaneous decay is completely negligible, the hyperfine state of the Caesium
atom can be changed by spontaneous Raman scattering. In our current setup, we
measure typically T1 3 s [33]. With the exception of the trap life time of order
1 min this time is longer than all other relaxation times. It can be further increased
by reducing the trapping laser power.
Several mechanisms contribute to transversal relaxation described by the time
constant T2 . Here, we distinguish reversible contributions with time constant T2∗
arising from inhomogeneities of the measured ensemble, and irreversible contributions (T2 ), which affect the ensemble homogeneously. The total transversal
relaxation time constant is thus composed of two different time constants with
T2−1 = T2∗ −1 + T2 −1 . Using Ramsey’s method of separated oscillatory fields [43]
we have experimentally determined the atomic coherence properties with regard
to dephasing in the dipole trap [44]. A detailed analysis can be found in [41].
Figure 7 shows an example of Ramsey spectroscopy, i.e., the evolution of the
mF = 0 hyperfine state under the action of two π/2 microwave pulses as a function of the delay time between the pulses. If the microwave is resonant with the
hyperfine transition, one expects perfect transfer from one to the other hyperfine
state. The “Ramsey-fringes” observed here result from a small, intentional detun-
F IG . 7. Population oscillation showing hyperfine coherences of optically trapped Caesium atoms:
Dephasing Ramsey fringes and spin echo signal. The |F = 3, mF = 0 state is coupled to the
|F = 4, mF = 0 state by 9.2 GHz microwaves. The solid line corresponds to a theoretical prediction
based on the thermal energy distribution of the atoms in the dipole trap only. For details see [41,44].
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ing from perfect resonance. The initially observed coherent oscillation collapses
after a dephasing time T2∗ ≈ T2 , where longer dephasing times are observed for
more shallow dipole potentials. This dephasing is caused by the thermal distribution of atomic motional states in the dipole trap which causes an inhomogeneous
distribution of light shifts: “Cold” atoms with low kinetic energy near the potential minimum, or intensity maximum of the dipole trap experience on average
stronger light shifts than “hot” atoms with larger kinetic energy.
The phase evolution of the internal atomic quantum state depends on the external, motional degrees of freedom since binding forces are caused by the light shift
of the internal energy levels. Since the two hyperfine states F = 3 and F = 4 experience a small but significant relative light shift of order νhfs /νD2 = η 10−4 ,
the phase evolution of any superposition state is affected by this difference and
causes dephasing depending on the trajectory of the atom in the trap. In a semiclassical model, we have assumed that the free precession phase accumulated by
an atomic superposition state between the two π/2-pulses depends on the average
differential light shift only and calculated the thermal ensemble average yielding
the solid line in Fig. 7. A quantum mechanical density matrix calculation of the
same observable reproduces this result within a few percent. The deviation can
be attributed to the occurrence of small oscillator quantum numbers nosc 7 in
the stiff direction of the trap. We find that the envelope of the collapse of the initial oscillation corresponds to the Fourier transform of the thermal oscillator state
distribution [41].
It is known that a “spin-echo” can be induced by application of a rephasing
pulse [45]. Application of a π-pulse at time Tπ induces an echo of the Ramsey
signal with a maximum amplitude at time 2Tπ . The revival of the oscillation is
also shown in Fig. 7. We have measured a 1/e decay time T2 0.15 s for the
revival amplitude. We have experimentally analyzed in detail the origin of this
irreversible decay. We have found that currently the dominating sources of decoherence are the lack of beam pointing stability as well as intensity fluctuations of
the trapping laser beams, while other effects such as magnetic field fluctuations
and heating are negligible [41]. All relevant relaxation and dephasing times are
recapitulated in Table I. Since no fundamental source of decoherence has been
found which could not be reduced by technical measures, it should be possible
to further increase the time span of coherent quantum evolution of the trapped
atoms.
Alternatively, we have also employed resonant two-photon Raman transitions
in order to introduce pseudo-spin rotations. In Fig. 8 we show a measurement
of population oscillations (Rabi oscillations) between the F = 4 and F = 3
Caesium hyperfine ground states [46]. Efficient two-photon Rabi rotations are already achieved with relatively low power levels below 1 mW in each laser beam,
e.g., in Fig. 8 the two-photon Rabi frequency exceeds 10 kHz. It is routine today
to use focused Raman laser beams in order to address an individual particle out
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Table I
Measured hyperfine relaxation times of atoms in our dipole trap
Trelax
Umax /k
(mK)
mF
T1
1
0, −4
8.6 s
T2∗
0.1
0.04
0.1
0
0
−4
3 ms
19 ms
270 µs
thermal motion, scalar light shift
thermal motion, scalar light shift
thermal motion, vector light shift
T2
0.1
0.04
0.1
0
0
−4
34 ms
150 ms
2 ms
0.1
−4
600 µs
beam pointing instability
beam pointing instability
without gradient: thermal motion,
vector light shift
with gradient: thermal motion,
inhomogeneous magnetic field
Value
Limiting mechanism
spontaneous Raman scattering
F IG . 8. Population (Rabi) oscillation showing hyperfine coherences of optically trapped Caesium
atoms induced by resonant two-photon Raman transitions [46]. On the left side, details of the Caesium
quantum states involved and the power levels of the Raman laser beams are given.
of a string of trapped ions [47] and to induce quantum coherences. This method,
which has significantly contributed to the first successful operations of fundamental quantum gates with in these systems [48,49], is straightforwardly transferred
to systems of neutral atoms. However, in Section 7 we will show that, with neutral
atoms, a gradient method providing spatial resolution via spectral resolution can
be applied which eliminates the need for focused laser beams.
6. Loading Multiple Atoms into the Dipole Trap
When atoms are transferred from the MOT into the dipole trap, they are distributed randomly across a 10 µm stretch of the standing wave, corresponding to
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F IG . 9. (a) After the transfer from the MOT, the atoms are trapped in the potential wells of the
standing wave dipole trap at random positions. The spatial period of the schematic potential wells
is stretched for illustration purposes. (b) Fluorescence image of five optically resolved atoms in the
standing wave dipole trap (trap axis is horizontal) after the 1D expansion detailed in the text. Integration time is 0.5 s.
about 20 antinodes or potential wells. With 5 atoms, the average separation is
only 2 µm, too small to be optically resolved by our imaging system.
In order to improve the addressability, we have adopted a modified transfer
procedure: After the transfer from the MOT into the standing wave dipole trap,
formed by the two counterpropagating laser beams, we switch off one of the two
beams within 1 ms. The potential of the resulting running wave dipole trap, created by one focussed laser beam, has Lorentzian shape with a FWHM of about
1 mm in the longitudinal direction. We let the atoms expand longitudinally for
1 ms such that they occupy a length of ≈100 µm. Then, we switch the second
trapping laser beam on again within 1 ms, so that the atoms are “arrested” by the
standing wave micropotentials at the position they have reached during the expansion. Exposure to the optical molasses warrants low temperatures of the trapped
atoms. The 5 fluorescent spots in Fig. 9 correspond to a single atom each, spread
out across 50 µm in this case with easily resolvable spatial separations.
As has been pointed out in Section 2.3, we have recently started to operate
a feedback scheme for loading a preset number of atoms into our DT. For this,
the MOT is rapidly loaded with a selectable mean number of atoms, which are
only transferred into the DT if the desired number of atoms is detected in the
MOT. This is particularly useful if one seeks to carry out experiments with a
larger number (>3) of atoms. In this case, loading the DT with a Poissonian distributed number of atoms and postselection of the events with the desired atom
number dramatically increases data acquisition time. First results obtained with
this scheme are presented in Fig. 10: Part (a) shows the accumulated unconditional MOT fluorescence histogram for a large number of MOT loading cycles
with a mean atom number of about 3. Part (b), on the other hand, corresponds to
those events, where three atoms have been detected in the MOT, loaded into the
DT, and retransferred into the MOT. The resulting conditional histogram clearly
shows that we manage to controllably load three atoms into the DT with a good
efficiency. In the course of these experiments, we have also found that single atom
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91
F IG . 10. Selectively loading 3 atoms. (a) Binned fluorescence signal detected by the APD after a
large number of MOT loading processes. Part (b) contains all events, where three atoms were detected
by the feedback loop. These atoms were then transferred into the DT and back into the MOT, see text
for details.
occupation of the 1D lattice sites is generally preferred over multiple occupation
favoring a regular, non-Poissonian distribution of the atoms. Details will be published in [32].
7. Realization of a Quantum Register
A quantum register consists of a well-known number of qubits that can be individually addressed and coherently manipulated. Our quantum register is composed
of a string of neutral atoms, provided by the procedures described in the previous
sections, which can be selectively prepared in arbitrary quantum states.
In ion traps selective addressing is achieved by means of focused Raman laser
beams [47]. As discussed in Section 5, we have shown that Raman pulses can
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F IG . 11. Sequence of operations to generate and detect a |01010 quantum register state in a string
of five atoms. The whole sequence lasts 1.5 s.
be used to create coherent superpositions of hyperfine states of the atoms trapped
in our experiment [46]. However, in the experiments presented here, we use an
alternative technique where we apply microwave radiation which is made resonant
with an atom at a selected site only by means of magnetic field gradients. In
this method, spatial selectivity is indeed realized in the same way as in magnetic
resonance imaging (MRI) [50].
We can currently operate our register in the following way [51], see Fig. 11:
We load between 2 and 10 atoms into our dipole trap. We then take a camera
picture and determine the positions of all atoms with sub-micrometer precision.
In the next step all atoms are optically pumped into the same |F = 4, mF = −4
quantum state as described in Section 4 to initialize the register.
Individual addressing is now realized by tuning the microwave frequency to
the exact transition frequency corresponding to the known individual atomic
sites where the relationship is controlled by an external B-field gradient of
B 0.15 µT/µm along the DT axis. The atomic resonance frequency is shifted
by the linear Zeeman effect according to ν = νhfs + 24.5 kHz/µT, and we find
a spatial frequency shift of dν/dz = 3.7 kHz/µm. We also apply a homogeneous magnetic field of about 0.4 mT in order to provide guiding for the angular
momenta and to reduce the influence of transversal magnetic field gradients. In
Fig. 11 we show the result of two selective inversion operations (π-pulses) carried
out with a string of five atoms stored in our dipole trap array.
We have furthermore measured the resolution of the magnetic field gradient
method. Figure 12 shows the result for the longest pulses applied (83 µs FWHM).
The solid line is obtained from a numerical solution of the Bloch equations and
reproduces the measurement very well. The spatial resolution is limited by the
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MANIPULATING SINGLE ATOMS
93
F IG . 12. Measured spatial resolution of the addressing scheme. The data were obtained by deliberately addressing positions offset from the actual atom site. For each point approximately 40 single
atom events were analyzed. The Gaussian microwave π -pulse has a FWHM length of 83 µs.
Fourier width of the microwave pulse. Our method clearly demonstrates that we
can address atoms for separations exceeding 2.5 µm (i.e., atoms are separated by
about 5 empty sites). The resolution of the magnetic method in our current set-up
is thus comparable to addressing by optical focusing. Neighboring atoms experience of course a phase shift due to non-resonant interaction with the microwave
radiation. However, this phase shift is known and can be taken into account in
further operations.
We have furthermore explored the coherence properties of atoms, now in the
magnetically most sensitive mF = −4 states instead of the mF = 0 states. The
results are displayed in Table I of Section 5. It is not surprising that dephasing
times are much shorter in this case and are indeed dominated by fluctuations and
inhomogeneities of the magnetic field. However, they are already now much larger
than simple operation times for, e.g., π-pulses and technical improvements will
further enhance the time available for coherent evolution.
The method described requires very precise timing of the microwave pulses in
order to guarantee a precise control of the evolution from one quantum state to another. As an alternative, we have also applied quantum state control by means of
rapid adiabatic passage [52]: In this case, the frequency of an intense microwave
pulse is swept through resonance thereby transforming an initial into a final eigenstate of the system, in our case realized for the |F = 4 and |F = 3 hyperfine
ground states. In a gradient magnetic field we have analyzed the transfer probability as a function of the resonance position of the sweep center frequency with
respect to the trapped atom for a fixed sweep width. The result in Fig. 13 shows
the expected flat top profile indicating the reduced sensitivity to the precise setting
of the center frequency and the sweep width [53]. The width of the edges which
drop to zero within 3 µm is a measure of the spatial resolution of this method and
comparable to the resonant addressing scheme described above.
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F IG . 13. Position-dependent adiabatic population transfer of individual atoms in an inhomogeneous magnetic field. The graph shows the population transfer as a function of the position offset x
along the trap axis. Each data point is obtained from about 40 single atom measurements. The solid
line is a theoretical fit [53].
Summarizing, in this section we have demonstrated procedures to experimentally realize both write and read operations at the level of a single neutral atom.
We have demonstrated individual addressing of the atoms within a string of stored
atoms with excellent resolution, and we are able to prepare arbitrary quantum superpositions on an individual atomic, or qubit site. In conclusion we have demonstrated the operation of a neutral atom quantum register, including the application
of spin rotations, i.e., Hadamard gates in the language of quantum information
processing.
8. Controlling the Atoms’ Absolute and Relative Positions
Considering the ratio between the experimentally measured 2.5 µm addressing
resolution presented above and the 1 mm Rayleigh zone of our standing wave
DT, our neutral atom quantum register could in principle operate on more than
100 individually addressable qubits. Methods for the regularization of the distribution of atoms by controlling their absolute positions in the trap must be realized,
however, in order to manage larger quantum registers. Tight position control is
furthermore essential to realize the necessary controlled atom–atom interaction.
In optical cavity QED, for example, this interaction is mediated by the field of
an ultrahigh finesse Fabry–Perot resonator [36,54]. The field mode sustained by
such a resonator has a typical transverse dimension of 10 µm so that the atom pair
will have to be placed into this mode with a submicrometer precision while the
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MANIPULATING SINGLE ATOMS
95
distance between the atoms has to be controlled at the same level. We have demonstrated such a submicrometer position control for individual neutral atoms [55].
8.1. A N O PTICAL C ONVEYOR B ELT
The position of the trapped atoms along the DT axis can be conveniently manipulated by introducing a relative detuning between the two counter-propagating
dipole trap laser beams. A detuning by ν causes the standing wave pattern to
move in the laboratory frame with a speed νλDT /2, where λDT is the wavelength of the DT laser. As a result, the trapping potential moves along the DT
axis and thereby transports the atoms [56–58]. In the experiment, the relative detuning between the DT beams can be easily set with radiofrequency precision by
acousto-optic modulators (AOMs, Fig. 4). They are placed in each beam and are
driven by a phase-synchronous digital dual-frequency synthesizer. A phase slip of
one cycle between the two trapping laser beams corresponds to a transportation
distance of λDT /2.
We can realize typical accelerations of a = 10,000 m/s2 and hence accelerate
the atoms to velocities of up to 5 m/s (limited by the 10 MHz bandwidth of the
AOMs) in half a millisecond. Thus, for typical parameters, a 1 mm transport takes
about 1 ms. At the same time, the displacement of the atoms is controlled with
a precision better than the dipole trap laser wavelength since this scheme allows
us to control the relative phase of the two trapping laser beams to a fraction of
a radian.
Using continuous illumination, we have imaged the controlled motion of one
and the same or several atoms (Fig. 14) transported by the conveyor belt [58]
with observation times exceeding one minute. Recently, it was shown that optical dipole traps similarly to our arrangement can be used to transport neutral
atoms into high finesse resonators for cavity-QED experiments with very good
precision [59,60].
8.2. M EASURING AND C ONTROLLING THE ATOMS ’ P OSITIONS
If one wants to take ultimate advantage of the optical conveyor belt transport
above in order to place atoms at a predetermined position, the atoms’ initial position along the dipole trap axis has to be known with the highest possible precision,
ideally better than the distance between two adjacent potential wells. This can be
achieved by recording and analyzing an ICCD fluorescence image of the trapped
atoms. We have shown that by fitting the corresponding fluorescence peaks with a
Gaussian, the atoms’ position can be determined with a ±150 nm precision from
an ICCD image with 1 s exposure time [55].
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F IG . 14. Transport of 3 atoms by an optical conveyor belt: Snapshots of the movie published
in [58]. In the first image, 3 atoms are stored in the MOT from where they are loaded into the conveyor
belt formed by two counterpropagating laser beams. The frequency difference of the laser beams is
controlled with two AOMs driven by a phase-coherent RF-source. At 40 s and 65 s the direction of
transport is reversed. The atoms are lost from the conveyor belt by random collisions with thermal
residual gas.
Furthermore, we have demonstrated that by means of our optical conveyor belt
technique, we can place an atom at a predetermined position along the dipole trap
axis with a ±300 nm accuracy. Such a position control sequence is exemplified in
8]
MANIPULATING SINGLE ATOMS
97
F IG . 15. Active position control. (a) After transferring a single atom from the MOT into the dipole
trap its initial position is determined from an ICCD image and its distance with respect to the target
position is calculated. (b) The atom is then transported to the target position and its final position is
again measured from an ICCD image.
Fig. 15. After loading one atom from the MOT into the dipole trap, its position has
a ±5 µm uncertainty, corresponding to the diameter of the MOT. We determine
the atom’s initial position from a first ICCD fluorescence image and calculate its
distance L from the desired target position. The atom is then transported to this
target position and the success of the operation is verified by means of a second
ICCD image.
In order to measure the distance between two simultaneously trapped atoms,
we determine their individual positions as above. From one such measurement
with
√ 1 s integration time, their distance can thus be inferred with a precision of
2 × 150 nm. This precision can even be further increased by taking more than
one image of the atom pair and by averaging over the measurements obtained
from these images. Now, since the atoms are trapped inside a periodic potential,
their distance d should be an integer multiple of the standing wave period: d =
nλDT /2; see Fig. 9(a). This periodicity is clearly visible in Fig. 16, where the
cumulative distribution of atomic separations is given when averaging over more
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D. Meschede and A. Rauschenbeutel
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F IG . 16. Cumulative distribution of separations between simultaneously trapped atoms inside the
standing wave potential. The discreteness of the atomic separations due to the standing wave potential
is clearly visible.
than 10 distance measurements for each atom pair. The resolution of this distance
measurement scheme is ±36 nm, much smaller than the standing wave period.
We directly infer this value from the width of the vertical steps in Fig. 16. This
result shows that we can determine the exact number of potential wells separating
the simultaneously trapped atoms [55].
8.3. T WO -D IMENSIONAL P OSITION M ANIPULATION
A single standing wave optical dipole trap allows to shift the position of a string
of trapped atoms as a whole in one dimension along the dipole trap axis using the
optical conveyor belt technique presented above. If one seeks to prepare strings
with a well-defined spacing or to rearrange the order of a string of trapped atoms,
however, a two-dimensional manipulation of the atomic positions is required. For
this reason, we have set up a second standing wave dipole trap, perpendicular to
the first one, which acts as optical tweezers and which allows us to extract atoms
out of a string and to reinsert them at another predefined position.
Figure 17 shows a first preliminary result towards this atom sorting and distance
control scheme [61]. We start with a string of three randomly spaced atoms which
has been loaded from the MOT into the horizontal (conveyor belt) dipole trap.
In Fig. 17(a), the string has already been shifted such that the rightmost atom is
placed at the position of the vertical (optical tweezers) dipole trap. This atom is
then extracted with the vertical dipole trap and, after shifting the remaining two
atoms along the horizontal dipole trap, we place it 15 µm to the left of the initially
leftmost atom of the string; see Figs. 17(b)–(d). Repeating this procedure a second
time, we prepare a string of three equidistantly spaced atoms, where the order of
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MANIPULATING SINGLE ATOMS
99
F IG . 17. Rearranging a string of three atoms using two perpendicular standing wave dipole traps.
See text for details.
the string has been modified according to (1, 2, 3) → (3, 1, 2) → (2, 3, 1); see
Figs. 17(e)–(h).
9. Towards Entanglement of Neutral Atoms
There is a plentitude of proposals of how to implement a two-qubit quantum
gate with neutral atoms which suggest the coherent photon exchange of two
atoms inside a high-finesse optical resonator [36,54,59,62]. The experimental
challenges for their realization are quite demanding. Although there has been a
number of successes in optical cavity-QED research recently, including the transport of atoms into a cavity [59,60], trapping of single atoms inside a cavity [63],
single photon generation [64,65], feedback control of the atomic motion in a cavity [66,67], and cooling of atoms inside a cavity [68–70], the realization of a
two-qubit quantum gate with ground state atoms remains to be shown.
9.1. A N O PTICAL H IGH -F INESSE R ESONATOR FOR S TORING P HOTONS
Our goal is the deterministic placement of two atoms inside an optical high-finesse
resonator. For this purpose, we have already set up and stabilized a suitable res-
100
D. Meschede and A. Rauschenbeutel
[9
onator [71]. We plan to transport atoms from the MOT, which is a few millimeters
away from the cavity, into the cavity mode using our optical conveyor belt. Employing the imaging techniques and the image analysis presented above, we were
recently able to control the position of the trapped atoms along the trap axis with
a precision of ±300 nm [55]. This should allow us to reliably place the atoms
into the center of the cavity mode, which has a diameter of 10 µm. Since the
microwave-induced one-qubit operations on the quantum register demonstrated
in Section 7 do not require optical access to the trapped atoms, they can even take
place inside the cavity.
9.2. A F OUR -P HOTON E NTANGLEMENT S CHEME
One of the most promising schemes to create entanglement between two atoms in
optical cavity QED was proposed by L. You et al. [54] and is the basis for the realization of a quantum phase gate [72]. It relies on the coherent energy exchange
between two atoms stimulated by a four-photon Raman process involving the cavity mode and an auxiliary laser field. We have determined optimized theoretical
parameters and calculated the expected fidelity according to this proposal for our
particular experimental conditions. With a maximum fidelity of F = 85%, which
can be expected from this calculation. The demonstration of entanglement and
the implementation of a quantum gate thus seems feasible with our experimental
apparatus.
9.3. C OLD C OLLISIONS IN S PIN -D EPENDENT P OTENTIALS
We plan to investigate small strings of collisionally interacting neutral atoms
for applications in quantum information processing. The atoms are stored, one
by one, in a standing wave dipole trap and the interaction between the atoms,
necessary for the implementation of quantum gates, will be realized through controlled cold collisions [39,40] which have been demonstrated with large sample
of ultracold atoms already but without addressability of the individual atomic
qubit [15]. For this purpose, we will employ the technique of spin dependent
transport [39,40] at the level of individual atoms. This technique will allows us to
“manually” split the wave functions of the trapped atoms in a deterministic and
fully controlled single atom Stern–Gerlach experiment, where the dipole trap provides the effective magnetic field. By recombining the atomic wave function, we
will then realize a single atom interferometer and directly measure the coherence
properties of the splitting process. A sequence of splitting operations, carried out
on a single atom, will result in a quantum analogue of the Galton board, where the
atom carries out a quantum walk. Such quantum walks have recently been proposed as an alternative approach to quantum computing [73]. Our ultimate goal
10]
MANIPULATING SINGLE ATOMS
101
is the implementation of fundamental quantum gates using controlled cold collisions within a register of 2–10 trapped neutral atoms. A parallel application of
such quantum gates should then open the route towards the preparation of small
cluster states [74] consisting of up to 10 individually addressable qubits.
10. Conclusions
In this overview, we have presented experimental techniques and results concerning the preparation and manipulation of single or a few optically trapped neutral
Caesium atoms. We have shown that a specially designed magneto-optical trap
(MOT) can store a countable number of atoms. Information about the dynamics
of these atoms inside the MOT can be gained at all relevant timescales by analyzing photon correlation in their resonance fluorescence. Furthermore, using active
feedback schemes, the Poissonian fluctuations of the number of atoms in the MOT
can be overcome, making such a MOT a highly deterministic source of an exactly
known number of cold atoms.
For coherent manipulation, we transfer the atoms with a high efficiency from
the dissipative MOT into the conservative potential of a standing wave dipole
trap (DT). The quantum state of atoms stored in this DT can be reliably prepared
and detected at the level of single atoms. We have examined the coherence properties of the atoms in the DT and identified the dephasing mechanisms in this
system. The experimentally measured long coherence times show that the atomic
hyperfine ground states are well suited for encoding and processing coherent information.
A string of such trapped Caesium atoms has thus been used to realize a quantum register, where individual atoms were addressed with microwave pulses in
combination with a magnetic field gradient. Using this method, we have demonstrated all basic register operations: initialization, selective addressing, coherent
manipulation, and state-selective detection of the individual atomic states.
We have furthermore demonstrated a high level of control of the atoms’ external degrees of freedom. Our DT can be operated as an “optical conveyor belt”
that allows to move the atoms with submicrometer precision along the DT. In addition, we have measured the absolute and relative positions of the atoms along
the dipole trap with a submicrometer accuracy. This high resolution allows us to
measure the exact number of potential wells separating simultaneously trapped
atoms in our 532 nm-period standing wave potential and to transport an atom to a
predetermined position with a suboptical wavelength precision.
Finally, using a second dipole trap operated as optical tweezers, we have obtained first results towards an active control of the atoms’ relative positions within
the string. This will allow us to prepare strings with a preset interatomic spacing
and to rearrange the order of atoms within the string at will.
102
D. Meschede and A. Rauschenbeutel
[12
The presented techniques are compatible with the requirements of cavity QED
and controlled cold collision experiments. In our laboratory, we now actively work
towards the implementation of such experiments in order to realize quantum logic
operations with neutral ground state atoms.
11. Acknowledgements
We wish to thank the Deutsche Forschungsgemeinschaft, the Studienstiftung des
Deutschen Volkes, the Deutsche Telekom Stiftung, INTAS, and the European
Commission for continued support. Furthermore, we are indebted to numerous
enthusiastic coworkers and students at the Diplom- and doctoral level who have
participated in this research: W. Alt, K. Dästner, I. Dotsenko, L. Förster, D. Frese,
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ADVANCES IN ATOMIC, MOLECULAR AND OPTICAL PHYSICS, VOL. 53
SPATIAL IMAGING WITH WAVEFRONT
CODING AND OPTICAL COHERENCE
TOMOGRAPHY∗
THOMAS HELLMUTH
Department of Optoelectronics, Aalen University of Applied Sciences, Germany
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2. Enhanced Depth of Focus with Wavefront Coding . . . . . . . . .
2.1. Abbe’s Theory of the Microscope and Wavefront Coding . .
2.2. Partial Coherent Illumination and Wavefront Coding . . . .
2.3. Wavefront Coding with Variable Phase Plates . . . . . . . .
3. Spatial Imaging with Optical Coherence Tomography . . . . . .
3.1. Time Domain Optical Coherence Tomography (TDOCT) . .
3.2. Linear Optical Coherence Tomography (LOCT) . . . . . . .
3.3. Spectral Domain Optical Coherence Tomography (SDOCT)
4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . .
6. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Abstract
With wavefront coding the wavefront in the pupil plane of an optical imaging system
is modified by introducing a phasemask. The resulting image intensity distribution
is processed with an inverse digital filter providing an image of the object with enhanced depth of focus. Optical coherence tomography provides three-dimensional
information about the object. The depth resolution is only determined by the coherence length of the light source. New applications and methods based on these
techniques are presented.
∗ I am very pleased to dedicate this review article to Prof. Herbert Walther. I have learnt from him
how to get an understanding of complicated phenomena both in quantum and in classical optics in
terms of simple pictures and key experiments. Happy birthday.
105
© 2006 Elsevier Inc. All rights reserved
ISSN 1049-250X
DOI 10.1016/S1049-250X(06)53004-6
106
T. Hellmuth
[1
1. Introduction
It is easy to measure the height of an object with a microscope by focusing first
onto the top plane and then onto the bottom plane of the object and measuring the
displacement of the probe stage. However, it is more difficult to find the position
of best focus the smaller the numerical aperture of the objective is because of
its large depth of focus. Furthermore, due to the low numerical aperture lateral
resolution is also low. Thus, high depth of focus seems to be associated with low
lateral resolution and low depth resolution.
In fact, according to linear optical system theory first established by Ernst Abbe
in 1881 with his theory of the microscope depth resolution is directly related to
lateral resolution of the optical system [4]. For a diffraction limited optical system
both depth of focus DOF = λ/NA2 and lateral resolution limit dmin = λ/2NA are
determined by the numerical aperture NA of the objective and the wavelength λ.
Many optical systems are intrinsically characterized by a large depth of focus.
For example, in ophthalmoscopy the optically usable pupil diameter of the eye is
only two millimeters. With dilation of the pupil higher pupil diameters are possible. But due to the optical aberrations of the eye the dilated pupil diameter cannot
be fully utilized for imaging the fundus. With the normal eye length of 24 millimeters and the mean refractive index of the aqueous and vitreous humor of n = 1.34
the effective numerical aperture of fundus imaging systems is practically limited
to NA = 0.06 with λ = 0.55 µm giving a depth of focus of 0.15 mm and a lateral
resolution limit of 5 µm. Although lateral resolution fulfills most diagnostic needs
depth of focus of 0.15 mm forbids resolution of the microscopically thin layer
structures of the retina with classical optical sectioning techniques like confocal
imaging.
Similar restrictions of the numerical aperture of imaging systems limit also
depth resolution in industrial metrology. The working distance has to be large to
avoid collisions and the aperture angle of the optical system has to be small to fit
into narrow apertures. On the other hand surface structures of workpieces have
to be measured within the manufacturing process with resolutions in the order of
microns.
In other applications like image processing large numerical apertures are
needed to get enough light to the CCD-target. The prize to pay is small depth
of focus. As a consequence the position of the object has to be controlled by
complex autofocus systems.
These practical examples show that the complementarity of depth of focus versus lateral resolution on one side and the complementarity of depth of focus versus
depth resolution on the other severely limit the performance of optical systems in
many situations.
2]
SPATIAL IMAGING
107
Within the last 15 years two new approaches have been brought up which have
in common to provide high depth of focus without restricting either lateral resolution or depth resolution.
An optical imaging technique with both enhanced depth of focus and high
lateral resolution was first invented by Dowski et al. in 1995 using wavefront coding [1–3]. With this approach the optical transfer function of an imaging system
is modified at the pupil plane. An inverse digital filter is applied to the image to
restore an image with high depth of focus and almost no loss of lateral resolution.
Optical coherence tomography invented by Fujimoto et al. in 1991 is an imaging technique providing both high depth of focus and high depth resolution [8].
This method makes use of short coherence interferometry where depth resolution
is no longer limited by the numerical aperture of the optical system but by the
coherence length of the light source.
Both wavefront coding and optical coherence tomography essentially depend
on the coherence properties of the illumination. Wavefront coding only works
with illumination characterized by a low degree of spatial coherence whereas optical coherence tomography is based on the low temporal coherence of the light
source.
In the following sections various new applications and extensions of the two
methods are discussed which have been developed and investigated at Aalen University of Applied Sciences.
2. Enhanced Depth of Focus with Wavefront Coding
Wavefront coding is a technique which provides high depth of focus without loss
of lateral resolution. A cubic phaseplate (Fig. 1) is located at the exit pupil of the
optical system.
The phaseplate is a transparent plate with one flat surface on one side and a
cubic surface on the other. The surface sag of the cubic surface can be described by
the sag function h(x, y) = α(x 3 +y 3 ). The parameter α determines the “strength”
of the phaseplate. Because the phaseplate modifies the wavefront Φ(x, y) in the
exit pupil the optical transfer function OTF of the optical system is modified. The
OTF is the autocorrelation function of the pupil function p(x, y) [6]:
p(x, y) = t (x, y)e−iΦ(x,y) ,
(1)
OTF = p(x, y) ⊗ p(x, y)
(2)
with “⊗” symbolizing the correlation operation. It is t (x, y) the stop function
which is 1 within the stop aperture and 0 outside. The modulation transfer function
MTF = |OTF| describes the image contrast as a function of the spatial frequency
of a periodic object.
108
T. Hellmuth
[2
F IG . 1. Measured surface of cubic phaseplate.
The cubic phaseplate reduces the image contrast but does not reduce the bandwidth of the optical system. In addition, with the cubic phaseplate in place the
OTF and the MTF do not change significantly when the object is defocused over
many depths of focus of the optical system. Because the OTF is invariant over a
large depth of field it is possible to apply an inverse filter OTF−1 to the Fourier
transform of the acquired image with the phaseplate in place to get an unblurred
image with a large depth of focus.
Figure 2 shows the MTF of a focused diffraction limited optical system with
and without phaseplate. Below the respective MTFs are shown for the defocused
object. Whereas the bandwidth of the MTF dramatically shrinks when the optical
system without phaseplate is defocused both the shape and the bandwidth of the
MTF of the optical system with cubic phaseplate remain constant. In addition the
MTF of the defocused optical system without phaseplate is zero for certain spatial
frequencies, in other words, these frequencies are not transmitted to the image at
all.
Figure 3 shows the point spread function (PSF) of the optical system with the
cubic phaseplate in place. The PSF is related to the OTF by the Fourier transform
OTF = FT{PSF}. The PSF can be measured by using a transilluminated pinhole
as an object and registering the image with a CCD camera.
Figure 4 (left) shows the blurred image of a defocused barcode pattern. The cubic phaseplate is not inserted. Figure 4 (right) shows the inversely filtered image
when the cubic phaseplate is inserted. This image is Fourier transformed to get
the spatial frequency spectrum. The complex spectrum function is inversely filtered by the inverse optical transfer function OTF−1 = 1/FT{PSF} of the optical
system with the cubic phaseplate in place. Finally the inversely filtered spectrum
is multiplied by the focused optical transfer function of the optical system with-
2]
SPATIAL IMAGING
109
F IG . 2. Ideal MTF of a focused diffraction limited system (top left). MTF of the same optical
system with phaseplate in place (top right). MTF of defocused optical system without phaseplate
(bottom left). MTF of defocused optical system with phaseplate (bottom right).
out phaseplate OTFideal . The restored image is received by the inverse Fourier
transform
imagerestored = FT−1 FT{image} · OTF−1 · OTFideal .
(3)
Because the OTF of the optical system with cubic phaseplate does not significantly change over a large depth of field the inverse filter is able to restore the
unblurred image even if the object is defocused.
A typical application shows Fig. 5. A barcode reading system needs a high
aperture to receive enough light from the object. Barcode readers are mainly used
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F IG . 3. Point spread function of the optical system with cubic phaseplate. It is registered with a
CCD target. The object is a transilluminated pinhole with a diameter considerably smaller than the
resolution of the objective.
F IG . 4. Image of defocused barcode. Objective without cubic phaseplate (left). Inversely filtered
image of defocused barcode. Objective with cubic phaseplate (right).
in logistic applications, for example, to identify pieces of luggage on conveyor
belts in airports. Due to the high aperture which is necessary to accept enough
light within a short image acquisition time the depth of focus of an ordinary camera becomes too small compared to the variable object distances one has to cope
with. Fast autofocus systems have to track the barcode. An alternative is depth of
focus enhancement with cameras equipped with cubic phaseplates.
The theory of wavefront coding can be explained in terms of ambiguity functions [2]. Although this approach is very useful for the optimization and the design
of cubic phaseplates it is of a more formal nature. Therefore, in the following section the theory shall be explained in terms of Abbe’s theory of the microscope.
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F IG . 5. Barcode reading system for luggage identification.
2.1. A BBE ’ S T HEORY OF THE M ICROSCOPE AND WAVEFRONT C ODING
Abbe’s theory of the microscope is originally based on coherent illumination
(Fig. 6) [11]. The light from a point light source is collimated by the condenser
lens illuminating a diffraction grating. The diffraction orders are plane waves
propagating into different directions which are focused into the back focal plane
of the microscope objective where the aperture stop of the objective is located.
The tube lens collimates again the various diffraction orders to bring them to interference at the image plane where a CCD may be located.
The cubic phaseplate is inserted at the exit pupil of the objective which is in this
case the aperture stop plane. However, the cubic phaseplate would not have much
influence on the imaging properties of the system in this coherent illumination
case. It would only cause a shift of the interference pattern in the image plane
because the phaseplate thickness is different at the locations where the diffraction
orders are focused.
In order to understand the behavior of an optical system with a cubic phaseplate
it is necessary to take the partial coherence properties of the illumination into
account.
Instead of a point light source the light source is now extended. The various
diffraction orders of the grating object generate images of the light source in the
back focal plane of the objective (Fig. 7). The light field illuminating the object is
no longer a coherent wave but has to be described by a spatial coherence function
G(x, y) in the object plane [36]. According to the van Zittert–Zernike theorem
this coherence function can be calculated as the Fourier transform of the intensity
distribution of the light source located in the front focal plane of the condenser
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F IG . 6. Microscope setup with coherent illumination. The object is a diffraction grating located
in the front focal plane of the objective. The diffraction orders are focused into the back focal plane
(aperture plane) of the objective. The tube lens collimates the waves emanating from the foci. The
plane waves interfere at the image plane generating the image of the grating object.
F IG . 7. Partial coherent illumination. The extended light source is imaged into the back focal plane
of the objective. Each diffraction order generates an image of the light source in the objective aperture
plane.
lens. The coherence function G(x, y) is modulated by the amplitude object transmission function Fobj (x, y). The objective lens generates the Fourier transform of
the coherence function multiplied by the transmission function of the objective
in the back focal plane of the objective lens. The final image is generated in the
image plane with the tubelens by a further Fourier transform.
If the shape of the incoherent light source can be described by f (r) = 1 within
a circle of radius 1 and f (r) = 0 outside of the circle the coherence function
of the illuminating light field in the object plane can be described by the shift
invariant function G(x, y) = J1 (r)/r (J1 (r) is the respective Bessel function of
first order and r is the radial variable). This is the same function which describes
the amplitude of a focused laser beam with a diameter equal to the source diameter
of the incoherent light source (Fig. 8). The diffraction grating splits the focused
laser beam into various diffraction orders which are collimated by the objective
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F IG . 8. Laserscanning microscope. The object is shifted laterally (object scan). The diffraction
orders interfere within the aperture. A detector (not shown) integrates the light within the aperture.
The detector signal is recorded as a function of the scan position.
lens into parallel light bundles laterally shifted relative to each other proportional
to the spatial frequency of the grating object.
The first orders interfere with the zeroth order in the overlapping areas indicated
as interference domains in Fig. 8. When the object is laterally shifted (object
scan) the phase of the first diffraction orders are shifted relative to the phase of
the zeroth order according to the shift theorem of Fourier transform theory. The
detector (not shown in Fig. 8) integrates the energy across the whole aperture.
The detector signal is recorded as a function of the scan position. The modulation
contrast of the detector signal decreases with increasing grating constant because
the area of the interference zone decreases. The MTF describes the contrast as a
function of the spatial frequency ν. The MTF is proportional to the area of the
interference zone. It is given by the formula [6]
ν
ν
2
2
1 − (ν/νmax ) ,
arccos
−
MTF(ν) =
π
νmax
νmax
where νmax = 2NA/λ is the resolution limit which is reached when the zeroth and
first order do not overlap any more, in other words, the OTF is the autocorrelation
function of the pupil function (see Eq. (2)).
If the object is defocused the first diffraction orders constitute plane waves
which are tilted relative to the plane wave of the zeroth order. Within the interference zones interference fringes appear. Their fringe density increases with
increasing defocus. When the object is shifted in the lateral direction (scan) the
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F IG . 9. Wavefront W (x) across the pupil with defocus without phasemask (top, left), focused with
phasemask (center, left) and with defocus and phasemask (bottom, left). Modulation frequency f (x)
of the interference pattern within the interference zone corresponding to the diagrams on the left. It
is f (x) the second derivative of W (x). On the right: Corresponding interference patterns within the
interference zone.
phase of the interference pattern changes but not the fringe density. The modulation contrast of the resulting signal decreases with defocus because the detector
integrates across the aperture. Figure 9 (first row, second diagram) shows the
frequency of the interference fringes across the interference zone. The spatial
modulation frequency of the interferogram f (x) is constant but increases with
increasing defocus.
Instead of the plane waves of the diffraction orders of a grating object the image
generation process can be also discussed in terms of the spherical wave emanating
from a point object. In the focused case the spherical wave is transformed into a
plane wave by the objective lens and becomes a spherical wave in the defocused
case. In the approximation of paraxial optics the spherical wave can be approximated by a paraboloid which is a parabola W (x) ∼ x 2 in one dimension (Fig. 9,
first row, left diagram). The spatial modulation frequency of the interferogram
f (x) is related to the wavefront W (x) by
f (x) ∼ d 2 W (x)/dx 2 .
If a cubic phaseplate is located at the aperture plane the wavefront W (x) is a
cubic function (Fig. 9, second row, left diagram). The second derivative is a linear
function (Fig. 9, second row, central diagram). That means that the modulation
frequency of the interference fringes within the interference zone between the ze-
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roth and first order is described by a linear chirp function. The zero crossing of
f (x) in Fig. 9 indicates the area where the interference pattern is not modulated.
The integrating detector averages the intensity across the interference zone. However, the averaged energy across the aperture is stronger modulated compared to
the case without cubic phaseplate because of the broad zone where the fringe density is low. The broad dark zone in Fig. 9 (second row, right diagram) contributes
much to the modulation contrast. It oscillates between bright and dark when the
object is scanned. The dense interference fringes do not contribute to image contrast because they are averaged out. The dark zone is laterally shifted when the
object is defocused but does not change in size. Thus, the MTF is insensitive to
defocus. Of course, the contrast of the detector signal modulation finally vanishes
when the defocus is so high that the zero crossing leaves the interference zone.
2.2. PARTIAL C OHERENT I LLUMINATION AND WAVEFRONT C ODING
The qualitative description of wavefront coding in terms of Abbe’s theory as
shown in the last section can be described quantitatively with Hopkins’ theory
of partial coherent imaging [5].
The Fourier transform (object spectrum) of the object transmission function
Fobj (x, y) is
F˜obj (f, g) =
+∞
Fobj (x, y)ei2π(f x+gy) dx dy
(4)
−∞
with f and g as the spatial object frequency in the x- and y-direction, respectively.
The image intensity distribution spectrum with partial coherent illumination is
J˜image (f, g) =
+∞
T˜ f + f, g + g, f , g −∞
∗ × F˜obj f + f, g + g F˜obj
f , g df dg
(5)
with the bilinear transfer function
T˜
f0 , g0 ; f0 , g0
+∞
=
J˜cond f¯, g¯ K˜ f¯ + f0 , g¯ + g0
−∞
× K˜ ∗ f¯ + f0 , g¯ + g0 d f¯ d g.
¯
(6)
˜
g) the
It is J˜cond (f, g) the circular pupil function of the condenser. It is K(f,
complex pupil function of the objective. It is
˜
K(f,
g) = t (f, g) exp iΦ(f, g)
(7)
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with the wavefront function Φ and the transmission function t of the objective
aperture which is zero outside the objective aperture and 1 inside. For onedimensional objects (e.g. edge) we get g = g = 0. Thus, the bilinear transfer
function in Eq. (6) becomes
T˜ f + f, 0, f , 0 = T˜ f, f +∞
=
J˜cond f¯, g¯ K˜ f¯ + f + f, g¯ K˜ ∗ f¯ + f , g¯ d f¯ d g.
¯
(8)
−∞
∗ (f , 0) = F˜ ∗ (f ) we finally get
Setting F˜obj (f + f, 0) = F˜obj (f + f ) and F˜obj
obj
for the image spectrum in Eq. (5),
+∞
∗ T˜ f, f F˜obj f + f F˜obj
f df .
J˜image (f ) =
(9)
−∞
The intensity distribution of the image is then
+∞
J x =
J˜image (f )e−i2πf x df.
(10)
−∞
Figure 10 shows the inversely filtered intensity distribution of the image of a
bar pattern illuminated with partial coherent light in comparison with the simulation [16]. The inverse filter function is derived from the pointspread function of a
pinhole object. The degree of coherence of the illumination does not influence the
PSF. However, the imaging of the bar pattern is significantly influenced. Therefore, the inverse filter based on the PSF cannot compensate the image artefacts
F IG . 10. The circular condenser aperture is smaller than the quadratic objective aperture (partial
coherent illumination). The inversely filtered image (with phaseplate) shows typical fringes both in
experiment and simulation which are due to the high degree of spatial coherence of illumination.
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introduced by the coherence properties of the illumination. As a consequence the
inversely filtered image exhibits artefacts both in the experiment and the simulation (edge fringes) as shown in Fig. 10.
2.3. WAVEFRONT C ODING WITH VARIABLE P HASE P LATES
So far the effect of cubic phasemasks with fixed strength parameter α has been
discussed. A large value of the parameter α is associated with a large depth of
focus but also with several drawbacks.
With a strong phaseplate the OTF decreases. Thus the inverse filter becomes
strong at higher frequencies where the signal is noisy. As a consequence the inverse filtered image shows a high noise level. Therefore, the strength of the cubic
phaseplate should be chosen only as high as it is necessary. A phaseplate with
variable parameter α which can be adapted to the required depth of focus is an
alternative [15]. Figure 11 shows the setup of a variable phaseplate system. It consists of two phaseplates. Phaseplate 1 has a convex surface which can be described
by the surface function
f (x, y) = κ x 4 + y 4
(11)
(the bottom side is flat). Phaseplate 2 has a concave surface which can be described by the surface function
g(x, y) = −κ x 4 + y 4 .
(12)
Both phaseplates can be shifted relative to each other by the displacement parameter in the diagonal direction (45◦ to the x-axis). As a result one obtains an
F IG . 11. Variable phaseplate system with surface functions f (x, y) = ±κ(x 4 + y 4 ). The displacement generates an effective optical performance of the system equivalent to the performance
of a cubic phasemask.
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effective wavefront Φ(x, y),
Φ(x, y) ∼ f (x − , y − ) + g(x + , y + )
= 8κ x 3 + y 3 + 8κ3 (x + y)
(13)
corresponding to the wavefront produced by a phaseplate with an effective cubic
parameter α = 8κ which can be adjusted by appropriately setting the displacement parameter . The “linear” term in Eq. (13) (x + y) leads only to a
displacement of the image which can be easily compensated by digital processing.
An alternative setup of a variable phaseplate system shows Fig. 12. It consists
of 4 phaseplates with “cylinder-like” surfaces.
Phaseplate 1 and 2 can be shifted in the x-direction (perpendicular to the optical
axis) in opposite directions. Thus, phaseplate 1 is shifted by − and phaseplate 2
by +. Phaseplate 1 has a convex surface on one side and a flat surface on the
other. The convex surface can be described by
f1 (x, y) = κ · x 4 .
(14)
Phaseplate 2 has a concave surface which can be described by
f2 (x, y) = −κ · x 4 .
(15)
Phaseplate 3 and 4 can be shifted along the y-axis. Phaseplate 3 has a convex
surface on one side and a flat surface on the other. The convex surface can be
described by
f3 (x, y) = κ · y 4 .
(16)
Phaseplate 4 has a concave surface which can be described by
f4 (x, y) = −κ · y 4 .
(17)
F IG . 12. Variable phaseplate system with cylinder like surface function f (x, y) = ±κx 4 and
±κy 4 , respectively.
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When the phaseplates are shifted by one obtains an effective wavefront Φ(x, y),
Φ(x, y) ∼ f1 (x − , y) + f2 (x + , y) + f3 (x, y − ) + f4 (x, y + )
∼ 8κ x 3 + y 3 + 83 κ(x + y).
(18)
With α = 8κ the strength of the cubic phaseplate can be adjusted via the shift
parameter . Again, the linear term x + y only causes a slight image shift.
This system consisting of 4 elements is more complicated than the two part solution described above. However, it is easier to manufacture cylinder like surfaces
than free-form surfaces required in the two part solution.
The phase plates discussed above are not rotational symmetric. Classical grinding and polishing processes which are used for spherical glass surfaces cannot be
used. At the Center of Optical Technology at Aalen University of Applied Science optical surfaces of arbitrary shape can be polished with a polishing robot
(Fig. 13) [17]. Figure 14 shows the interferometer measurements of the convex
F IG . 13. Polishing robot.
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F IG . 14. Interferometer measurement of the concave surface κ(x 4 + y 4 ) and its convex counterpart.
and concave surface of the (x 4 + y 4 )-phaseplate in glass. Similar results are obtained by making phaseplates in PMMA with a diamond turning machine.
3. Spatial Imaging with Optical Coherence Tomography
Optical coherence tomography (OCT) is a noninvasive imaging technique providing subsurface imaging of biological tissue with micrometer-scale resolution.
OCT was first used for imaging of the retina [8,9,21] and is now applied to a
variety of medical fields to gain morphological [23,24,26,38,41] and functional
data [27]. All OCT sensors either work in the time or Fourier domain. In the time
domain the depth gating of the sample is achieved by using a low coherence light
source, a Michelson interferometer setup and a reference optical delay line. An
OCT image (B-scan) is built up of several scans of the optical delay line in the reference arm (A-scans) [32]. In the Fourier domain depth information of the sample
is obtained by investigating the spectrum of the interferometer output [33] or by
using a tunable laser and a single photodiode as sensor [34,35].
We have investigated a third approach [10,12,13]. It is an OCT-sensor without
using a variable reference optical delay line, a spectrometer or a tunable laser. The
main item of the interferometer is a two-pinhole device built of two monomode
fibers aligned in parallel. Light emerging from these two fibers interferes on a
linear CCD-array similar to Young’s two-pinhole experiment. For this reason the
setup is called linear OCT sensor (LOCT). Depth gating is achieved by detecting
the interference signal on the CCD-array. Different positions of the interference
signal on the CCD-array correspond to different depths inside the sample. Therefore a complete A-scan can be derived from a single readout of the CCD-array.
3.1. T IME D OMAIN O PTICAL C OHERENCE T OMOGRAPHY (TDOCT)
Figure 15 shows the classical setup of a time domain optical coherence tomograph. The light source is a superluminescent diode with a short coherence length
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F IG . 15. Time domain OCT.
F IG . 16. Linear OCT interference signal.
in the order of 10 µm. Because the light from a SLD is diffraction limited like
laser light it can be focused into a monomode fiberoptic interferometer with high
efficiency. The interferometer consists of a 3 dB coupler splitting the incoming
light from the SLD into a reference path and a probe path. The light of the reference arm is reflected back by a retroreflecting prism into the fiber. The prism is
mounted on an electromechanical scanner (galvoscanner) moving the prism back
and forth. The light of the probe arm hits the sample which may be a multilayer
structure as, for example, the different tissue layers of the retina of a patient’s
eye. The light reflected from an individual layer interferes with the light from
the reference arm only if the arm length of the reference arm corresponds to the
distance between the probe arm fiber exit and the respective layer of the sample.
Because of the moving retroreflecting prism the interference signal occurs as a
burst (Fig. 16). The signal can be also recorded with a demodulating logarithmic
amplifier providing the envelope function of the interference signal. Another option is to calculate the envelope function of the interferogram with the Hilbert
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transform [25]. The analog OCT-signal in Fig. 16 can be regarded as the real part
of an analytical function h+ (t) = hRe (t) + i · hIm (t). The imaginary part hIm (t)
is related to the real part by the Hilbert transform
1
hIm (t) =
π
+∞
−∞
hRe (t ) dt .
t − t
(19)
The envelope function of the interference signal is
f (t) = h2Re + h2Im
as shown in Fig. 17. The OCT signal can be interpreted as the optical echo from
the object. It is analogous to the A-scan signal in ultrasound imaging. However,
the resolution of OCT is better than ultrasound by at least a factor 10. If the light
beam is focused onto the object and scanned laterally the subsequent A-scans can
be arranged to a B-scan map which displays a tomographic view of the object.
Figure 18 shows a cross-sectional view of the fundus of an eye [31]. The signal
intensity is shown in false color contrast.
F IG . 17. Envelope of OCT signal calculated from signal shown in Fig. 16 with Hilbert transform.
F IG . 18. OCT tomogram from retina.
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We have studied various applications and new methods based on OCT at Aalen
University of Applied Sciences which shall be discussed in the following sections.
3.1.1. Determination of Blood Oxygenization with OCT
The main application of OCT is the tomographic imaging of the retinal layers of
the human eye for diagnostic purposes. But this provides only morphologic information. Many diseases of the retina occur before any morphologic changes are
observable. As an example glaucoma is usually detected by perimetry where the
visual function of the retina is registered. Another age related disease of the macular region of the fundus is macular degeneration. Treatment with lasers is difficult
and can only retard the progression of the disease. Both glaucoma and macular
degeneration are supposed to be related to metabolic and blood supply disorders
of the fundus which can only be understood if functional imaging techniques are
available which can identify changes of concentration of metabolic substances
like oxygen, glucose or cholesterol concentrations. Whereas it is difficult to identify glucose and cholesterol spectra without fluorescent markers oxygenization
of the red blood cells can be detected by measuring the blood spectrum. Figure 19 shows schematically the absorption coefficients of oxygenized hemoglobin
(HbO2 ) and deoxygenized hemoglobin (Hb) (see also [18]). The crossover of the
two spectra is the isobestic point. Its wavelength is around 800 nm.
Figure 20 shows the OCT signal for two wavelengths (680 nm and 815 nm)
from oxygenized and deoxygenized blood pumped through a transparent tube [40].
The light from two OCT interferometers equipped with a 680 nm SLD and a
815 nm SLD, respectively, is combined with a dichroic mirror. The two wavelength beams are collinear to collect the OCT signal simultaneously from the
F IG . 19. Absorption coefficient of oxygenized (HbO2 ) and deoxygenized (Hb) blood.
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F IG . 20. In vitro measurement of OCT signal with 680 nm and 815 nm for oxygenized and deoxygenized blood.
F IG . 21. In vivo OCT tomograms of the retina.
same object point. It can be seen from Fig. 20 that the OCT signals from the deoxygenized blood probe and the oxygenized blood probe exhibit the same signal
strength at 815 nm because this wavelength is near the isobestic point of the blood
spectrum. At 680 nm the OCT signal from the oxygenized blood probe is higher
than the signal from the deoxygenized blood in agreement with Fig. 19. It can also
be seen that the penetration of the SLD light into the blood sample is higher for
the longer wavelength 815 nm. This is because the backscattering cross-section
of the blood cells is lower at 815 nm than at 680 nm.
Figure 21 shows an in vivo OCT measurement of the retina of a human eye at
815 nm and 680 nm [40]. The OCT beam is scanned across a vein and an artery
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of the retinal bloodvessel system. It can be seen that the OCT signals at 815 nm
from the artery and the vein do not differ significantly in comparison to the OCT
signals at 680 nm which show a much smaller signal from the vein. This is in
agreement with the spectral properties of HbO2 and Hb shown in Fig. 19.
3.1.2. Bloodflow Measurement and OCT
The blood supply of the retina consists of two separate vascular systems. The
choroid provides the blood supply for the outer one half of the sensory retina. The
choroid consists of a dense network of capillaries. It is supplied from the posterior
ciliary artery. The inner portion of the retina is supplied by the branches of the
central retinal artery which enters the eye at the nerve head (blind spot). These
two blood supply systems are independent. Both circulations must be intact to
maintain retinal function.
Laser Doppler velocimetry is a technique which allows to measure the blood
flow at the fundus of the eye in vivo [19,20,27]. A laser goes through one half
of the eye pupil (Fig. 22). The angle of incidence of the laser at the fundus is
proportional to the lateral offset of the laser at the pupil. Due to the finite angle
of incidence there is a finite component of the k-vector of the laser parallel to the
fundus. The laser light is scattered at the moving blood cells. Thus, the frequency
of the reflected k-vector component parallel to the fundus is shifted due to the
Doppler effect. The frequency shift is ν = ν0 · v/c, where ν0 is the nominal
frequency of the laser, v is the velocity of the blood cells and c is the speed of
light. But the same k-vector component is also scattered at the retinal tissue which
is at rest. The light backscattered from the retina passes the pupil and is finally
detected with a photodiode. The light scattered at the moving blood cells and at
F IG . 22. Setup for laser Doppler velocimetry of the retinal blood flow.
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F IG . 23. OCT-signal from the retina (A-scan).
the stationary tissue interferes at the detector. Because of the Doppler shift of the
light scattered at the moving blood cells the interference signal is modulated with
the Doppler shift frequency ν which is in the range of some kHz depending on
the angle of incidence at the fundus and on the bloodflow velocity. The strength
of the Doppler signal is proportional to the density of the blood cells at the laser
focus on the retina. Therefore the product of the strength and the frequency shift
of the Doppler signal is proportional to the blood flow [19].
However, this technique measures the integral bloodflow of both the choroidal
circulation (supplied by the posterior ciliary artery) and the inner retinal circulation (supplied by the central retinal artery). OCT is a technique which can
differentiate between the two circulation systems because the various parts of the
OCT signal from the retina can be attributed to the various retinal layers.
The OCT beam is coupled into the pupil in the same decentered way as the laser
beam with laser Doppler velocimetry. The OCT signal is shown in Fig. 23. The
first peak originates from the retinal sheet nurtured by the ciliary artery the second
from the choroid. The two signal peaks are Fourier transformed separately to provide the Fourier spectra of the two signals shown in Fig. 24 [40]. The choroidal
signal frequency is shifted by 8 kHz relative to the signal frequency from a stationary mirror as a reference object. The retinal signal frequency is shifted by
5 kHz. Taking the angle of incidence of the OCT beam at the fundus into account
a bloodflow velocity of 4 cm/s can be estimated from the Doppler shift of the
choroidal signal and 2.5 cm/s for the retinal bloodflow.
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F IG . 24. OCT Doppler velocimetry of the retinal blood flow (in vivo). Both the spectra of the
choroidal signal and of the retinal signal are shifted relative to the stationary mirror signal. The
choroidal spectrum is shifted more than the retinal spectrum indicating a higher bloodflow velocity in
the choroidal tissue. It can also be seen that both the retinal and the choroidal spectrum is broader than
the mirror spectrum which is caused by the velocity distribution of the blood cells.
3.1.3. Eye Length Measurement with OCT
Short coherence length interferometry can be used to measure the length of the
human eye which is an important parameter for cataract surgery [22]. There, it is
necessary to select the correct intraocular lens before removing the eye lens. The
eye length has to be measured through the turbid eye lens in a contactless mode.
The method described in [22] brings the reflex from the cornea and the retina to
interference by sending the light from the eye into a Michelson interferometer
with different arm lengths. The length difference is chosen so that it corresponds
to the optical length of the eye bringing the corneal reflex and the retinal reflex
to interference. The length difference of the interferometer is mechanically varied
until the interference between the retinal reflex and the corneal reflex occurs.
We have investigated an alternative approach making use of a time domain
OCT setup as shown in Fig. 25 [14]. Of course the range for the movement of the
retroreflector in the TDOCT interferometer described above would be too long
to cover the whole eye length in a short time. Short acquisition time is necessary
to avoid artefacts due to the saccadic eye movements. Another problem is the
dispersion of the aqueous and vitreous humor of the eye causing a spreading of
the interferogram and thus a reduction of the signal to noise ratio.
In the setup of Fig. 25 the retroreflector is scanned periodically in the range of
2 mm. In addition a PMMA rod with an optical length corresponding to the optical
length of the standard human eye is periodically flipped in and out (as shown in
Fig. 25) of the reference path of the OCT interferometer. With the rod flipped
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F IG . 25. Setup of OCT interferometer for eye length measurement.
F IG . 26. OCT signal for eye length measurement. The corneal reflex consists of the front reflex
(left diagram, left peak) and the reflex at the rear side of the cornea (left diagram, right peak). The
fundus reflex consists of the signal scattered by the retinal nerve fiber layer (right diagram, left peak)
and by the choroid (right diagram, right peak).
out the signal from the cornea (corneal reflex) is recorded (Fig. 26, left diagram).
With the rod flipped in the fundus reflex is recorded (Fig. 26, right diagram).
The corneal reflex consists of two peaks. The first peak corresponds to the reflection at the front surface of the cornea the other to the reflection at the interface
surface between cornea and the anterior chamber of the eye. The retinal reflex
also consists of two peaks. One is the reflex from the nerve fiber layer in front of
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the receptor layer the other is generated by the choroid tissue behind the receptor
layer providing the blood supply for the retina. For the eye length measurement
the distance between the front corneal reflex and the receptor layer is relevant as
shown in Fig. 26. Due to the similar dispersion properties of the PMMA rod the
dispersion of the aqueous and vitreous humor is compensated. The optical eye
length can be determined with an accuracy of 20 microns which corresponds to
an error in refractive power of the optical system of the eye of less than 0.05 D.
The residual uncertainty of the eyelength measurement is due to the variability of
the dispersion of the aqueous and vitreous humor because the measurement wavelength is 830 nm but the wavelength of the visible spectrum is around 550 nm.
The clinical practice, however, shows that these uncertainties are negligible.
3.1.4. OCT in Optical Manufacturing
Within the last 15 years computer numerical controlled (CNC) machines have
completely changed optical manufacturing. In particular the manufacturing of aspheres has become an interesting alternative to spherical lenses both under cost
and functional aspects. Aspheric surfaces provide additional degrees of freedom
in lens design where several spherical surfaces would be necessary. Therefore,
aspheres permit lighter and smaller objectives for projection systems, sensors or
photographic systems. Aspheres are manufactured in three steps. First, the grinding machine has to shape the lens with an accuracy of 1 µm. In a second step
the lens is polished. The polishing process smoothes the surface. In a third step
the polished surface is measured interferometrically with an accuracy in the order of 10 nm. Deviations of the surface from the nominal design data are locally
corrected in the polishing process. For that purpose a polishing robot is used to
guide the polishing tool directly to the zone of the aspheric surface which is to be
corrected.
The grinding of the lens is the most crucial step in the whole process because no
measurement feedback permits inline corrections. Because the surface of the lens
is not reflecting standard interferometric techniques cannot be applied. Geometrical optical techniques as used in autofocus sensors are not applicable because
they need large numerical apertures and small working distances.
Depth resolution of an OCT sensor depends only on the coherence length of the
light source. Therefore, an objective with low numerical aperture and long working distance can be used. Figure 27 shows the experimental setup [30]. It consists
of a fiber optic TDOCT-setup as described above. The objective in the probe arm
is fixed to the tool mount (not shown) which is moved across the surface of the object which is to be measured. When the tool is moved the fiber is bent causing an
index change in the fiber. The OCT signal is shifted due to the index change in the
order of several tens of microns corrupting the measurement results. Therefore,
a reference mirror is installed which is also fixed to the tool mount of the grinding machine. This reference signal is shifted in the same way as the probe signal
130
T. Hellmuth
[3
F IG . 27. Setup of OCT interferometer for aspheric profile measurement in grinding machine.
induced by the bending fiber. The movement of the probe signal relative to this reference signal finally provides the information about the surface topography when
the objective is moved across the asphere by the CNC grinding machine. Figure 28
shows the profile of a grinded plane surface derived from the peak positions of the
OCT signals. The result is compared with the measurement of the surface with a
ultraprecision tactile mechanical measurement machine (Zeiss UPMC).
The signal noise of the OCT-measurement is primarily due to the surface roughness. The lateral resolution of the OCT-measurement is in the order of 100 micron
(SLD-focus) whereas the lateral resolution of the tactile measurement machine is
limited by the probe ball with a diameter of about 2 mm which is scanned across
the surface. The working distance between the objective and the surface is 10 mm.
Because of the scattering properties of the rough surface the OCT signal can be
registered also from tilted surfaces with a tilt angle up to 20 degrees.
3.2. L INEAR O PTICAL C OHERENCE T OMOGRAPHY (LOCT)
Time domain OCT uses a galvoscanner as a moving device for the retroreflector. Mechanical devices are limited in their frequency bandwidth thus limiting
the image acquisition speed. Another important drawback is the limited lifetime
3]
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F IG . 28. Profile measurement of a grinded plane surface derived from the peak positions of the
OCT signals. The result is compared with the reference measurement of the surface with a ultraprecision tactile mechanical measurement machine (Zeiss UPMC).
F IG . 29. Linear optical coherence tomograph.
of electromechanical parts. An alternative approach comprises an interferometric
setup shown in Fig. 29 [12,13].
132
T. Hellmuth
[3
F IG . 30. The position of the interferogram on the CCD-line depends on the pathlength difference.
Again a superluminescent diode is used as low coherence source. The light
passes a first coupler with a splitting ratio of 90:10. In order to improve the sensitivity of the setup, 90% of the light is directed to the sample arm, while 10% of
the light is routed to the reference arm. Additional 50:50 couplers are placed in
the sample and reference arm, respectively. In each interferometer arm, the light
passes the two 50:50 couplers. The reference beam is reflected from a reference
mirror whereas the sample beam is reflected from the sample that is to be imaged.
The position of the reference mirror is fixed, no optical delay line scanning is
needed. The backreflected light from the reference mirror and the sample again
passes the 50:50 couplers and is then routed to a fiber two-pinhole device. The
distance between the two fibers can be varied between 250 µm and 8 mm. Outside
the two fibers, the light propagates in a solid angle determined by the numerical
aperture of the monomode fibers (NA = 0.11). A linear CCD-array is located
in a distance of 25 cm from the two fibers and is illuminated with the light from
the two fibers. An efficient illumination of the CCD-array is achieved by using
a cylindrical lens optics. Therefore the circular light cone emerging from each
fiber is transformed into two overlapping lines at the position of the CCD-array,
collinear with the active area. The signal processing is done using a bandpass
filter, a logarithmic amplifier, a demodulator, an AD-converter and a computer.
Figure 30 shows how the position of the interference structure on the CCD-line
depends on the path length difference of the probe arm and the reference arm
of the interferometer. If these path lengths are equal the interference pattern is
3]
SPATIAL IMAGING
133
located at the center of the CCD line. Then the path lengths between the two fiber
outputs are equal.
If the probe arm length differs from the reference arm length by the interference pattern appears on the CCD line where the distance between the maximum of
the interference pattern and fiber output A differs from the corresponding distance
to fiber output B by .
It can be shown that the theoretical limit of the signal to noise ratio is equivalent
to that of TDOCT [10]. Another special aspect of LOCT is that the fringe oscillation frequency can be set independently from the image acquisition time and
the wavelength by choosing an appropriate separation of the two fiber outputs A
and B.
3.3. S PECTRAL D OMAIN O PTICAL C OHERENCE T OMOGRAPHY (SDOCT)
Besides TDOCT and LOCT a third method has become an interesting alternative
which is known as spectral domain optical coherence tomography (SDOCT) [22,
33,37]. The principle is shown in Fig. 31. The light from the superluminescent
diode is split into two beams by a beamsplitter (or a 3dB-coupler). The reference beam is reflected from the reference mirror. The probe beam is reflected
from the sample. The two reflected beams are superimposed and hit a diffraction
grating. The dispersed beam is finally focused onto a CCD-line. The registered
spectrum represents basically the spectrum of the light source. However, the spectrum is modulated due to the interference of the probe beam and the reference
beam (Fig. 32) [39]. The modulation frequency is proportional to the armlength
difference of the reference arm and the probearm. Computing the Fourier trans-
F IG . 31. Spectral domain optical coherence tomograph.
134
T. Hellmuth
[3
F IG . 32. Modulated spectrum of SLD.
F IG . 33. Fourier transform of signal in Fig. 32. The position of the peak depends on the armlength
difference of the interferometer.
form finally provides the signal (Fig. 33). The signal is equivalent to the signal of
TDOCT or LOCT. The position of the peak depends on the armlength difference
and thus on the position of the sample object. However, as it will be shown in
the next section the SDOCT signal is superior to the TDOCT and LOCT signal
concerning the signal to noise ratio.
3.3.1. Comparison of Noise in Spectral Domain OCT and Time Domain OCT
In [28,29] TDOCT and SDOCT are compared in respect of their noise performance. It is shown that if photon shot noise is the relevant noise source SDOCT
is superior to TDOCT and LOCT. The following simulation shall illustrate this
remarkable result. The results can also be directly applied to the noise analysis of
line spectrometers and Fourier transform spectrometers because of their analogy
to SDOCT and TDOCT. Only if amplifier noise is the dominant noise source line
3]
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135
F IG . 34. Comparison of Poissonian noise in spectral domain OCT and time domain OCT.
spectrometers and Fourier transform spectrometers are equivalent regarding noise
performance.
In the following simple model it is assumed that the SDOCT signal is generated
on a CCD line with 512 pixels with 50 photons per pixel on the average. The
whole photon budget is therefore 25,600 photons. The modulated spectrometer
signal is given by
fm = A · sin2 (2πνs m)
with m = −N/2 . . . + N/2. The simulated signal is shown in Fig. 34 (top left)
with νs = 0.4, the number of pixels N = 512 and A = 100. The values fm
exhibit Poissonian noise generated by a standard Poissonian noise generation algorithm [7]. The Fourier transform of fm is given by the N values
f˜n =
+N/2
e−i2πnm fm
m=−N/2
with n = −N/2 . . . + N/2. The SDOCT-Signal is then given by the N values
2
+N/2
2
e−i2πnm fm sn = f˜n = −N/2
136
T. Hellmuth
[5
as shown in Fig. 34 (bottom, left). For comparison the corresponding TDOCTsignal is given by the N values
1
2
1 + e−(m/b) sin(2πνt m)
2
shown in Fig. 34 (top, right) with νt = 0.2 and the coherence length of the light
source b = 20. Also Poissonian noise is introduced. The mean photon budget is
as in the SDOCT case 25,600. The envelope function can be calculated with the
Hilbert transform (Fig. 34 bottom, right). Comparing the SDOCT and TDOCT
signal it is obvious that the signal to noise ratio is better for SDOCT than for
TDOCT although the same number of signal photons has been taken for the simulation.
tm = A
4. Conclusion
Both wavefront coding and optical coherence tomography are commercially used
techniques. Although in principle many fields of applications are open to these
methods wavefront coding is mainly used for specific sensor applications like
barcode readers. OCT is used mainly in ophthalmology.
The limiting factors for wavefront coding are mainly artefacts introduced by
the inverse filtering. Thus, this technique is not yet found in the field of high quality imaging techniques like microscopy and digital photography although first
commercial trials have been undertaken in microscopy. Because of the low manufacturing costs of the phaseplates and the progress in the field of fast signal and
image processing many new applications in medium quality photography (e.g.
mobile phone cameras or surveillance cameras) and in sensor technology can be
expected.
The limiting factor in OCT is mainly the coherence length and the brightness
of the light source. It has been proven in various applications that high resolution OCT tomograms can be obtained by using femtosecond lasers, rapidly swept
tunable laser sources [42] or supercontinuum laser sources generated with femtosecond laser pulses in photonic crystals [43]. As soon as these light sources are
available at low cost in high quantities OCT will have a great future not only in
ophthalmology as today but also in such important fields of medical applications
like endoscopy.
5. Acknowledgements
The projects have been supported by the “Landesstiftung Baden-Württemberg”
and “Bundesministerium für Bildung und Forschung”. We gratefully appreciate
6]
SPATIAL IMAGING
137
valuable discussions and cooperation with Dr. C. Hauger and Dr. H. Gross (both
Carl Zeiss AG, Oberkochen).
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ADVANCES IN ATOMIC, MOLECULAR AND OPTICAL PHYSICS, VOL. 53
THE QUANTUM PROPERTIES OF
MULTIMODE OPTICAL AMPLIFIERS
REVISITED
G. LEUCHS1,* , U.L. ANDERSEN1 and C. FABRE2
1 Max Planck Research Group of Optics, Information and Photonics,
University of Erlangen-Nürnberg, Erlangen, Germany
2 Laboratoire Kastler-Brossel, Université Pierre et Marie Curie et Ecole Normale Supérieure,
Place Jussieu, cc74, 75252 Paris cedex 05, France
1.
2.
3.
4.
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6.
7.
General Linear Input–Output Transformation for a Linear Optical Device
The Phase-Insensitive Amplifier . . . . . . . . . . . . . . . . . . . . . . .
The Multimode Phase Insensitive Amplifier . . . . . . . . . . . . . . . . .
The Nature of the Ancilla Modes . . . . . . . . . . . . . . . . . . . . . . .
An Optical Amplifier Working at the Quantum Limit . . . . . . . . . . . .
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Abstract
There are a number of physically different realizations of an optical amplifier and
yet they all share the same fundamental quantum limit as far as their noise characteristics are concerned. We review the underlying mathematical formalism without the
restriction of a minimum number of modes being involved, its physical implications
and relate it to the phenomenological models for the various amplifiers.
The study of optical amplifiers and their properties started with the invention of
the maser and the laser, based on stimulated emission [1–5]. These first optical
amplifiers belong to the class of phase insensitive amplifiers. Modern semiconductor optical amplifiers, and especially the Erbium doped fiber amplifiers which
are widely used in telecommunications [6], belong to the same category [7,8].
There is a second class of optical amplifiers which are based on non-linear optical
processes such as stimulated Brillouin, Rayleigh or Raman processes [9] or three* E-mail: [email protected].
139
© 2006 Elsevier Inc. All rights reserved
ISSN 1049-250X
DOI 10.1016/S1049-250X(06)53005-8
140
G. Leuchs et al.
[1
and four-wave mixing [10,11], which are also, in most cases, phase-insensitive.
However, there are some configurations where the amplification depends on the
phase of the input signal. The degenerate parametric amplifier, or the parametric
amplifier with the input signal is injected on both the signal and idler modes, are
examples of such phase-sensitive amplifiers [12].
In the simplest case, well below saturation, all these amplifiers are linear devices, i.e. the field operators describing the output beams are linear combinations
of the field operators at the input side. Nevertheless, these seemingly simple mathematical relations allow for a fairly complex scenario as described in the papers
on the quantum behavior of linear amplifiers by Haus and Mullen [13] and by
Caves [14]. All the properties including the quantum aspects can be traced back
to these linear field operator transformations. We will discuss these relations in a
general framework, with the possibility of involving many modes in the device.
Next we will treat the amplifier more phenomenologically and identify the experimental nature of the modes the existence of which is required by the unitarity of
the field operators. To some extent the different types of amplifiers can be related
to the different types of attenuators.
In a recent demonstration of quantum cloning with continuous variables, the
essential ingredient was an optical amplifier working at the quantum limit [15].
This amplifier does not use any non-linear optical process nor stimulated emission
but just linear optical elements, detectors, modulators and electronic feed-forward
circuits. The amplifier set-up with its modular structure makes it easy to identify
the various origins of the noise figure of the optical amplifier and to compare it
with the general performance limitation.
1. General Linear Input–Output Transformation for a Linear
Optical Device
Let us consider a linear optical device, which can be an amplifier, an attenuator
or a quantum gate. From a basic point of view it is well known that there can
be no such device which transforms an input field described by the field operator
aˆ to an output field described by a field operator aˆ = β aˆ with |β| = 1. This
would violate the requirement that all free space field operators fulfill the bosonic
commutation relation
a,
ˆ aˆ † = aˆ , aˆ † = 1.
(1)
The conservation of such a commutation relation leaves two possibilities:
• The device is single mode, but of the form
aˆ → aˆ = β1 aˆ + β2 aˆ †
(2)
2]
MULTIMODE OPTICAL AMPLIFIERS
141
with
|β1 |2 − |β2 |2 = 1.
(3)
• The device is multimode and couples N > 1 input modes to N output modes.
We will first consider here the simplest and basic case of a two mode device,
and call bˆ the annihilation operator of this second, “ancilla”, mode. The most
general transformation fulfilling the requirement (1) is the two-mode Bogoliubov transformation [16]
a,
ˆ bˆ → aˆ = β1 aˆ + β2 aˆ † + γ1 bˆ + γ2 bˆ †
(4)
|β1 |2 − |β2 |2 + |γ1 |2 − |γ2 |2 = 1.
(5)
with
The terms proportional to β2 and γ2 correspond to “spontaneous emission”
processes, where photons are produced at the output even in absence of any photon in the corresponding input mode.
In the first, single mode case, the device is necessarily phase-sensitive: it
multiplies the input mean field by a factor which depends on the quadrature
Xˆ φ = eiφ aˆ + e−iφ aˆ † of the input signal. When the input is a coherent or vacuum state, the output is a squeezed state [17]. Although being formally a linear
process, the phase sensitive amplifier associated with the squeezing operation is
an intrinsically non-linear process and requires an optically non-linear interaction [17].
In the two-mode case, relation (4) leads to phase insensitivity of a new type
which is present even if β2 = 0. In this latter case,
|γ1 |2 − |γ2 |2 = 1 − |β1 |2 .
(6)
The device is an amplifier when |β1 | > 1, and an attenuator when |β1 | < 1.
Relation (6) implies that at least γ2 must be different from zero in the amplifier
case, and that at least γ1 must be different from zero in the attenuator case. But in
the general case both coefficients are different from zero and mode aˆ may, e.g.,
be coupled to a squeezed vacuum. Attenuation and amplification are merely two
limiting cases.
2. The Phase-Insensitive Amplifier
The general phase insensitive amplifier of energy gain G > 1 contains an unknown mixture of γ1 and γ2 . It is characterized by the transformation
√
a,
ˆ bˆ → aˆ = G aˆ + g1 bˆ + g2 eiα bˆ † ,
(7)
142
G. Leuchs et al.
[2
√
where all quantities G, g1 , g2 and α are real. The phase factors of β1 = G
ˆ The term ‘phase
and γ1 have been included in the annihilation operators aˆ and b.
insensitive’ relates to the fact that with β2 = 0 the amplification of the mean
values of the quadrature components is phase insensitive which still leaves room
for phase sensitive noise. Relation (4) implies that g1 and g2 have to fulfill the
condition
g22 − g12 = G − 1.
(8)
The quantum statistical properties of this amplifier can be described by the
moments of the field quadratures. An important scenario is the case where the
input mode aˆ is in a coherent state |α and mode bˆ is in the vacuum state |0 which
we will write as |α, 0. The input signal is a modulation at a given frequency of
the amplitude or the phase of the input wave, or more generally of any quadrature
Xˆ φ = eiφ aˆ + e−iφ aˆ † of the input signal. The second moment of this quadrature
at the output of the device is
2
δ Xˆ φ = G + g12 + g22 + 2g1 g2 cos(α + 2φ).
(9)
Let us recall that the variance for any quadrature of a coherent state is 1 with the
present notations. Using relation (8), this gives the following value for the noise
figure F , which is in the present case the output noise δ Xˆ φ 2 divided by the gain
g
+
g12 + G − 1 cos(α − 2φ)
1
1
.
F = 2 − + 2g1
(10)
G
G
In the usual parametric amplifier case, the ancilla mode is the idler mode and
g1 = 0. The noise figure reduces to [13,14]
1
(11)
,
G
and in the limit of large gain we recover the familiar result F = 2, well known as
the 3 dB quantum limit of the phase insensitive optical amplifier.
One notices that when g1 = 0 the noise of the amplifier, and therefore its noise
figure, is phase sensitive, whereas the gain for the mean value is phase insensitive.
Its minimum value is obtained for φ = π−α
2 ,
g1 − g12 + G − 1
1
Fmin = 2 − + 2g1
G
G
G−1
1
g1
=2− −2
(12)
.
G
g + g2 + G − 1 G
F =2−
1
1
√
An interesting limiting case is when g1 G, for which, according to (12),
Fmin is getting close to 1. One has in this case a two-mode noiseless amplifier.
3]
MULTIMODE OPTICAL AMPLIFIERS
143
A possible implementation is to use a regular non-degenerate parametric amplifier
and to insert a perfect squeezer at the input of the idler mode. This configuration
has been experimentally studied in [18].
So far we discussed the case where either the signal mode or the ancilla mode
are supposed to be unique in the amplification or attenuation process. We will consider in the following sections the possibility of linear coupling between multiple
modes.
3. The Multimode Phase Insensitive Amplifier
Another important class of optical amplifiers is the multi-mode amplifier, that is
likely to amplify simultaneously several orthogonal modes, for example, image
amplifiers [19–21]. For the simplicity of the discussion, we will take here the
simple example of a two-mode phase insensitive amplifier having the same gain
for any combination of the two modes aˆ 1 and aˆ 2 . As in the single mode case, reˆ Let us first assume
lation (1) requires the existence of at least one ancilla mode b.
that there is only one such mode. We can then write
aˆ 1 , aˆ 2 , bˆ → aˆ 1 = β aˆ 1 + γ1 bˆ + γˆ2 bˆ † ,
aˆ 2 = β aˆ 2 + γ1 bˆ + γˆ2 bˆ †
(13)
and using the commutators of the various field operators
2 2
|γ2 |2 − |γ1 |2 = γ2 − γ1 = |β|2 − 1,
γ2 γ2 ∗ − γ1 γ1 ∗ = 0.
(14)
A straightforward derivation shows that the relations (14) cannot be simultaneously fulfilled, so that a second ancilla mode is needed. The demonstration can
easily be extended to the N -mode amplifier. The conclusion of this simple but
general reasoning is that one needs at least as many ancilla modes as there are
input signal modes in a multimode amplifier.
In the two-mode amplifier, calling bˆ1 and bˆ2 these two required modes, and in
the simple case where only creation operators for the ancilla modes are involved
in the input-output relation, one has
aˆ 1 , aˆ 2 , bˆ → aˆ 1 = β aˆ 1 + γ11 bˆ1† + γ12 bˆ2† ,
aˆ 2 = β aˆ 2 + γ21 bˆ1† + γ22 bˆ2† ,
(15)
and
|γ11 |2 + |γ12 |2 = |γ21 |2 + |γ22 |2 = |β|2 − 1,
∗
∗
+ γ12 γ22
= 0,
γ11 γ21
(16)
144
G. Leuchs et al.
[4
which correspond to a unitary transformation in the two-mode ancillary space:
γ11 = |β|2 − 1 cos θ eiψ ,
γ12 = |β|2 − 1 sin θ e−iψ ,
=
−
|β|2 − 1 sin θ eiψ ,
γ21
γ22 = |β|2 − 1 cos θ e−iψ ,
(17)
where θ and ψ are arbitrary angles. If one performs the inverse of this transformation on the two-mode signal space, the gain will not be changed for the two
new modes, as the gain matrix is proportional to the identity, and the ancilla terms
will be diagonalized, a single ancilla mode being associated to each amplified signal mode: the only possible configuration for a two-mode amplifier is therefore
two independent single mode amplifiers with identical gains. As a result the noise
figure will be the same as in the single mode case, and independent of the combinations of input modes used as a signal. These conclusions are no longer valid
in the more complicated case where the gain is different for the two amplified
modes.
4. The Nature of the Ancilla Modes
If one now turns to physical implementations of the optical amplifier one might
ask the question: what is the additional mode bˆ which is so essential in the mathematical description of the amplifier?
In the case of the parametric amplifier the answer is straightforward. Mode
bˆ is the idler mode which has to be in the vacuum state for the standard phase
insensitive amplifier (Fig. 1).
Next one asks the same question for the prototype of all amplifiers, the medium
with population inversion which gives rise to amplification by stimulated emission. Here the identification of mode bˆ is not straightforward. There are many
F IG . 1. Sketch of a parametric amplifier. The pump is taken to be a classical field and the paraˆ
metric amplifier couples mode aˆ and b.
4]
MULTIMODE OPTICAL AMPLIFIERS
145
F IG . 2. Panel (a) shows the signal field mode amplified by the medium and two of the N − 1
vacuum modes scattering into the signal mode. Panel (b) represents a formal sketch of the situation.
and β
In the text two (N × N ) matrices are used, β+mn couples aˆ n† to aˆ m
mm couples aˆ n to aˆ m .
more than just two modes which one has to consider. A related multi-mode formalism is used to describe multiple scattering in inhomogeneous media [22]. All
but the signal mode are taken to be in the vacuum state. Each of these vacuum
modes may couple into the signal mode (or any other vacuum mode) by scattering off one of the inverted molecules in the medium via spontaneous emission
(Fig. 2a). The linear coupling of the N input modes to the N output modes is
described by an N × N matrix
aˆ m
= βmm aˆ m +
∞
βmn aˆ n + β+mn aˆ †n
(18)
n=m
with
|βmm |2 +
n=m
|βmn |2 −
n=m
|β+mn |2 = 1.
(19)
If m is the signal mode then |βmm |2 = G and n=m |β+mn |2 − n=m |βmn |2 =
G − 1. Under these conditions the noise figure for the stimulated emission ampli-
146
G. Leuchs et al.
fier is
F =1+
1 ∗
∗
βmn β+mn + βmn
β+mn
+ |βmn |2 + |β+mn |2 .
G
[4
(20)
n=m
In the case of the ideal amplifier (i.e. βmn = 0 if m = n), the formula reduces to
1
1 |β+mn |2 = 2 − ,
F =1+
(21)
G
G
n=m
and with the sum rule (19) we retrieve the familiar results for the single mode
case (11). This shows that a multimode manifold of vacuum modes effectively has
the same impact as the one vacuum mode in the single mode case. The quantum
limit of the ideal amplifier results from the admixture of the creation operator of
ˆ which is actually the linear combination
one “super” vacuum mode b,
ˆn
n=m βmn a
.
bˆ = (22)
2
n=m |βmn |
In the case of the stimulated emission amplifier, the aˆ n modes being the plane
wave modes in which spontaneous photons are likely to be emitted. For a single
F IG . 3. Sketch of a 2 × 2 beam splitter (a) and a lossy element (b) such as a neutral density filter
in which case the loss channels n are absorbed inside the medium (not shown).
5]
MULTIMODE OPTICAL AMPLIFIERS
147
atom or molecule of the amplifying medium this super mode bˆ is nothing but the
dipole wave which is emitted by the atom or which couples best to the atom [23,
24]. The discussion also shows that confining the amplifying medium, e.g., to the
core of a photonic crystal fiber [25] will reduce the number of modes but will not
affect the quantum noise limit of the amplifier.
It is worth noting that the amplifier and the phenomenological description
above can be mapped to the familiar case of an optical attenuator. It is immediately
clear that the single mode case, i.e. the counterpart to the parametric amplifier, is
the ubiquitous beam splitter. The stimulated emission amplifier however resembles a neutral density attenuator where the light is also coupled to many modes
(see Fig. 3). Again with a similar line of arguments these many modes can be
treated effectively as one mode for the purpose of noise consideration.
5. An Optical Amplifier Working at the Quantum Limit
In a recent experiment it was shown that an optical amplifier working at the quantum limit can be demonstrated using just linear optical elements, detectors, an
amplifying electronic circuit and optical modulators for amplitude and phase [15].
The scheme, which is an extension of the intensity modulation amplifier [26], is
sketched in Fig. 4.
The signal input field is split at the first beam splitter and mixed with the auxiliary mode v1 . The split off part is measured according to the scheme of Arthur
and Kelly [27] where x and p denote the amplitude and phase quadratures of the
field measured for example with a local oscillator (not shown). The detected signals are amplified and fed forward to an amplitude and a phase modulator. If the
amplification factor λ is chosen properly in dependence on the splitting ratio of
F IG . 4. Sketch of optical amplification with beam splitters, detectors and modulators. The detector
signals have to be amplified and fed forward to the detector (see [15]).
148
G. Leuchs et al.
[7
the first beam splitter, then the input v1 does not influence the noise characteristics
of the amplifier. Note that this is a universal scheme. For a theoretical quantum
treatment of the electro-optic feed forward see [28–30]. In the spirit of the above
discussion this amplifier can be described with the single additional mode which
is readily identified with the vacuum input field v2 .
6. Conclusion
We have discussed the fundamental noise limit of optical amplification in which
many modes are involved, either for the input signal or for the ancilla modes.
We have considered various physical implementations: the parametric amplifier, the stimulated emission amplifier and the quantum electro-optic feedforward
amplifier. Other amplifiers such as a Raman amplifier [31,32] can be discussed
following the same arguments. Quantum noise considerations are relevant, e.g.,
to optical communication [33,34]. The main point here was to physically identify
the ancillary field modes required for the mathematical description.
7. References
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[22] C.W.J. Beenakker, Thermal radiation and amplified spontaneous emission from a random
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[25] J.C. Knight, T.A. Birks, P.J.S. Russell, et al., All-silica single-mode optical fiber with photonic
crystal cladding, Opt. Lett. 21 (1996) 1547.
[26] A.V. Masalov, A.A. Putilin, Quantum noise of a modulation optical amplifier, Opt. Spectrosc. 82
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[28] H.M. Wiseman, G.J. Milburn, All-optical versus electro-optical quantum-limited feedback, Phys.
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[29] B. Julsgaard, J. Sherson, J.I. Cirac, J. Fiurasek, E.S. Polzik, Experimental demonstration of quantum memory for light, Nature 432 (2004) 482, see supplementary note.
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[31] E.B. Tucker, Amplification of 9.3-kMc/sec ultrasonic pulses by MASER action in Ruby, Phys.
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This page intentionally left blank
ADVANCES IN ATOMIC, MOLECULAR AND OPTICAL PHYSICS, VOL. 53
QUANTUM OPTICS OF ULTRA-COLD
MOLECULES
D. MEISER, T. MIYAKAWA, H. UYS and P. MEYSTRE
Department of Physics, The University of Arizona, 1118 E. 4th Street, Tucson, AZ 85705, USA
1. Introduction . . . . . . . . . . . . . . . . . . . .
2. Molecular Micromaser . . . . . . . . . . . . . .
2.1. Model . . . . . . . . . . . . . . . . . . . .
2.2. Results . . . . . . . . . . . . . . . . . . . .
3. Passage Time Statistics of Molecule Formation
4. Counting Statistics of Molecular Fields . . . .
4.1. BEC . . . . . . . . . . . . . . . . . . . . .
4.2. Normal Fermi Gas . . . . . . . . . . . . .
4.3. Fermi Gas with Superfluid Component . .
5. Molecules as Probes of Spatial Correlations . .
5.1. Model . . . . . . . . . . . . . . . . . . . .
5.2. BEC . . . . . . . . . . . . . . . . . . . . .
5.3. Normal Fermi Gas . . . . . . . . . . . . .
5.4. BCS State . . . . . . . . . . . . . . . . . .
6. Conclusion . . . . . . . . . . . . . . . . . . . .
7. Acknowledgements . . . . . . . . . . . . . . .
8. References . . . . . . . . . . . . . . . . . . . .
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152
153
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182
Abstract
Quantum optics has been a major driving force behind the rapid experimental developments that have led from the first laser cooling schemes to the Bose–Einstein
condensation (BEC) of dilute atomic and molecular gases. Not only has it provided
experimentalists with the necessary tools to create ultra-cold atomic systems, but it
has also provided theorists with a formalism and framework to describe them: many
effects now being studied in quantum-degenerate atomic and molecular systems find
a very natural explanation in a quantum optics picture. This article briefly reviews
three such examples that find their direct inspiration in the trailblazing work carried
out over the years by Herbert Walther, one of the true giants of that field. Specifically, we use an analogy with the micromaser to analyze ultra-cold molecules in a
double-well potential; study the formation and dissociation dynamics of molecules
using the passage time statistics familiar from superradiance and superfluorescence
studies; and show how molecules can be used to probe higher-order correlations in
ultra-cold atomic gases, in particular bunching and antibunching.
151
© 2006 Elsevier Inc. All rights reserved
ISSN 1049-250X
DOI 10.1016/S1049-250X(06)53006-X
152
D. Meiser et al.
[1
1. Introduction
Quantum optics plays a central role in the physics of quantum-degenerate atoms
and molecules. Laser light and its coherent and incoherent interactions with
atoms are ubiquitous in these experiments, and the tools that have culminated in
the achievement of Bose–Einstein condensation (BEC) (Anderson et al., 1995;
Bradley et al., 1995; Davis et al., 1995) were first studied and understood in
quantum optics. Indeed, the deep connection between quantum optics and cold
atom physics was realized well before the first experimental realizations of BEC,
both at the experimental and theoretical levels. On the theory side, there are
(at least) two important reasons why quantum optics methods are well suited
for the study of cold atoms systems. First, bosonic fields have direct analogs
in electromagnetic fields, which have been extensively studied in quantum optics. Second, for fermions the Pauli Exclusion Principle restricts the occupation of a given mode to zero or one, and these two states—mode occupied or
empty—can often be mapped onto a two-level system, as we shall see. As a
result, many situations familiar from quantum optics are also found in coldatom systems, including matter–wave interference (Andrews et al., 1997), atom
lasers and matter–wave amplifiers (Inouye et al., 1999b; Ketterle and Miesner,
1997; Kozuma et al., 1999; Law and Bigelow, 1998) matter–wave beam splitters
(Burgbacher and Audretsch, 1999) four-wave mixing (Christ et al., 2003; Lenz et
al., 1993; Meiser et al., 2005a; Miyakawa et al., 2003; Moore et al., 1999; Rojo
et al., 1999; Search et al., 2002b), and Dicke superradiance (Inouye et al., 1999a;
Moore and Meystre, 1999), to name a few.
At the same time the physics of ultra-cold atoms is much richer than its
quantum-optical counterpart since atoms can be either fermions (DeMarco and
Jin, 1998, 1999; Hadzibabic et al., 2003) or bosons and have a rich internal
structure. In addition, the interaction between atoms can be tuned relatively
easily on fast time scales using for instance Feshbach resonances (Duine and
Stoof, 2004; Dürr et al., 2004; Inouye et al., 1998, 2004; Stan et al., 2004;
Timmermans et al., 1999) or two-photon Raman transitions (Theis et al., 2004;
Wynar et al., 2000). Indeed, some of the most exciting recent developments in
the physics of ultra-cold atoms are related to the coherent coupling of atoms
to ultra-cold molecules by means of Feshbach resonances (Dürr et al., 2004;
Regal et al., 2003), and photo-association (Kerman et al., 2004; Wynar et al.,
2000). Both bosons and fermions have been successfully converted into molecules. In both cases BEC of molecules has been observed (Donley et al., 2002;
Greiner et al., 2003; Jochim et al., 2003; Zwierlein et al., 2003), and the longstanding question of the BEC-BCS crossover is being investigated experimentally
and theoretically in those systems (Bartenstein et al., 2004; Holland et al., 2001;
Ohashi and Griffin, 2002; Regal et al., 2004; Timmermans et al., 2001; Zwierlein
2]
QUANTUM OPTICS OF ULTRA-COLD MOLECULES
153
et al., 2004). Other developments with close connections with quantum optics include the trapping of atoms in optical lattices (Greiner et al., 2002a, 2002b; Jaksch
et al., 1998), which play a role closely related to a high-Q resonator in cavity QED
(Search et al., 2004; Walther, 1992) and leads in addition to fascinating connections with condensed matter physics and quantum information science.
With so many close connections between the physics of quantum-degenerate
atomic and molecular systems and quantum optics, it is natural and wise to
go back to the masters of that field to find inspiration and guidance, and this
is why Herbert Walther’s intellectual imprint remains so important. This brief
review illustrates this point with three examples. Section 2 shows that the conversion of pairs of fermions into molecules in a double-well potential can be
described by a generalized Jaynes–Cummings model. Using this equivalence,
we show that the dynamics of the molecular field at each site can be mapped
to that of a micromaser, one of Herbert Walther’s most remarkable contributions (Meschede et al., 1985). Section 3 further expands on the mapping of
ultra-cold fermion pairs onto two-level atoms to study the role of fluctuations
in the association and dissociation rates of ultra-cold molecules. We show that
this system is closely related to Dicke superradiance, and with this analogy as
a guide, we discuss how the passage time fluctuations depend sensitively on the
initial state of the system. In a third example, inspired by Herbert Walther’s work
on photon statistics and antibunching (Brattke et al., 2001; Krause et al., 1989;
Rempe et al., 1990; Rempe and Walther, 1990) Section 4 analyzes how the statistics of their constituent atoms affects the counting statistics of molecules formed
by photo-association. We compare the three cases where the molecules are formed
from a BEC, an ultra-cold Fermi gas and a Fermi system with a superfluid component. The concept of quantum coherence developed by R.J. Glauber and exploited
in many situations by H. Walther and his coworkers, in particular in their studies of resonance fluorescence, are now applied to characterizing the statistical
properties of the coupled atom–molecule system. Finally, Section 5 further elaborates on these ideas to probe spatial correlations and coherent properties of atomic
samples, and we find that the momentum distribution of the molecules contains
detailed information about the second-order correlations of the initial atomic gas.
2. Molecular Micromaser
Ultra-cold atoms and molecules trapped in optical lattices provide an exciting
new tool to study a variety of physics problems. In particular, they provide remarkable connections with the condensed matter of strongly correlated systems
and with quantum information science, a very well controlled environment to
study processes such as photo-association (Ryu et al., 2005), and, from a pointof-view more directly related to quantum optics, can be thought of as matter–
154
D. Meiser et al.
[2
wave analog of photons trapped in high-Q cavities. In particular, the high degree
of real-time control of the system parameters offers the opportunity to directly
experimentally study some of the long-standing questions of condensed matter
physics, such as the ground state structure of certain models and many-body
dynamic properties (Jaksch and Zoller, 2005). The coherent formation of molecules in an optical lattice via either Feshbach resonances and two-photon Raman photo-association has been studied both theoretically (Jaksch et al., 2002;
Damski et al., 2003; Esslinger and Molmer, 2003; Molmer, 2003; Moore and
Sadeghpour, 2003) and experimentally (Köhl et al., 2005; Rom et al., 2004;
Ryu et al., 2005; Stöferle et al., 2005). In particular, the experiment of Ref. (Ryu
et al., 2005) observed reversible and coherent Rabi oscillations in a gas of coupled
atoms and molecules.
The idea of the molecular micromaser (Search et al., 2003) relies on the observation that, as a consequence of Fermi statistics, the photo-association of fermionic atoms into bosonic molecules can be mapped onto a generalized Jaynes–
Cummings model. This analogy allows one to immediately translate many of
the results that have been obtained for the Jaynes–Cummings model to atom–
molecule systems. In addition, the molecular system possesses several properties
that have no counterpart in the quantum optics analog, giving rise to interesting generalizations of the original micromaser problem (Filipowicz et al., 1986;
Guzman et al., 1989; Meschede et al., 1985; Rempe et al., 1990). One of these
new features is the inter-site tunneling of atoms and molecules between adjacent
lattice sites, leading to a system that can be thought of as an array of molecular
micromasers (Search et al., 2004).
To see how this works, rather than treating a full lattice potential we consider the dynamics of the molecular field in the simpler model of a coupled
atom–molecule system in a double-well potential. We first show that inter-well
tunneling enhances number fluctuations and eliminates trapping states in a manner similar to thermal fluctuations. We also examine the buildup of the relative
phase between the two molecular states localized at the two wells due to the combined effect of inter-well tunneling and two-body collisions. We identify three
regimes, characterized by different orders of magnitude of the ratio of the twobody collision strength to the inter-well tunneling coupling. The crossover of the
non-equilibrium steady state from a phase-coherent regime to a phase-incoherent
regime is closely related to the phase locking of condensates in Josephson-type
configurations (Leggett, 2001), while we consider an open quantum system with
incoherent pump and molecular loss which results in a dissipative steady state.
2.1. M ODEL
We consider a mixture of two hyperfine spin states |σ =↑, ↓ of fermionic atoms
of mass mf trapped in a double-well potential at temperature T = 0, which can be
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QUANTUM OPTICS OF ULTRA-COLD MOLECULES
155
coherently combined into bosonic molecules of mass mb via two-photon Raman
photo-association. If the band-gap of the lattice potential is much larger than any
other energy scale in the system, the fermions and molecules occupy only the
lowest energy level of each well and the number of fermions of a given spin state
is at most one in each well.
In the tight binding approximation, the effective Hamiltonian describing the
coupled atom–molecule system is
Hˆ 0i + Hˆ I i + Hˆ T ,
Hˆ =
(1)
i=l,r
where
1
Hˆ 0i = h(ω
¯ b + δ)nˆ i + h¯ ωf (nˆ ↑i + nˆ ↓i ) + h¯ Ub nˆ i (nˆ i − 1)
2
n
ˆ
n
ˆ
+ h¯ Ux nˆ i (nˆ ↑i + nˆ ↓i ) + hU
¯ f ↑i ↓i ,
Hˆ I i = hχ(t)
bˆi† cˆ↑i cˆ↓i + H.c.,
¯
†
†
cˆ↑r + cˆ↓l
cˆ↓r + H.c.
Hˆ T = −h¯ Jb bˆl† bˆr − h¯ Jf cˆ↑l
(2)
(3)
(4)
Here cˆσ i and bˆi , i = l, r, are the annihilation operators of fermionic atoms and
bosonic molecules in the left (l) and right (r) wells, respectively. The corresponding number operators nˆ i = bˆi† bˆi and nˆ σ i = cˆσ† i cˆσ i have eigenvalues ni and nσ i ,
respectively, and hω
¯ b and h¯ ωf are the energies of the molecules and atoms in the
isolated wells.
The terms proportional to Ub , Ux , and Uf in Hˆ 0i describe on-site two-body interactions between molecules, between atoms and molecules, and between atoms,
respectively. The interaction Hamiltonian Hˆ I i describes the conversion of atoms
into molecules via two-photon Raman photo-association. The photo-association
coupling constant χ(t) is proportional to the far off-resonant two-photon Rabi frequency associated with two nearly co-propagating lasers (Heinzen et al., 2000),
and δ is the two-photon detuning between the lasers and the energy difference of
the atom pairs and the molecules. The tunneling between two wells is described
by the parameters Jb and Jf in the tunneling Hamiltonian Hˆ T .
The molecular field is “pumped” by a train of short photo-association pulses of
duration τ , separated by long intervals T τ during which the molecules are
subject only to two-body collisions and quantum tunneling between the potential
wells, as well as to losses due mainly to three-body collisions and collisional relaxation to low-lying vibrational states. In the absence of inter-well tunneling, this
separation of time scales leads to a situation very similar to that encountered in
the description of traditional micromasers, with the transit of individual two-level
atoms through the micromaser cavity replaced by the train of photo-association
pulses.
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D. Meiser et al.
[2
The dynamics of the molecular field in the double-well system is governed by
the following four mechanisms:
(i) Coherent pumping by injection of pairs of fermionic atoms inside the
double-well. This process is the analog of the injection of two-level atoms into
a micromaser cavity. The injection of pairs of fermionic atoms into the doublewell potential can be accomplished, e.g., by Raman transfer of atoms from an
untrapped internal state (Jaksch et al., 1998; Mandel et al., 2004). This results in
the pumping of fermions into the double well at a rate Γ (Search et al., 2002a).
We assume that for times T Γ −1 , a pair of fermions has been transferred to the
two wells with unit probability, that is, the state of the trapped fermions in well i
is
† †
cˆ↑i |0.
|ei = cˆ↓i
(5)
(ii) Molecular damping, which is the analog of cavity damping. During the time
intervals T when the photo-association lasers are off, the molecular field decays
at rate γ (Search et al., 2003). The decay of the molecules is due to Rayleigh
scattering from the intermediate molecular excited state, three-body inelastic collisions between a molecule and two fermions, and collisional relaxation from a
vibrationally excited molecular state to deeply bound states. These loss mechanisms can be modeled by a master equation (see, e.g., Meystre and Sargent III,
1999; Miyakawa et al., 2004; Scully and Zubairy, 1997).
(iii) The application of a train of photo-association pulses. This mechanism is
formally analogous to the Jaynes–Cummings interaction between the single-mode
field and a sequence of two-level atoms traveling through the microwave cavity
in the conventional micromaser. As already mentioned, we assume that these are
square pulses of duration τ and period T + τ , with τ much shorter than all other
−1
, γ −1 . This assumption is essential if we are
time scales in this model, τ Jb,f
to neglect damping and tunneling while the photo-association fields are on. The
change in the molecular field resulting from atom–molecule conversion is given
by
Fi (τ )ρˆb ≡ Tra Ui (τ )ρˆab (t)Ui† (τ ) ,
(6)
where ρˆab is the total density operator of the atom–molecule system and Tra [ ]
denotes the trace over the atomic variables, Ui (τ ) = exp (−i hˆ i τ/h¯ ) being the
evolution operator for a single-well Hamiltonian,
hˆ i = Hˆ 0i + Hˆ I i .
The key observation that allows us here and below to build a bridge from the cold
atoms and molecular system to quantum optics systems is that by means of the
mapping (Anderson, 1958)
σˆ −i = c↑i c↓i ,
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QUANTUM OPTICS OF ULTRA-COLD MOLECULES
157
† †
σˆ +i = c↓i
c↑i ,
†
†
c↑i + c↓i
c↓i − 1,
σˆ zi = c↑i
(7)
the atomic degrees of freedom take the form of a fictitious two-level system. The
operators σˆ +i , σˆ −i , and σˆ zi can be interpreted as the raising and lowering operators of the fictitious two-level atom and the population difference, respectively.
The single-well Hamiltonian takes the form
hˆ i = h¯ (ωb + Ux ) nˆ i + h¯ (ωf + Ux nˆ i )σˆ zi
h¯
+ h¯ χ(t)bˆi† σˆ −i + χ ∗ (t)bˆi σˆ +i + Ub nˆ i (nˆ i − 1),
(8)
2
where we have dropped constant terms and we have redefined ωb and ωf according to ωb + δ → ωb and ωf + Uf /2 → ωf .
The Hamiltonian hˆ i is Jaynes–Cummings-like, and for χ = const, the resulting dynamics can be determined within the two-state manifolds of each well
{|ei , ni , |gi , ni + 1} by a simple extension of the familiar solution to the Jaynes–
Cummings model. Within each manifold the system undergoes Rabi oscillations.
Since tunneling is neglected during the photo-association steps, the two wells
are independent of each other and identical to each other. The resultant molecular
gain is then modeled by independent coarse-grained equations of motion for the
reduced density matrices of each molecular mode.
(iv) The unitary time evolution of the molecular field under the influence of
two-body collisions and quantum tunneling, a process absent in conventional micromasers. During the intervals T it is governed by
i
∂ ρˆb
= − Hˆ b , ρˆb ,
(9)
∂t
h¯
where the Hamiltonian
Ub
Hˆ b = −h¯ Jb bˆl† bˆr + bˆr† bˆl + h¯
(10)
(nˆ l − nˆ r )2
4
contains tunneling and collisions. In Eq. (10), we have neglected terms that are
functions only of Nˆ = nˆ l + nˆ r , a step justified as long as the initial density matrix
is diagonal in the total number of molecules in the two wells.
Combining the coherent and incoherent processes (i) to (iv), we obtain the full
evolution of the molecular field
1 i
∂ ρˆb
Fi (τ ) − Ii ρˆb − [Hb , ρˆb ],
Li ρˆb +
=
(11)
∂t
T
h¯
l,r
l,r
where ρˆb is the reduced density matrix of the molecules. The initial condition for
the molecules is taken to be the vacuum state. Because the molecular pumping
and decay is the same in both wells, the density matrix ρ remains diagonal in the
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D. Meiser et al.
[2
total number of molecules in the two wells for all times. This is a generalization
of the micromaser result that the photon density matrix will remain diagonal if it
is initially diagonal in a number state basis (Filipowicz et al., 1986).
2.2. R ESULTS
The master equation describing the molecular micromaser dynamics contains six
independent parameters: the number of photo-association
√ cycles per lifetime of
the molecule, Nex = 1/γ T ; the “pump parameter” Θ = Nex |χ|τ ; the two-body
collision strength and tunneling coupling strength per decay rate, ub = Ub /γ and
tJ = Jb /γ ; and finally, the detuning parameter η = (2ωf − ωb )/2|χ| and the
nonlinear detuning parameter β ≡ (2Ux − Ub )/2|χ|.
In our model, the atomic and molecular level separations in the wells are required to be much larger than the relevant interaction energies,
h¯ ωb Ub nˆ i nˆ i − 1 , |χ| nˆ i ,
(12)
nˆ i being the average number of molecules in well i. A comparison with actual
experimental parameters (Greiner et al., 2002a; Jaksch et al., 1998; Miyakawa
et al., 2004) shows that these conditions are satisfied as long as the number of
molecules does not exceed 10. In addition, the neglect of inter-well tunneling and
damping effects during the photo-association pulses requires that
τ Jb−1 , γ −1 .
(13)
This condition is satisfied in typical experiments.
In the remainder of this section, we discuss the dynamics of the molecular field
obtained by direct numerical integration of the master equation with a Runge–
Kutta algorithm until a dissipative steady state is reached. For simplicity, we
confine our discussion to exact resonance only, η = β = 0, and a fixed value
of Nex = 10.
2.2.1. Single-Well Molecular Statistics
We first discuss the statistics of a single-well molecular mode, which is given by
tracing over the full density matrix with respect to degrees of freedom of the other
localized mode as
ρ(nl , nr ; nl , nr ).
P (nl{r} ) = Trr{l} ρ(nl , nr ; ml , mr ) =
(14)
nr{l}
We note that off-diagonal elements of the density matrix for a single well are zero.
Since the initial state of the molecules in each well is the same, i.e. the vacuum
state, and Hˆ b is invariant with respect to the interchange l ↔ r, the molecule
statistics for left and right wells are identical, P (nl ) = P (nr ) ≡ P (n).
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QUANTUM OPTICS OF ULTRA-COLD MOLECULES
159
F IG . 1. nˆ i versus Θ/π for ub = 0 and Nex = 10, and for (a) tJ = 0 and (b) tJ = 5.
Figure 1 shows the steady-state average number nˆ i plotted as a function of
the pump parameter Θ for ub = 0, Nex = 10. In the absence of inter-well
tunneling, corresponding to Fig. 1(a), the result reproduces that of conventional
micromasers, with a “lasing” threshold behavior at around Θ ≈ 1 and an abrupt
jump to a higher mean occupation at about the first transition point, Θ 2π.
The former effect is not affected by the tunneling coupling as shown in Fig. 1(b).
However, the latter abrupt jump disappears in the presence of inter-well tunneling.
This is because the coupling to the other well leads to fluctuations in the number
of molecules in each well and has an effect similar to thermal fluctuations in the
traditional micromaser theory. The enhancement of fluctuations can also be seen
in Fig. 2 where the Mandel Q-parameter
Q=
nˆ 2i − nˆ i 2
−1
nˆ i is plotted as a function of Θ.
It is known that in the usual micromaser the sharp resonance-like dips in nˆ i and Q are attributable to trapping states, which√are characterized by a sharp photon number. For the specific value of Θ = 5 π, as shown in Fig. 3(a), the
number probability does not reach beyond number state |ni = 1 in the case of
tJ = 0. As shown in Fig. 3(b), the tunneling coupling makes possible transitions
into higher number states and eliminates the trapping state in a manner similar to
thermal fluctuations in the conventional micro maser.
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D. Meiser et al.
[2
F IG . 2. Q parameter versus Θ/π for ub = 0 and Nex = 10, and for (a, solid line) tJ = 0 and
(b, dashed line) tJ = 5.
F IG . 3. Molecular number statistics P (ni ) for Θ =
and (b) tJ = 5.
√
5 π , ub = 0 and Nex , and for (a) tJ = 0
2]
QUANTUM OPTICS OF ULTRA-COLD MOLECULES
161
2.2.2. Phase Coherence between Two Micromasers with Tunneling Coupling
So far we have discussed the single-well molecule statistics and how it is affected by inter-well tunneling. Now we turn to a more detailed discussion of
the phase coherence between the two localized modes. It is very useful to divide the parameter space of the ratio of the two-body collision strength to the
inter-well tunneling coupling into three regimes (Leggett, 2001): “Rabi-regime”
(ub /tJ Nˆ −1 ); “Josephson-regime” (Nˆ −1 ub /tJ Nˆ ); and “Fockregime” (Nˆ ub /tJ ); where Nˆ denotes the average total molecule number.
The analysis of the relative coherence of the molecular fields in the two wells
is most conveniently discussed in terms of the angular momentum representation
Jˆ+ = Jˆx + i Jˆy = bˆl† bˆr ,
Jˆ− = Jˆx − i Jˆy = bˆr† bˆl ,
1
Jˆz = bˆl† bˆl − bˆr† bˆr ,
2
Nˆ Nˆ
2
ˆ
+1 .
J =
2 2
(15)
The symmetry of the density matrix with respect to the two wells furthermore
implies that Jˆz = Jˆy = 0. The first-order coherence between the molecular
fields in the left and right potential wells is then given by Jˆx . Figure 4 shows the
normalized steady-state first-order coherence Jˆx /nˆ j as a function of ub /tJ for
Θ = π and tJ = 2.5. Jˆx is suppressed in both the Rabi and Fock regimes and
has an extremum at |ub |/tJ ∼ 0.6. In the Fock regime, |ub |/tJ Nˆ ∼ 10, the
F IG . 4. Jˆx /nj versus ub /tJ for Θ = π and tJ = 2.5.
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D. Meiser et al.
[2
non-linearity in Hˆ b dominates and reduces the coherence between the localized
states of each well. We note that the average occupation numbers for each well
are relatively unaffected by ub /tJ , with nˆ j = Nˆ /2 = 4.78–4.87 for |ub |/tJ =
102 –10−2.5 .
The reason why the first-order coherence is suppressed in the weak coupling
limit, ub = 0, can be understood as follows. The expectation value Jˆx corˆ
responds to
√ the difference in occupation numbers
√ between the in-phase, bs =
ˆ
ˆ
ˆ
ˆ
ˆ
(bl + br )/ 2, and out-of-phase, ba = (bl − br )/ 2, states of the localized states
of each well, Jˆx = bˆs† bˆs − bˆa† bˆa . Since the bandwidth of the photo-association
pulse is larger than their energy splitting, 1/τ Jb , those states are equally populated, resulting in Jˆx = 0 for ub = 0. Thus, the origin of the mutual coherence
between two molecular modes is due solely to two-body collisions. Furthermore,
we remark that a semiclassical treatment results in Jˆx = 0 for all times and all
values of ub /tJ (Miyakawa et al., 2004). Hence, we conclude that the build-up of
Jˆx is a purely quantum-mechanical effect due to quantum fluctuations.
The phase distribution of the two wells can be studied using the Pegg–
Barnett phase states (Barnett and Pegg, 1990; Javanainen and Ivanov, 1999;
Luis and Sanchez-Soto, 1993; Pegg and Barnett, 1988). Since the density matrix
is diagonal in the total number of molecules it is sufficient to consider the relative
phase. Figure 5 shows the time evolution of the relative phase distribution in three
different regimes: (a) Rabi, ub /tJ = 0.0032, (b) Josephson, ub /tJ = 0.5623,
and (c) Fock ub /tJ = 56.23, for Θ = π, tJ = 2.5. Since the vacuum state
is taken as the initial state, the relative phase at t = 0 is randomly distributed,
P (φn ) = const.
In the Rabi regime, corresponding to Fig. 5(a), bimodal phase distribution with
peaks around both 0 and ±π builds up in the characteristic time γ −1 needed to
reach a steady state (Filipowicz et al., 1986). In the Josephson regime, the relative
phase locks around 0 (±π), for repulsive (attractive) two-body interactions, see
Fig. 5(b). In contrast to these two regimes, in the Fock regime the relative phase
distribution becomes almost random for all times, and the localized modes in the
two wells evolve independently of each other.
F IG . 5. Time evolution of P (φn ) for Θ = π , tJ = 2.5 and for (a) ub /tJ = 0.0032,
(b) ub /tJ = 0.5623, (c) ub /tJ = 56.23.
3]
QUANTUM OPTICS OF ULTRA-COLD MOLECULES
163
The three regimes of phase distributions correspond to different orders of magnitude of the ratio ub /tJ . The crossover of the non-equilibrium steady state from
a phase-coherent regime to the random-phase situation is reminiscent of the
superfluid-Mott insulator phase transition for the ground state of an optical lattice (Fisher et al., 1989; Jaksch et al., 1998). Since we consider just two sites,
however, there is no sharp transition between these regimes.
3. Passage Time Statistics of Molecule Formation
We now turn to a second-example that illustrates the understanding of the dynamics of quantum-degenerate atomic and molecular systems that can be gained from
quantum optics analogies. Here, we consider the first stages of coherent molecular
formation via photo-association. Since in such experiments the molecular field is
typically in a vacuum initially, it is to be intuitively expected that the initial stages
of molecule formation will be strongly governed by quantum noise, hence subject to large fluctuations. One important way to characterize these fluctuations is
in terms of the so-called passage time, which is the time it takes to produce, or
dissociate, a predetermined number of molecules. Quantum noise results in fluctuations in that time, whose probability distribution can therefore be used to probe
the fluctuations in the formation dynamics.
Because of the analogy between pairs of fermionic atoms and two-level systems
that we already exploited in the discussion of the molecular micromaser, one can
expect that the problem at hand is somewhat analogous to spontaneous radiation
from a sample of two-level atoms, the well-know problem of superradiance. In
this section we show that this is indeed the case, and use this analogy to study the
passage time statistics of molecular formation from fermionic atoms.
We consider again a quantum-degenerate gas of fermionic atoms of mass mf
and spin σ =↑, ↓, coupled coherently to bosonic molecules of mass mb = 2mf
and zero momentum via photo-association. Neglecting collisions between fermions and assuming that for short enough times the molecules occupy a single-mode
of the bosonic field, this system can be described by the boson–fermion model
Hamiltonian
1
†
†
H =
cˆk↑ + cˆ−k↓
cˆ−k↓
h¯ ωk cˆk↑
2
k
†
† †
bˆ † cˆk↑ cˆ−k↓ + bˆ cˆ−k↑
,
cˆk↓
+ hω
(16)
¯ b bˆ bˆ + h¯ χ
k
bˆ † , bˆ
are molecular bosonic creation and annihilation operators and
where
†
, cˆkσ are fermionic creation and annihilation operators describing atoms of
cˆkσ
momentum hk
¯ and spin σ . The first and second terms in Eq. (16) describe the
164
D. Meiser et al.
[3
kinetic energy h¯ ωk /2 = h¯ 2 k 2 /(2mf ) of the atoms and the detuning energy of the
molecules respectively, and the third term describes the photo-association of pairs
of atoms of opposite momentum into molecules.
Introducing the pseudo-spin operators (Anderson, 1958) analogous to Eq. (7),
1 †
†
cˆ−k↓ − 1 ,
cˆ cˆk↑ + cˆ−k↓
2 k↑
†
†
= (σˆ k− )† = cˆ−k↓
cˆk↑
,
σˆ kz =
σˆ k+
(17)
the Hamiltonian (16) becomes, within an unimportant constant (Barankov and
Levitov, 2004; Meiser and Meystre, 2005),
bˆ † σˆ k− + bˆ σˆ k+ .
h¯ ωk σˆ kz + h¯ ωb bˆ † bˆ + hχ
H =
(18)
¯
k
k
This Hamiltonian is known in quantum optics as the inhomogeneously broadened (or non-degenerate) Tavis–Cummings model (Tavis and Cummings, 1968).
It describes the coupling of an ensemble of two-level atoms to a single-mode
electromagnetic field. Hence the mapping (17) establishes the formal analogy between the problem at hand and Dicke superradiance, with the caveat that we are
dealing with a single bosonic mode (Andreev et al., 2004; Barankov and Levitov,
2004; Javanainen et al., 2004; Meiser and Meystre, 2005; Miyakawa and Meystre,
2005; Pazy et al., 2005). Instead of real two-level atoms, pairs of fermionic atoms
are now described as effective two-level systems whose ground state corresponds
to the absence of a pair, |gk = |0k↑ , 0−k↓ and the excited state to a pair of atoms
of opposite momenta, |ek = |1k↑ , 1−k↓ , in close analogy to the treatment of the
atoms in the previous section.
The initial condition consists of the molecular field in the vacuum state and a
filled Fermi sea of atoms
σˆ k+ |0,
|F =
(19)
k|kF |
where kF is the Fermi momentum. As such, the problem at hand is in direct analogy to the traditional superradiance problem where one starts from an ensemble
of excited two-state atoms, as expected from our previous comments. Later on we
will also consider an initial state containing only molecules and no atoms. This
is an important extension of the traditional Dicke superradiance system, where
the two-level atoms are coupled to all modes of the photon vacuum a situation,
thereby precluding the possibility of an initial state containing a single, macroscopically occupied photon mode unless the system us prepared in a high-Q
cavity.
We assume from now on that the inhomogeneous broadening due to the spread
in atomic kinetic energies can be ignored. This so-called degenerate approximation is justified provided that the kinetic energies are small compared to the
3]
QUANTUM OPTICS OF ULTRA-COLD MOLECULES
165
atom–molecule coupling energy, β = F /(h¯ χ) 1, where F is the Fermi energy. It is the analog of the homogeneous broadening limit of quantum optics,
and of the Raman–Nath approximation in atomic diffraction. A comparison with
typical experimental parameters (Heinzen et al., 2000) shows that the degenerate
approximation is justified if the number of atoms does not exceed ∼102 –103 (Uys
et al., 2005).
Limiting thus our considerations to small atomic samples, we approximate all
ωk ’s by ωF and introduce the collective pseudo-spin operators
Sˆz =
(20)
σˆ kz ,
Sˆ ± =
σˆ k± ,
k
k
obtaining the standard Tavis–Cummings Hamiltonian (Miyakawa and Meystre,
2005; Tavis and Cummings, 1968)
H = h¯ ωF Sˆz + h¯ ωb bˆ † bˆ + h¯ χ bˆ Sˆ + + bˆ † Sˆ − .
(21)
2
This Hamiltonian conserves the total spin operator Sˆ . The total number of atoms
is twice the total spin and hence is also a conserved quantity. Sˆz measures the
difference in the numbers of atom pairs and molecules.
Equation (21) can be diagonalized numerically with reasonable computation
times even for relatively large numbers of atoms. One can, however, gain significant intuitive insight in the underlying dynamics by finding operator equations
of motion and then treating the short-time molecular population semiclassically,
nˆ b → nb . To this end we introduce the “joint coherence” operators
Tˆx = bˆ Sˆ + + bˆ † Sˆ − /2,
Tˆy = bˆ Sˆ + − bˆ † Sˆ − /2i,
(22)
and find the Heisenberg equations of motion
n˙ˆ b = −2χ Tˆy ,
T˙ˆ x = δ Tˆy ,
T˙ˆ y = −δ Tˆx − χ 2Sˆz nˆ b + Sˆ + Sˆ − ,
(23)
(24)
(25)
where δ = ωb − ωF , so that 2χ Tˆx + δ nˆ b is a constant of motion.
In the following, we confine our discussion to the case of δ = 0 for simplicity.
We thus neglect the contribution of Tˆx in Eq. (25). In order to better understand
the short time dynamics we reexpress Sˆ + Sˆ − as
Sˆ + Sˆ − = −nˆ 2b + (2S − 1)nˆ b + N.
This shows that the operator Sˆ + Sˆ −
(26)
is non-vanishing when the molecular field is in
a vacuum and hence can be interpreted as a noise operator. Indeed Eqs. (23)–(25)
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D. Meiser et al.
[3
F IG . 6. Short-time dynamics of nˆ b . From left to right, the curves give the linearized solution
(28) and the full quantum results for N = 500, N = 250, and N = 100, respectively. Figure taken
from Ref. (Uys et al., 2005).
show that the buildup of the molecular field is triggered only by noise if nˆ b = 0
initially. By keeping only the lowest-order terms in nˆ b we can eliminate Tˆy to
obtain the differential equation
n¨ˆ b ≈ 2N χ 2 (2nˆ b + 1)
which, for our initial state, may be solved to yield
√ nˆ b (t) ≈ sinh2 χ N t .
(27)
(28)
Figure 6 compares the average molecule number nˆ b obtained this way, with the
full quantum solution obtained by direct diagonalization of the Hamiltonian (21)
for various values of N . The semiclassical approach agrees within 5% of the full
quantum solution until about 20% of the population of atom pairs has been converted into molecules in all cases.
Next we turn to the passage time statistics. In Fig. 7 we show (solid line)
the distribution of times required to produce a normalized molecule number
nref
b /N = 0.05 from a sample initially containing N = 500 pairs of atomic fermions, as found by direct diagonalization of the Hamiltonian (21). This distribution
differs sharply from its counterpart for the reverse process of photodissociation
from a molecular condensate into fermionic atom pairs, which is plotted as the
dashed line in Fig. 7. In contrast to photo-association, this latter process suffers
significantly reduced fluctuations.
To understand the physical mechanism leading to this reduction in fluctuations
we again turn to our short time semi-classical model. Within this approximation,
the Heisenberg equations of motion (23)–(24) can be recast in the form of a Newtonian equation (Miyakawa and Meystre, 2005)
d 2 nb
dU (nb )
=−
,
dnb
dt 2
(29)
3]
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167
F IG . 7. Passage time distribution for converting 5% of the initial population consisting of only
atoms (molecules) into molecules (atoms) for N = 500. For initially all atoms: solid line, for initially
all molecules: dashed line.
F IG . 8. Effective potential for a system with N 1. The circle (square) corresponds to an initial
state with all fermionic atoms (molecules). The part of the potential for nb < 0 is unphysical. Figure
taken from Ref. (Uys et al., 2005).
where the cubic effective potential U (nb ) is plotted in Fig. 8. (Note we have now
kept all orders in nb .) In case the system is initially composed solely of fermionic atoms, nb (0) = 0, the initial state is dynamically unstable, with fluctuations
having a large impact on the build-up of nb . In contrast, when it consists initially
solely of molecules, nb = N , the initial state is far from the point of unstable
equilibrium, and nb simply “rolls down” the potential in a manner largely insensitive to quantum fluctuations. This is a consequence of the fact that the bosonic
initial state provides a mean field that is more amenable to a classical description.
Hence, while the early stages of molecular dimer formation from fermionic atoms
are characterized by large fluctuations in formation times that reflect the quantum
fluctuations in the initial atomic state, the reverse process of dissociation of a condensate of molecular dimers is largely deterministic. The diminished fluctuations
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D. Meiser et al.
[4
in this reversed process is peculiar to the atom–molecule system and not normally
considered in the quantum optics analog of Dicke superradiance.
4. Counting Statistics of Molecular Fields
An important quantum mechanical characteristic of a quantum field is its counting
(or number) statistics. In this section we show how the similarity of the coherent
molecule formation with quantum optical sum-frequency generation can be used
to determine the counting statistics of the molecular field. In particular we show
how the counting statistics depends on the statistics of the atoms from which the
molecules are formed. Besides being interesting in its own right, such an analysis
is crucial for an understanding of several recent experiments that used a “projection” onto molecules to detect BCS superfluidity in fermionic systems (Regal et
al., 2004; Zwierlein et al., 2004). Our work shows that the statistical properties
of the resulting molecular field indeed reflect properties of the initial atomic state
and are a sensitive probe for superfluidity.
As before, we restrict our discussion to a simple model in which all the molecules are generated in a single mode. We use time dependent perturbation theory
to calculate the number of molecules formed after some time t, n(t), as well as
the equal-time second-order correlation function g (2) (t, t). We also integrate the
Schrödinger equation numerically for small numbers of atoms, which allows us
to calculate the complete counting statistics Pn .
4.1. BEC
Consider first a cloud of weakly interacting bosons well below the condensation
temperature Tc . It is a good approximation to assume that all atoms are in the
condensate, described by the condensate wave function ψ0 (x). The coupled system of atoms and molecules is described by the effective two-mode Hamiltonian
(Anglin and Vardi, 2001; Javanainen and Mackie, 1999)
† 2
†2
Hˆ BEC = h¯ δ bˆ † bˆ + hχ
(30)
¯ bˆ cˆ + bˆ cˆ ,
ˆ bˆ † and c,
ˆ cˆ† are the bosonic annihilation and creation operators for the
where b,
molecules and for the atoms in the condensate, respectively, δ is the detuning
between the molecular and atomic level, and h¯ χ is the effective coupling constant.
Typical experiments start out with all atoms in the condensate and no molecules, corresponding to the initial state,
cˆ†Na
|0,
|ψ(t = 0) = √
Na !
(31)
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169
F IG . 9. Number statistics of molecules formed from a BEC with Nmax = 30 and δ = 0.
where Na = 2Nmax is the number of atoms, Nmax is the maximum possible number of molecules and |0 is the vacuum of both molecules and atoms. We can
numerically solve the Schrödinger equation for this problem in a number basis
and from that solution we can determine the molecule statistics Pn (t). The results
of such a simulation are illustrated in Fig. 9, which shows Pn (t) for 30 initial
atom pairs and δ = 0. Starting in the state with zero molecules, a wave-packetlike structure forms and propagates in the direction of increasing n. Near Nmax
the molecules begin to dissociate back into atom pairs.
We can gain some analytical insight into the short-time dynamics of molecule
formation by using first-order perturbation theory (Kozierowski and Tana´s, 1977;
Mandel, 1982). We find for the mean molecule number
n(t) = (χt)2 2Nmax (2Nmax − 1) + O (χt)2
(32)
and for the second factorial moment
−2 2
.
+ O Nmax
g (2) (t1 , t2 ) = 1 −
Nmax
(33)
For Nmax large enough we have g (2) (t1 , t2 ) → 1, the value characteristic of a
Glauber coherent field. From g (2) and n(t) we also find the relative width of the
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D. Meiser et al.
molecule number distribution as
(nˆ − n)2 = g (2) + n−1 − 1.
n
[4
(34)
It approaches n−1/2 in the limit of large Nmax , typical of a Poisson distribution.
This confirms that for short enough times, the molecular field is coherent in the
sense of quantum optics.
4.2. N ORMAL F ERMI G AS
We now turn to the case of photo-association from two different species of noninteracting ultra-cold fermions. The two species are again denoted by spin up and
down. At T = 0, the atoms fill a Fermi sea up to an energy μ. Weak repulsive
interactions give rise only to minor quantitative modifications that we ignore. We
refer to this system of non-interacting Fermions as a normal Fermi gas (NFG)
(Landau et al., 1980).
As before we assume that atom pairs are coupled only to a single mode of
the molecular field, which we assume to have zero momentum for simplicity.
Then, using the mapping to pseudo spins Eq. (17) we find that the system is
again described by the inhomogeneously broadened Tavis–Cummings Hamiltonian Eq. (18). However, in contrast to the previous case, we do not assume that
the fermionic energies are approximately degenerate, in order to be able compare
the results to the BCS case, where the kinetic energies are essential.
Figure 10 shows the molecule statistics obtained this way. The result is clearly
both qualitatively and quantitatively very different from the case of molecule
formation from an atomic BEC. From the Tavis–Cummings model analogy we
expect that for short times the statistics of the molecular field should be chaotic,
or “thermal”, much like those of a single-mode chaotic light field. This is because each individual atom pair “emits” a molecule independently and without
any phase relation with other pairs. That this is the case is illustrated in the inset
of Fig. 10, which fits the molecule statistics at selected short times with chaotic
distributions of the form
e−n/n
Pn,thermal = −n/n .
ne
(35)
The increasing ‘pseudo-temperature’ n corresponds to the growing average
number of molecules as a function of time.
As before we determine the short-time properties of the molecular field in firstorder perturbation theory. We find for the mean number of molecules
n(t) = (χt)2 2Na .
(36)
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171
F IG . 10. Number statistics of molecules formed from a normal Fermi gas. This simulation is for
Na = 20 atoms, the detuning is δ = 0, the Fermi energy is μ = 0.1h¯ χ and the momentum of the
ith pair is |ki | = (i − 1)2kF /(Na /2 − 1). The inset shows fits of the number statistics to thermal
distributions for various times as marked by the thick lines in the main figure.
It is proportional to Na , in contrast to the BEC result, where n was proportional
to Na2 , see Eq. (32). This is another manifestation of the independence of all the
atom pairs from each other: While in the BEC case the molecule production is a
collective effect with contributions from all possible atom pairs adding constructively, there is no such collective enhancement in the case of Fermions. Each atom
can pair up with only one other atom to form a molecule. For the second factorial
moment we find
1
g (2) (t1 , t2 ) = 2 1 −
(37)
2Na
which is close to two, typical of a chaotic or thermal field.
4.3. F ERMI G AS WITH S UPERFLUID C OMPONENT
Unlike repulsive interactions, attractive interactions between fermions have a profound impact on molecule formation. It is known that such interactions give rise
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D. Meiser et al.
[4
F IG . 11. Number statistics of molecules formed from a Fermi gas with pairing correlations. For
this simulation the detuning is δ = 0, the Fermi energy is μ = 0.1g and the background scattering
strength is V = 0.03χ resulting in Na ≈ 9.4 atoms and a gap of ≈ 0.15χ. The momenta of the
atom pairs are distributed as before in the normal Fermi gas case.
to a Cooper instability that leads to pairing and drastically changes the qualitative
properties of the atomic system. The BCS reduced Hamiltonian is essentially the
inhomogeneously broadened Tavis–Cummings Hamiltonian (18) with an additional term accounting for the attractive interactions between atoms (Kittel, 1987),
σˆ k+ σˆ k− . (38)
bˆ † σˆ k− + bˆ σˆ k+ − V
h¯ ωk σˆ kz + h¯ ωb bˆ † bˆ + hχ
H =
¯
k
k
k,k The approximate mean-field ground state |BCS is found by minimizing Hˆ BCS −
ˆ in the standard way. The dynamics is then obtained by numerically inteμN
grating the Schrödinger equation with |BCS as the initial atomic state and the
molecular field in the vacuum state.
Figure 11 shows the resulting molecule statistics for V = 0.03h¯ χ, which corresponds to a gap of = 0.15h¯ χ for the system at hand. Clearly, the molecule
production is much more efficient than it was in the case of a normal Fermi gas.
The molecules are produced at a higher rate and the maximum number of molecules is larger. The evolution of the number statistics is reminiscent of the BEC
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QUANTUM OPTICS OF ULTRA-COLD MOLECULES
173
case. This also shows that the qualitative differences seen between the normal
Fermi gas and a BEC in the previous section cannot be attributed to inhomogeneous broadening and the resulting dephasing alone but are instead a result of the
different coherence properties of the atoms.
The short-time dynamics is again obtained in first-order perturbation theory,
which gives now
2
n(t) ≈ (χt)2
(39)
+ Na .
V
In addition to the term proportional to Na representing the incoherent contribution
from the individual atom pairs that was already present in the normal Fermi gas,
there is now an additional contribution proportional to (/V )2 . Since (/V ) can
be interpreted as the number of Cooper pairs in the quantum-degenerate Fermi
gas, this term can be understood as resulting from the coherent conversion of
Cooper pairs into molecules in a collective fashion similar to the BEC case. The
coherent contribution results naturally from the non-linear coupling of the atomic
field to the molecular field. This non-linear coupling links higher-order correlations of the molecular field to lower-order correlations of the atomic field. For
the parameters of Fig. 11 /V ≈ 6.5 so that the coherent contribution from the
Cooper pairs clearly dominates over the incoherent contribution from the unpaired
fermions. Note that no signature of that term can be found in the momentum distribution of the atoms themselves. Their momentum distribution is very similar to
that of a normal Fermi gas. The short-time value of g (2) (t1 , t2 ), shown in Fig. 12,
decreases from the value of Eq. (37) for a normal Fermi gas at = 0 down to one
as increases, underlining the transition from incoherent to coherent molecule
production.
5. Molecules as Probes of Spatial Correlations
The single-mode description of the molecular field of the previous section results
in the loss of all information about the spatial structure of the atomic state. In
this final section we adopt a complementary view and study the coupled atom–
molecule system including all modes of the molecular and atomic field so as to
resolve their spatial structure. This problem is too complex to admit an exact
solution, hence we rely entirely on perturbation theory.
One of the motivations for such studies are the on-going experimental efforts
to study the so-called BEC-BCS crossover. A difficulty of these studies has been
that they necessitate the measurement of higher-order correlations of the atomic
system. While the momentum distribution of a gas of bosons provides a clear
signature of the presence of a Bose–Einstein condensate, the Cooper pairing between fermionic atoms in a BCS state hardly changes the momentum distribution
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D. Meiser et al.
[5
F IG . 12. g (2) (0+ , 0+ ) as a function of the gap parameter . (Figure taken from Ref. (Meiser and
Meystre, 2005).)
or spatial profile as compared to a normal Fermi gas. This poses a significant
experimental challenge, since the primary techniques for probing the state of an
ultra-cold gas are either optical absorption or phase contrast imaging, which directly measure the spatial density or momentum distribution following ballistic
expansion of the gas. In the strongly interacting regime very close to the Feshbach
resonance, evidence for fermionic superfluidity was obtained by projecting the
atom pairs onto a molecular state by a rapid sweep through the resonance (Regal
et al., 2004; Zwierlein et al., 2004). More direct evidence of the gap in the excitation spectra due to pairing was obtained by rf spectroscopy (Chin et al., 2004) and
by measurements of the collective excitation frequencies (Bartenstein et al., 2004;
Kinast et al., 2004). Finally, the superfluidity of ultra-cold fermions in the strongly
interacting regime has recently been impressively demonstrated via the generation
of atomic vortices (Zwierlein et al., 2005).
Still, the detection of fermionic superfluidity in the weakly interacting BCS
regime remains a challenge. The direct detection of Cooper pairing requires the
measurement of second-order or higher atomic correlation functions. Several researchers have proposed and implemented schemes that allow one to measure
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QUANTUM OPTICS OF ULTRA-COLD MOLECULES
175
higher-order correlations (Altman et al., 2004; Bach and Rza˙zewski, 2004; Burt
et al., 1997; Cacciapuoti et al., 2003; Hellweg et al., 2003; Regal et al., 2004) but
those methods are still very difficult to realize experimentally. While the measurement of higher-order correlations is challenging already for bosons, the theory of
these correlations has been established a long time ago by Glauber for photons
(Glauber, 1963a, 1963b; Naraschewski and Glauber, 1999). For fermions however, despite some efforts (Cahill and Glauber, 1999) a satisfactory coherence
theory is still missing.
From the previous section we know that one can circumvent these difficulties by making use of the non-linear coupling of atoms to a molecular field.
The non-linearity of the coupling links first-order correlations of the molecules
to second-order correlations of the atoms. Furthermore the molecules are always
bosonic so that the well-known coherence theory for bosonic fields can be used
to characterize them. Considering a simplified model with only one molecular
mode, it was found that the molecules created that way can indeed be used as a
diagnostic tool for second-order correlations of the original atomic field.
We consider the limiting case of strong atom–molecule coupling as compared
to the relevant atomic energies. The molecule formation from a Bose–Einstein
condensate (BEC) serves as a reference system. There we can rather easily study
the contributions to the molecular signal from the condensed fraction as well as
from thermal and quantum fluctuations above the condensate. The cases of a normal Fermi gas and a BCS superfluid Fermi system are then compared with it.
We show that the molecule formation from a normal Fermi gas and from the unpaired fraction of atoms in a BCS state has very similar properties to those of the
molecule formation from the non-condensed atoms in the BEC case. The state of
the molecular field formed from the pairing field in the BCS state on the other
hand is similar to that resulting from the condensed fraction in the BEC case. The
qualitative information gained by the analogies with the BEC case help us gain a
physical understanding of the molecule formation in the BCS case where direct
calculations are difficult and not nearly as transparent.
5.1. M ODEL
We consider again the three cases where the atoms are bosonic and initially form a
BEC, or consist of two species of ultra-cold fermions (labeled again by σ =↑, ↓),
with or without superfluid component. In the following we describe explicitly the
situation for fermions, the bosonic case being obtained from it by omitting the
spin indices and by replacing the Fermi field operators by bosonic field operators.
Since we are primarily interested in how much can be learned about the secondorder correlations of the initial atomic cloud from the final molecular state, we
keep the physics of the atoms themselves as well as the coupling to the molecular field as simple as possible. The coupled fermion–molecule system can
176
D. Meiser et al.
[5
be described by the Hamiltonian (Chiofalo et al., 2002; Holland et al., 2001;
Timmermans et al., 1999)
Hˆ =
ωk
k,σ
2
†
cˆkσ +
cˆkσ
k
ωk aˆ k† aˆ k + V −1/2
k1 ,k2 ,σ
U0
cˆk†1 +q↑ cˆk†2 −q↓ cˆk2 ↓ cˆk1 ↑
2V
q,k1 ,k2
†
+ hg
a
ˆ
c
ˆ
c
ˆ
+
H.c.
.
¯
q q/2+k↓ q/2−k↑
+
U˜ tr (k2 − k1 )cˆk†2 σ cˆk1 σ
(40)
q,k
The kinetic energies
ωk are defined as before, V is the quantization volume,
U˜ tr (k) = V −1/2 V d 3 xe−ikx Utr (x) is the Fourier transform of the trapping potential Utr (r) and U0 = 4π h¯ 2 a/mf is the background scattering strength with
a the background scattering length. The coupling constant g between atoms and
molecules is, up to dimensions, equal to χ of the previous sections.
We assume that the trapping potential and background scattering are relevant
only for the preparation of the initial state before the coupling to the molecules
is switched on at t = 0 and can √
be neglected in the calculation of the subsequent
dynamics. This is justified if h¯ g N U0 n, h¯ ωi , where n is the atomic density,
N the number of atoms, and ωi are the oscillator frequencies of the atoms in the
potential Utr (r) that is assumed to be harmonic. Experimentally, the interaction
between the atoms can effectively be switched off by ramping the magnetic field
to a position where the scattering length is zero, so that this assumption is fulfilled.
Regarding
the strength of the coupling constant g, two cases are possible:
√
h¯ g N can be much larger or much smaller than the characteristic kinetic energies
involved. For fermions the terms broad and narrow resonance have been coined
for the two cases, respectively, and we will use these for bosons as well. Both
situations can be realized experimentally, and they give rise to different effects.
For strong coupling the conversion process needs not satisfy energy conservation
because of the energy time uncertainty relation. For weak coupling energy conservation is enforced. This energy selectivity can be useful in certain situations
because it allows one to resolve additional structures in the atomic state. The
analysis of this case is fairly technical, however. Therefore we only consider the
case of strong coupling and refer the interested reader to (Meiser et al., 2005b) for
details of the calculations and the case of weak coupling.
First-order time-dependent perturbation theory requires that the state of the
atoms does not change significantly and consequently, only a small fraction of
the atoms are converted into molecules. It is reasonable to assume that this is true
for short interaction times or weak enough coupling. Apart from making the system tractable by analytic methods there is also a deeper reason why the coupling
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QUANTUM OPTICS OF ULTRA-COLD MOLECULES
177
should be weak: Since we ultimately wish to get information about the atomic
state, it should not be modified too much by the measurement itself, i.e. the coupling to the molecular field. Our treatment therefore follows the same spirit as
Glauber’s original theory of photon detection, where it is assumed that the light–
matter coupling is weak enough that the detector photocurrent can be calculated
using Fermi’s Golden rule.
5.2. BEC
We consider first the case where the initial atomic state is a BEC in a spherically
symmetric harmonic trap. We assume that the temperature is well below the BEC
transition temperature and that the interactions between the atoms are not too
strong. Then the atomic system is described by the field operator
ˆ
ˆ
ψ(x)
= ψ0 (x)cˆ + δ ψ(x),
(41)
where ψ0 (x) is the condensate wave function and cˆ is the annihilation operator for
an atom in the condensate. In accordance with the assumption of low temperatures
and weak interactions we do not distinguish between the total number of atoms
ˆ
and the number of atoms in the condensate. The fluctuations δ ψ(x)
are small and
those with wavelengths much less than RTF will be treated in the local density
approximation while those with wavelengths comparable to RTF can be neglected
(Bergeman et al., 2000; Hutchinson and Zaremba, 1997; Reidl et al., 1999).
We are interested in the momentum distribution of the molecules
n(p, t) = bˆp† (t)bˆp (t)
(42)
which for short times, t, can be calculated using perturbation theory. In the broad
resonance limit we ignore the kinetic energies and find
2
nBEC (p, t) = (gt)2 N (N − 1)V ψ˜ 02 (p)
3
d x †
2
+ (gt) 4N
(43)
δ cˆp (x)δ cˆp (x) ,
V
where the expectation value in the last term includes a√thermal average. From this
expression we see that our approach is justified if ( N gt)2 1 because for
such times the initial atomic state can be assumed to remain undepleted. The first
term in Eq. (43) is the contribution from condensed atoms and the second term
comes from uncondensed atoms above the condensate. The contribution from the
condensate can be evaluated in closed form in the Thomas–Fermi approximation
for a spherical trap. The contribution from the thermal atoms can be calculated
using the local density approximation. The details of this calculation can be found
in Ref. (Meiser et al., 2005b).
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D. Meiser et al.
[5
F IG . 13. Momentum distribution of molecules formed from a BEC (dashed line) with a = 0.1aosc
and T = 0.1Tc and a BCS type state with kF a = 0.5 and aosc = 5kF−1 (0) (solid line), both for
N = 105 atoms. The BCS curve has been scaled up by a factor of 20 for easier comparison. The
inset shows the noise contribution for BEC (dashed) and BCS (solid) case. The latter is simply the
momentum distribution of molecules formed from a normal Fermi gas. The local density approximation treatment of the noise contribution in the BEC case is not valid for momenta smaller than 2π/ξ
(indicated by the dotted line in the inset). Note that the coherent contribution is larger than the noise
contribution by five orders of magnitude in the BEC case and three orders of magnitude in the BCS
case.
The momentum distribution (43) is illustrated in Fig. 13. The contribution from
the condensate is a collective effect, as indicated by its quadratic scaling with the
atom number. It clearly dominates over the incoherent contribution from the fluctuations, which is proportional to the number of atoms and only visible in the
inset. The momentum width of the contribution from the condensate is roughly
h¯ 2π/RTF which is much narrower than the contribution from the fluctuations,
whose momentum distribution has a typical width of h/ξ
¯ , where ξ = (8πan)−1/2
is the healing length. This is a case where coherence properties of the atoms can
be read off the momentum distribution of the molecules: The narrow momentum distribution of the molecules is only possible if the atoms were coherent over
distances ∼RTF . At this point this is a fairly trivial observation and the same information could have been gained by looking directly at the momentum distribution
of the atoms, which is after all how Bose–Einstein condensation was detected already in the very first experiments (Anderson et al., 1995; Bradley et al., 1995;
5]
QUANTUM OPTICS OF ULTRA-COLD MOLECULES
179
Davis et al., 1995). Still we mention it because it will be very interesting (indeed
interesting enough to motivate this whole work!) to contrast this situation to the
BCS case below.
Using the same approximation scheme we can calculate the second-order correlation. If we neglect fluctuations we find
6
(44)
+ O N −2 .
N
For N → ∞ this is very close to 1, which is characteristic of a coherent state.
This result implies that the number fluctuations of the molecules are very nearly
Poissonian. The fluctuations lead to a larger value of g (2) , making the molecular
field partially coherent, but their effect is only of order O(N −1 ).
(2)
gBEC
(p1 , t1 ; p2 , t2 ) = 1 −
5.3. N ORMAL F ERMI G AS
We treat the gas in the local density approximation where the atoms locally fill a
Fermi sea
cˆk† |0
|NFG =
(45)
|k|<kF (x)
with local Fermi momentum hk
¯ F (x) and |0 being the atomic vacuum. It is related to the local density of the atoms in the usual way (Butts and Rokshar, 1997;
Landau et al., 1980).
The momentum distribution and second-order correlation function are readily
found in perturbation theory. The momentum distribution is shown in the inset
in Fig. 13. The total number of molecules scales only linear with the number
of atoms, meaning that, in contrast to the BEC case, the molecule formation is
non-collective. Each atom pair is converted into a molecule independently of all
the others and there is no collective enhancement. Furthermore the momentum
distribution of the atoms is much wider than in the BEC case. It’s width is of
1/3
the order of hn
¯ 0 indicating that the atoms are correlated only over distances
comparable to the inter atomic distance.
Similarly, we find for the local value of g (2) at position x,
1
(2)
(2)
,
gloc (p, x, t) ≡ gloc (p, t; p, t, x) = 2 1 −
(46)
Neff (p, x)
where Neff is the number of atoms that are allowed to form a molecule on the basis
of momentum conservation. For large Neff g (2) approaches 2 which is characteristic of a thermal field. Indeed, using the analogy with an ensemble of two-level
atoms coupled to every mode of the molecular field provided by the Tavis–
Cummings model, it is easy to see that the entire counting statistics is thermal.
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D. Meiser et al.
[5
5.4. BCS S TATE
Let us finally consider a system of fermions with attractive interactions, U0 < 0,
at temperatures well below the BCS critical temperature. It is well known that
for these temperatures the attractive interactions give rise to correlations between
pairs of atoms in time reversed states known as Cooper pairs. We assume that
the spherically symmetric trapping potential is sufficiently slowly varying that the
gas can be treated in the local density approximation. More quantitatively, the
local density approximation is valid if the size of the Cooper pairs, given by the
correlation length
λ(r) = vF (r)/π(r),
is much smaller than the oscillator length for the trap. Here, vF (r) is the velocity
of atoms at the Fermi surface and (r) is the pairing field at distance r from the
origin, which we take at the center of the trap. Loosely speaking, in the local density approximation the ground state of the atoms is determined by repeating the
variational BCS-calculation of the previous section in a small volume at every position x. A thorough discussion of this calculation can be found in Ref. (Houbiers
et al., 1997).
We find the momentum distribution of the molecules from the BCS-type state
by repeating the calculation done in the case of a normal Fermi gas. For the BCS
wave function, the relevant atomic expectation values factorize into normal and
anomalous correlations. The normal terms are proportional to densities and are
already present in the case of a normal Fermi gas while the anomalous contributions are proportional to the gap parameter. The momentum distribution of the
molecules becomes
2
2
nBCS (p, t) ≈ (gt) cˆp/2+k,↓ cˆp/2−k,↑ + nNFG (p, t).
(47)
k
The first term is easily shown to be proportional to the square of the Fourier
transform of the gap parameter. Since the gap parameter is slowly varying over
the size of the atomic cloud, this contribution has a width of the order of h/R
¯ TF , in
complete analogy with the BEC case above. The total number of atoms in the first
contribution is proportional to the square of the number of Cooper pairs, which
is a macroscopic fraction of the total atom number well below the BCS transition
temperature. That means that this contribution is a collective effect. The second
term is the wide and incoherent non-collective contribution already present in the
case of a normal Fermi gas. It is very similar to the thermal noise in the BEC case
as far as its coherence properties are concerned.
For weak interactions such that the coherent contribution is small compared to
the incoherent contribution, the second-order correlations are close to those of a
normal Fermi gas given by Eq. (46), g (2) (p, x, t) ≈ 2. However, in the strongly
6]
QUANTUM OPTICS OF ULTRA-COLD MOLECULES
181
interacting regime, kF |a| ∼ 1, and large N , the coherent contribution from the
paired atoms dominates over the incoherent contribution from unpaired atoms. In
this limit one finds that the second-order correlation is close to that of the BEC,
g (2) (p, x, t) ≈ 1. The physical reason for this is that at the level of even-order
correlations the pairing field behaves just like the mean field of the condensate.
This is clear from the factorization property of the atomic correlation functions in
terms of the normal component of the density and the anomalous density contribution due to the mean field. In this case, the leading-order terms in N are given
by the anomalous averages.
To summarize, molecules produced from an atomic BEC show a rather narrow
momentum distribution that is comparable to the zero-point momentum width of
the BEC from which they are formed. The molecule production is a collective
effect with contributions from all atom pairs adding up constructively, as indicated by the quadratic scaling of the number of molecules with the number of
atoms. Each mode of the resulting molecular field is to a very good approximation coherent (up to terms of order O(1/N )). The effects of noise, both due
to finite temperatures and to vacuum fluctuations, are of relative order O(1/N ).
They slightly increase the g (2) and cause the molecular field in each momentum
state to be only partially coherent.
In contrast, the momentum distribution of molecules formed from a normal
Fermi gas is much broader with a typical width given by the Fermi momentum of
the initial atomic cloud, i.e. the atoms are only correlated over an interatomic distance. The molecule production is not collective as the number of molecules only
scales like the number of atoms rather than the square. In this case, the secondorder correlations of the molecules exhibit super-Poissonian fluctuations, and the
molecules are well characterized by a thermal field.
The case where molecules are produced from paired atoms in a BCS-like state
shares many properties with the BEC case: The molecule formation rate is collective, their momentum distribution is very narrow, corresponding to a coherence
length of order RTF , and the molecular field is essentially coherent. The noncollective contribution from unpaired atoms has a momentum distribution very
similar to that of the thermal fluctuations in the BEC case.
6. Conclusion
In this paper we have used three examples to illustrate the profound impact of
quantum optics paradigms, tools and techniques, on the study of low-density,
quantum-degenerate atomic and molecular systems. There is little doubt that the
remarkably fast progress witnessed by that field results in no little part from the
experimental and theoretical methods developed in quantum optics over the last
decades. It is therefore fitting, on the occasion of Herbert Walther’s seventieth
182
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[8
birthday, to reflect on the profound impact of the field that he has helped invent,
and where he has been and remains so influential, on some of the most exciting
developments in AMO science.
7. Acknowledgements
This work was supported in part by the US Office of Naval Research, by the National Science Foundation, by the US Army Research Office, and by the National
Aeronautics and Space Administration.
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ADVANCES IN ATOMIC, MOLECULAR AND OPTICAL PHYSICS, VOL. 53
ATOM MANIPULATION IN OPTICAL
LATTICES*
GEORG RAITHEL and NATALYA MORROW
FOCUS Center, Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2. Theoretical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1. Light Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2. Atom-Field Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3. Quantum Monte-Carlo Wave-Function Simulations . . . . . . . . . . . . . .
3. Review of One-Dimensional Lattice Configurations for Rubidium . . . . . . . . .
3.1. Red-Detuned Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2. “Gray” Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3. Magnetic-Field-Induced Lattices . . . . . . . . . . . . . . . . . . . . . . . .
3.4. Related Laser-Cooling Methods . . . . . . . . . . . . . . . . . . . . . . . . .
4. Periodic Well-to-Well Tunneling in Gray Lattices . . . . . . . . . . . . . . . . . .
4.1. Experimental and Simulation Results . . . . . . . . . . . . . . . . . . . . . .
4.2. Analysis Based on Band-Structure . . . . . . . . . . . . . . . . . . . . . . .
5. Influence of Magnetic Fields on Tunneling . . . . . . . . . . . . . . . . . . . . . .
5.1. Motivation and Experimental Observations . . . . . . . . . . . . . . . . . . .
5.2. Interpretation of the Results Based on Two Models . . . . . . . . . . . . . .
6. Sloshing-Type Wave-Packet Motion . . . . . . . . . . . . . . . . . . . . . . . . .
6.1. Wave-Packets Localized in Single Lattice Wells . . . . . . . . . . . . . . . .
6.2. Experimental Study of Sloshing-Type Motion in a Magnetized Gray Lattice
7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8. Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction
Optical lattices are periodic light-shift potentials for cold atoms created by the interference of multiple laser beams. Atoms can be laser-cooled and localized in the
* This chapter has been prepared in dedication to Professor Herbert Walther and his 70th birthday.
His vision and excellence as an experimental physicist has lead to many discoveries in atomic physics
and quantum optics, and has inspired our research in many ways.
187
© 2006 Elsevier Inc. All rights reserved
ISSN 1049-250X
DOI 10.1016/S1049-250X(06)53007-1
188
G. Raithel and N. Morrow
[1
sub-micrometer-sized potential wells of optical lattices, leading to exciting possibilities in fundamental studies of quantum mechanics and applications of quantum
theory. The field has been reviewed by Jessen and Deutsch (1996). The localization of atoms in lattice wells has initially been shown through spectroscopic
studies by Verkerk et al. (1992) and Jessen et al. (1992). Due to the flexibilities in
the choice of atomic transitions, the number of lattice laser beams, their angles,
intensities and detunings, a wide variety of lattice geometries and potentials can
be realized, as shown in a systematic manner by Petsas et al. (1994). Applications
of optical lattices are many-fold. In many laboratories, optical lattices are employed to laser-cool atoms and to localize and store these atoms in microscopic
wells. Optical lattices have been utilized in a wide range of experiments on transient phenomena, including experiments on Landau–Zener tunneling (Niu et al.,
1996), Bloch oscillations (Dahan et al., 1996), Wannier–Stark states (Wilkinson
et al., 1996), wave-packet revivals (Raithel et al., 1998), and tunneling in nearresonant lattices (Dutta et al., 1999) and far-off-resonant lattices (Haycock et al.,
2000). A recent account of applications of optical lattices in atom lithography
is provided by Bradley et al. (1999). Optical lattices have further been proposed
as platforms for quantum information processing by Brennen et al. (1999) and
Jaksch et al. (1999). The dynamics of quantum gases in optical lattices has become a field of high interest. This area and some of its implications on quantum
information processing have recently been reviewed by Bloch (2004).
In this chapter, we focus on applications of optical lattices on the field of wavepacket preparation and manipulation. In all experimental schemes presented, the
lattices are used two-fold, namely as an initialization tool to prepare suitable
initial states of the quantum system, and as a platform on which the actual wavepacket experiments are performed. Therefore, the utilized lattices provide fast and
robust laser-cooling of cold atoms into the lowest few quantum states of the lattice, and the decoherence rate of the trapped atoms caused by the fluorescence of
the atoms in the lattice light is sufficiently low that coherent wave-packet motion
can be observed over a number of periods of the motion.
In the presented work, we deal with two types of wave-packet motion. In one
type, referred to as sloshing-type motion, wave-packets oscillate back and forth
in individual lattice wells. This type of oscillation has been observed by Kozuma
et al. (1996) and Raithel et al. (1998) using a photon-exchange method. The
sloshing-type motion occurs within the spatial range of a single lattice well, with
vanishing coupling between neighboring wells, on a length scale of a few tenths of
the laser wavelength used to form the lattice. The motion can be excited by a sudden small displacement of the laser-cooled and localized atoms from the minima
of the lattice wells or, equivalently, a sudden displacement of the lattice underneath the atoms. In a simplified harmonic model of the lattice wells and under the
assumption that the atoms are initially in the lattice ground state, this procedure
corresponds to the generation of coherent wave-packet states by a shift operation.
1]
ATOM MANIPULATION IN OPTICAL LATTICES
189
The sloshing-type wave-packet oscillation that ensues after the shift operation
can be measured in a non-destructive manner, revealing the lattice oscillation frequency, the anharmonicity of the lattice potential and decoherence rates (Raithel
et al., 1998). Also, measurements on the wave-packet motion can be employed
to apply real-time feedback onto the wave-packet motion, allowing one to study
feedback-controlled cold-atom systems (Morrow et al., 2002). The investigations
dealing with sloshing-type wave-packet motion require lattices that support at
least three to four bound states of the center-of-mass motion in lattice wells. Further, in order to be able to observe the wave-packet motion, the anharmonicity of
the wells should be sufficiently low to avoid wave-packet dispersion during the
first couple of oscillation periods. Also, the fluorescence-induced decoherence
rate should be much less than the oscillation frequency. These requirements are
satisfied by a quite large class of lattices.
The second type of wave-packet motion we discuss in this chapter is periodic
well-to-well tunneling. Tunneling measurements have, for instance, been used by
Dutta et al. (1999) to investigate gauge potentials that were predicted by Dum
and Olshanii (1996). In this chapter, we present measurements on the influence
of magnetic-field-induced level shifts on the tunneling behavior. For several reasons, the observation of tunneling is considerably more demanding than the study
of wave-packets evolving in single wells. The tunneling frequency of heavy atoms
such as rubidium tends to be very low, as can be seen both by simple estimates
of tunneling frequencies using approximate Gamow factors as well as by accurate
band-structure calculations. Tunneling frequencies of at least 103 s−1 are required
so that the tunneling can be observed on an experimentally feasible time scale.
Further, the lattice must provide efficient laser-cooling, because cooling is required to initialize the atoms in the lattice before the tunneling is measured. Also,
during the tunneling process the fluorescence-induced decoherence rate must be
lower than the tunneling rate. Among the lattice types we consider in this chapter,
only gray optical lattices, discussed in Section 3.2, satisfy all three conditions.
In Section 2 we describe the theoretical methods that we use to evaluate lattice
types with regard to their suitability for wave-packet experiments. These methods
are also used to model experimental data in detail and to obtain physical insight.
In Section 3 we provide an overview over the lattice types that are, in principle, at
our disposal. There, we compare the laser-cooling performance of the lattice types
and discuss their suitability for wave-packet and tunneling experiments. This section of the chapter can serve as a guide of how to select an optical-lattice type for a
wave-packet or tunneling experiment. In Section 4 we then describe experiments
on well-to-well tunneling of atoms in a one-dimensional gray optical lattice. In
Section 5 we investigate the modifications in tunneling behavior that result from
the addition of weak magnetic fields to the lattice. It is discussed how magnetic
fields in the range of a few tens of milli-Gauss can be used to tune the system
through several tunneling resonances. We also find that somewhat stronger mag-
190
G. Raithel and N. Morrow
[2
netic fields in the range of 100 mG suppress tunneling and induce more tightly
bound levels in a subset of lattice wells. Consequently, as shown in Section 6,
magnetic fields can be used to enable sloshing-type wave-packet motion in gray
optical lattices. The presented results are summarized and future prospects are
discussed in the conclusion (Section 7).
2. Theoretical Considerations
2.1. L IGHT F IELDS
In this chapter, we are concerned with one-dimensional lattice structures formed
by pairs of counter-propagating laser beams with wave-vectors kL = ±(2π/λ)ez ,
where we use ex , ey and ez for the Cartesian unit vectors. The electric field can
be written in the form
E(Z, t) = exp(−iωt) e+ A++ exp(ikL Z) + A+− exp(−ikZL )
+ e− A−+ exp(ikL Z) + A−− exp(−ikL Z) + c.c.
(1)
√
with spherical unit vectors e± = ∓(ex ± iey )/ 2, atomic center-of-mass coordinate Z, and c.c. referring to the complex conjugate. The field amplitudes A carry
two superscript indices, namely a first one identifying the circular-polarization
component, and a second one for propagation direction.
Throughout most of this chapter, we consider counter-propagating fields with
orthogonal polarizations, E(Z, t) = E0 [ex cos(kL Z − ωt) + ey sin(−kL Z − ωt)],
where E0 denotes the electric-field amplitude of a single beam. For this field, it is
seen that
√ √ A++ = −E0 / 2 2 ,
(2)
A+− = A−+ = A−− = E0 / 2 2 .
This field generates the most widely known type of sub-Doppler laser cooling,
“Sisyphus cooling”, which was experimentally observed by Lett et al. (1988)
and by Shevy et al. (1989), and explained by Dalibard and Cohen-Tannoudji
(1989) and further analyzed in detail by Finkelstein et al. (1992) and Guo and
Berman (1993). Another case of interest is that of two counter-propagating
circularly polarized beams with helicities that are the same in a fixed (beamindependent) frame. Such beams form a standing wave of circular polarization
(e.g., A++ = A+− = E0 /2 and A−+ = A−− = 0). In combination with
a transverse static magnetic field, this field can generate magnetic-field-induced
laser-cooling (Sheehy et al., 1990) and localization of atoms in a lattice structure.
Counter-propagating circularly polarized beams with helicities that are opposite
in a fixed (beam-independent) frame form a light-field with spatially rotating
linear polarization (e.g., A++ = A−− = E0 /2 and A−+ = A+− = 0). This
2]
ATOM MANIPULATION IN OPTICAL LATTICES
191
field does not produce modulated lattice potentials or intensities, but is characterized by a linear-polarization vector whose tip outlines the shape of a corkscrew.
The corkscrew configuration generates sub-Doppler laser cooling (Dalibard and
Cohen-Tannoudji, 1989) and, under certain conditions, velocity-selective coherent population trapping (Aspect et al., 1988).
2.2. ATOM -F IELD I NTERACTION
In the following, we review the methods that we use throughout this chapter in
order to model optical lattices. In the electric-dipole approximation, the atomˆ where the electric-dipole operator D
ˆ = −|e|(−ˆr+ e− −
field interaction is −ED,
√
√
rˆ− e+ + rˆz ez ). The operators rˆ+ = −(xˆ +i y)/
ˆ
2 and rˆ− = (xˆ −i y)/
ˆ
2 and rˆz act
on the internal (electronic) degree of freedom. Using the orthogonality relations
e+ e+ = e− e− = 0, e+ e− = e− e+ = −1, and noting that e∗+ = −e− and
e∗− = −e+ , the atom-field interaction at a center-of-mass location Z is
ˆ = |e| exp(−iωt) rˆ+ A++ exp(ikL Z) + A+− exp(−ikL Z)
−ED
+ rˆ− A−+ exp(ikL Z) + A−− exp(−ikL Z) + h.c.,
(3)
where h.c. is the Hermitian conjugate. Using the excited-state and ground-state
wave-functions |ψe and |ψg , the Schrödinger equation reads
hΓ
¯
ˆ
ih¯ ∂t |ψe = −ED|ψg + h¯ ω0 − i
|ψe ,
2
ˆ e ,
ih¯ ∂t |ψg = −ED|ψ
(4)
where h¯ ω0 is the energy of the excited state, and the term −i h¯2Γ accounts for
the decay of the excited state (Γ is the excited-state decay rate). Note that this
term represents a weak anti-Hermitian contribution in the effective Hamiltonian,
causing the norm of the wave-function to decay. The norm decay reflects the
spontaneous emission associated with the probability of finding the atom in the
excited state. |ψe and |ψg are spinor wave-functions that have 2F + 1 and
2F + 1 magnetic sublevels, where F and F denote the respective excited-state
and ground-state angular momenta.
To transform into a rotating frame, we use new wave-functions |Ψe =
exp(iωt)|ψe and |Ψg = |ψg . After making the rotating-wave approximation, in
which terms ∝ exp(±i2ωt) are neglected, the transformed Schrödinger equation
reads
ih¯ ∂t |Ψe = |e| rˆ+ A++ exp(ikL Z) + A+− exp(−ikL Z)
+ rˆ− A−+ exp(ikL Z) + A−− exp(−ikL Z) |Ψg 192
G. Raithel and N. Morrow
h¯ Γ
− hδ
|Ψe ,
¯ +i
2
ih¯ ∂t |Ψg = |e| −ˆr− A++∗ exp(−ikL Z) + A+−∗ exp(ikL Z)
− rˆ+ A−+∗ exp(−ikZ) + A−−∗ exp(ikZ) |Ψe ,
[2
(5)
where the laser-atom detuning δ = ω − ω0 . Also, note that rˆ− is the Hermitian
conjugate of −ˆr+ . Usually, the excited-state part of the wave-function reaches a
quasi-steady-state as the atom moves (slowly) through the lattice. In this case,
the excited state can be adiabatically eliminated by setting ∂t |Ψe = 0. We can
then express |Ψe in terms of |Ψg and insert the result in the lower part of the
Schrödinger equation (5). We obtain
ih¯ ∂t |Ψg = Vˆ (Z)|Ψg (6)
with an effective Hamiltonian Vˆ (Z). For the case of counter-propagating beams
with orthogonal polarizations, which we are mostly interested in, it is
Vˆ (Z) =
−e2 E02
rˆ+ rˆ− cos2 (kL Z) + rˆ− rˆ+ sin2 (kL Z)
2(h¯ δ + ih¯ Γ /2)
2
2
− rˆ+
sin(kL Z) cos(kL Z) .
+ i rˆ−
(7)
Considering the center-of-mass position Z a fixed classical parameter, Vˆ (Z)
can be evaluated in the internal ground-state Hilbert space {|g, F, m | m =
−F, −F + 1, . . . , F }. Thereby, the internal-state operators rˆ+ and rˆ− occur in
products that couple a ground-state vector into an excited-state one and back into
the ground state. All non-zero matrix elements of the operator rˆ+ are of the type
e, F , m = m + 1|ˆr+ |g, F, m or g, F, m|ˆr+ |e, F , m = m − 1, with the
exited-state Hilbert space being {|e, F , m | m = −F , −F + 1, . . . , F }.
The matrix elements of rˆ+ and rˆ− are products of a radial matrix element and
a Clebsch–Gordan coefficient. The Clebsch–Gordan coefficients can be calculated for all hyperfine transitions of interest and normalized such that the “cycling
transition” |g, F, m = F ↔ |e, F = F + 1, m = F + 1 has a Clebsch–
Gordan coefficient of one. They can then be arranged in matrices that represent
operators cˆ+ and cˆ− which essentially are the same as the rˆ+ and rˆ− except that
the radial matrix element has been factored out. We also reverse the sign of cˆ+
(meaning that cˆ+ and cˆ− are the Hermitian conjugate of each other). This procedure allows one to write Vˆ (Z) in the convenient form
Vˆ (Z) =
hΩ
¯ 12
cˆ+ cˆ− cos2 (kL Z) + cˆ− cˆ+ sin2 (kL Z)
2(δ + iΓ /2)
2
2
+ i cˆ+
sin(kL Z) cos(kL Z) ,
− cˆ−
(8)
2]
ATOM MANIPULATION IN OPTICAL LATTICES
193
where the Rabi frequency Ω1 = Γ 2II1sat . There, I1 is the single-beam intensity
and Isat the saturation intensity of the transition |g, F, m = F ↔ |e, F =
F + 1, m = F + 1. For 87 Rb, which we use in the experiments described
in Sections 4–6, Isat = 1.6 mW/cm2 and F = 2. As before, the internalstate operators cˆ+ and cˆ− occur in products that couple a ground-state vector
into an excited-state one and back into the ground state. Further, cˆ+ (cˆ− ) always increases (decreases) the magnetic quantum number. Therefore, the terms in
Eq. (8) involving cˆ+ cˆ− and cˆ− cˆ+ yield contributions that are diagonal in the basis
{|g, F, m | m = −F, . . . , F } and can be regarded as the primary light-shift ef2 and cˆ2 produce
fect of the lattice on the atomic m-levels. The terms involving cˆ+
−
contributions on the second off-diagonal that can be interpreted as stimulated Raman transitions driven by the lattice beams that cause mixing between states the
m-values of which differ by 2. Finally, recalling that cˆ+ and cˆ− are Hermitian
conjugates of one another, it is also seen that the effective Hamiltonian Vˆ (Z) is
Hermitian for the case Γ = 0 (as required).
Since in alkali atoms typically more than one excited-state hyperfine level F are important, we usually sum the Hamiltonian Vˆ (Z) over the relevant values
of F . Thereby, for different F different values of δ apply, given by the laser
frequency and the hyperfine splittings of the utilized transition. Also, the Clebsch–
Gordan coefficients entering into the cˆ+ and cˆ− are different for different F .
The effect of a small magnetic field B can also be included by adding a term
Pˆg gF μB Fˆ · B to Eq. (8), where gF is the ground-state g-factor, μB the Bohr
magneton, Fˆ the angular-momentum vector operator, and Pˆg a projector on the
ground-state manifold.
The eigenvalues and eigenvectors of Vˆ (Z) yield the adiabatic lattice potentials Vα (Z), plotted frequently in this chapter, and adiabatic internal states Ψα (Z)
of the lattice (the label α = 1, 2, . . . , 2F + 1). In a classical description of
the center-of-mass motion, the atoms move on these lattice potentials under
d
the influence of an AC electric-dipole force given by the gradient dZ
Vα (Z).
Thereby, the internal state of the atom adiabatically follows the internal-state
vector Ψα (Z). This notion can be used to qualitatively explain atom motion
in optical lattices, but it does not apply in spatial regions where the adiabatic
potentials Vα (Z) are not well separated and exhibit narrow anti-crossings. At
narrow anti-crossings between Vα (Z), the lattice-light-induced Raman coupling
between different m-levels is quite inefficient, and atoms traveling through the
anti-crossing region tend to move on the so-called diabatic potentials, Vm (Z) =
g, F, m|Vˆ (Z)|g, F, m (i.e. the diagonal components of the Hamiltonian Vˆ (Z)
in Eq. (8) in m-state representation). In this situation, atoms undergo Landau–
Zener transitions between different adiabatic potentials.
It is fairly straightforward to obtain the band structure and the Bloch states of
the lattices. We quantize the center-of-mass motion by considering the variable Z
194
G. Raithel and N. Morrow
[2
in Eq. (8) an operator acting on center-of-mass momentum states. Noting that
Eq. (8) only couples momentum states that differ by multiples of 2kL , a product
Hilbert space that is well suited to represent the atom-field interaction in Eq. (8)
with center-of-mass quantization is {|g, F, m | m = −F, −F + 1, . . . , F } ⊗
{|(2n + q)kL | n = 0, ±1, ±2 . . .}. There, the momentum states |(2n + q)kL include a fixed quasimomentum q restricted to a range −1 q 1. The
atom-field interaction potential Eq. (8), with Z taken as an operator, is the potential part Hˆ pot of the full Hamiltonian. The kinetic-energy term is Hˆ kin =
¯ 2 (2n + q)2 kL2 /(2M)]|m, (2n + q)kL m, (2n + q)kL |, where M is the
m,n [h
atom mass and an abbreviated notation for the product states of internal and external degrees of freedom is used. We represent Hˆ = Hˆ pot + Hˆ kin in the above
product Hilbert space, diagonalize the resultant matrix, and sort the real parts of
the eigenvalues. Note that q is held fixed in any given diagonalization. The band
structure of lattice is then obtained by plotting the lowest eigenvalue vs q, the
second-lowest eigenvalue vs q, and so on. Following an analogous procedure,
the decay rates of the Bloch states, given by twice the imaginary parts of the
energy eigenvalues, can be plotted. Numerous examples of band structures and
corresponding decay-rate plots are shown in Section 3 of this chapter. If desired,
the band-structure calculation also yields the periodic Bloch functions of the lat
(q,k)
(q,k)
tice, Z|Ψ (q,k) = exp(iqZ) n,m cn,m exp(i2nkL Z)|m, where the cn,m are
the Fourier coefficients of the Bloch function and k is a band label. Note that these
functions are spinor functions, as we sum over the magnetic quantum number m.
2.3. Q UANTUM M ONTE -C ARLO WAVE -F UNCTION S IMULATIONS
The atom dynamics in optical lattices can be simulated using the quantum MonteCarlo wave-function method (QMCWF), which was introduced by Dalibard et al.
(1992) and used by Marte et al. (1993) to model laser cooling. The simulations
employ a fully quantum-mechanical description of the internal and center-of-mass
degrees of freedom of the atoms. We employ QMCWF to gain insight into the
laser-cooling dynamics, wave-packet evolution and coherence decay times. The
simulations also allow us to determine the spatial and momentum distributions of
the atoms, including the degree to which the atoms become localized in the wells
of the optical-lattice potentials. We have found that the QMCWF simulations provide perfect modeling for our lattice experiments.
In the QMCWF method, the evolution of the density matrix describing the
atoms in the lattice is obtained by forming averages over N quantum trajectories
|Ψi (t), each of which is a realization of a single-atom wave-function evolution:
ρ(t)
ˆ =
N
1 |Ψi (t)Ψi (t)|
.
N
Ψi (t)||Ψi (t)
i=1
(9)
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ATOM MANIPULATION IN OPTICAL LATTICES
195
Averages are taken over ensembles of typically N = 104 to N = 105 quantum
trajectories. We usually do not store the full density matrix but only the expectation values of observables of interest vs time, such as kinetic energy, degree of
localization in the wells, average momentum and average magnetization, position
and momentum distributions, etc.
Each quantum trajectory |Ψi (t) consists of periods of deterministic Hamiltonian wave-function evolution, connected by discrete quantum jumps. The
Hamiltonian evolution is governed by an effective Hamiltonian Hˆ = Hˆ kin + Hˆ pot
with kinetic and potential operators as explained in Section 2.2. The effective
Hamiltonian describes the coherent interaction between atoms and light fields,
as well as wave-function damping caused by photon scattering. The damping is
implemented by the imaginary part in the energy denominator of Eq. (8), which
leads to a gradual decay of the wave-function norm. The quantum trajectories
are represented in the basis {|m, (2n + q)kL | m = −F, −F + 1, . . . , F and
n = 0, ±1, ±2, . . . ± nmax }, where nmax determines the cutoff value of the
momentum states included in the simulation. The range of momentum states
that becomes populated depends on the lattice parameters. Therefore, we choose
from cutoff numbers nmax = 16, 32 or 64, dependent on the physical situation. The continuous quasimomentum q, which satisfies −1 < q < 1, does not
change during the Hamiltonian portions of the wave-function evolution, as can
be seen by inspection of Eq. (8). The Hamiltonian evolution is carried out numerically in discrete time steps. We use a split-operator method (Kosloff, 1988;
Leforestier et al., 1991), in which the kinetic-energy operator, which is diagonal
in the momentum basis, is applied in the momentum basis, while the atom-field
interaction, which is diagonal in position, is applied in a position basis. Thus,
at each time step of the integration the quantum trajectory is transformed back
and forth between position and momentum representations using fast Fourier
transformations. Consequently, both the position and the momentum probability distributions of the quantum trajectory |Ψ can be obtained without numerical
overhead at any time of the wave-function evolution.
The periods of Hamiltonian evolution are interrupted by discrete, instantaneous
quantum jumps, which simulate the effect of the spontaneous scattering of lattice
photons, and polarization- and direction-sensitive photon detection. A quantum
jump is invoked when the norm of a quantum trajectory, which continuously decays during the Hamiltonian portions of the evolution, drops below a random
number that is picked at the beginning of each time segment of Hamiltonian
evolution. The time instants and effects of the quantum jumps are governed by
quantum-mechanical probability laws. In each quantum jump, random numbers
are drawn to select the type of transition and the direction of the spontaneously
emitted photon. In each quantum jump, the quasimomentum value can change by
any value −1 < q < 1. The applied value of q follows from a random number
and the radiation pattern that corresponds to the type of scattered photon. In each
196
G. Raithel and N. Morrow
[3
jump, the wave-function is modified in a well-defined manner, determined by the
wave-function prior to the jump, the rules of quantum measurement, and several
random numbers. After quantum jumps, the quantum trajectory |Ψ is normalized
and entered into the next segment of Hamiltonian evolution.
We have found that a considerable degree of detail in the QMCWF is required in order to reproduce experimental data. Our QMCWF take all groundand excited-state hyperfine levels of the system into account. This implies that
leak transitions and the effect of a re-pumping laser, which is usually required
in the experiments, are fully taken into account in the QMCWF. Magnetic fields,
gravity, and additional laser beams that are used to optically pump the atoms in
certain lattices (see Section 4) can also be included in the simulations.
3. Review of One-Dimensional Lattice Configurations
for Rubidium
In this chapter, we are concerned with one-dimensional lattice structures of rubidium (transition wavelengths λ = 795 nm for the D1-line 5S1/2 ↔ 5P1/2 and
λ = 780 nm for the D2-line 5S1/2 ↔ 5P3/2 ). We concentrate on 87 Rb, which has
ground-state hyperfine components F = 1 and F = 2. The lattices are mostly but
not always operated on the F = 2 level. In the presented calculations, all coupled
hyperfine levels of the excited states are taken into account. The results translate
to many atomic species with similar hyperfine structure and transition linewidths.
Since the lattices are to be used to both cool the atoms and to perform wavepacket and tunneling experiments on them, two figures of merits exist:
• Cooling time scale. Since we use one-dimensional lattices, spatial diffusion
transverse to the lattice-beam directions causes the atoms to escape the atomfield interaction region, which is defined by the diameter of the laser beams.
Typically, the atoms remain in the atom-field interaction region for 1–2 milliseconds. Therefore, the lattice type selected for each experiment needs to cool
and localize the atoms in the lattice wells within about 1 ms. Since generally
the time scale of laser cooling is given by the photon scattering rate, we seek
configurations that yield high initial photon scattering rates.
• Low steady-state temperature and long coherence time. These conditions,
which are a pre-requisite for wave-packet and tunneling experiments, are primarily achieved by choosing configurations in which the photon scattering rate
converges towards small values once most of the atom cooling has occurred.
One may also exploit coherence preservation due to the Lamb–Dicke effect
(Dicke, 1953).
In the following, we use potential and band-structure calculations as well as quantum Monte-Carlo simulations in order to evaluate various one-dimensional lattice
3]
ATOM MANIPULATION IN OPTICAL LATTICES
197
configurations with regard to their atom-cooling speed, steady-state temperature,
degree of atom localization, and steady-state photon scattering rate. Also, the
well-to-well tunneling rates of atoms cooled deeply into the lattices are estimated
based on Gamow factors and derived from the band structure of the lattices.
3.1. R ED -D ETUNED L ATTICES
The most common type of sub-Doppler Sisyphus cooling occurs in counterpropagating fields of orthogonal polarization, referred to as “lin-perp-lin” configuration (Dalibard and Cohen-Tannoudji, 1989). We first consider a lattice that
is red-detuned with respect to a closed transition of the type F ↔ F + 1, such
as the 87 Rb 5S1/2 F = 2 ↔ 5P3/2 F = 3 transition. This field is equivalent
to two circularly polarized standing waves of opposite helicity and a λ/4 spatial
displacement.
In Fig. 1 we show simulation results for a red-detuned lattice on the 87 Rb
5S1/2 F = 2 ↔ 5P3/2 F = 3 transition obtained from 104 quantum trajectories. The single-beam intensity is I1 = 10 mW/cm2 and the laser detuning −6Γ
relative to the utilized atomic transition (upper-state decay rate Γ = 2π ×6 MHz).
Figure 1(a) shows the adiabatic potentials Vα (Z) of the lattice in units of the recoil
energy, ERec = h¯ 2 kL2 /(2M) (for the 87 Rb D1-transition, ERec = h × 3.77 kHz).
The lowest adiabatic potential is of particular interest, because atoms brought
into the lattice become rapidly optically pumped onto that potential. The minima
of the lowest adiabatic potential correspond to locations of maximal intensity of
the circularly polarized standing-wave components of the lattice field. Since each
standing wave has a λ/2 period, and since the two standing waves are shifted
relative to each other by λ/4, the field maxima have λ/4 separation and alternating σ + and σ − -polarizations. At the σ + -maxima, the internal atomic state |Ψα=1 associated with the lowest adiabatic potential Vα=1 is practically identical with
|F = 2, m = 2, and at the σ − -maxima it is |F = 2, m = −2. Since the σ ± transitions 5S1/2 |F = 2, m = ±2 ↔ 5P3/2 |F = 3, m = ±3 have the largest
Clebsch–Gordan coefficient (namely 1), the value of Vα=1 at the field maxima is
maximal and negative.
The band structure, displayed in Fig. 1(b), shows that under the conditions of
Fig. 1 the lowest adiabatic potential supports about five tightly bound oscillatory states. For energies above the maxima of the lowest adiabatic potential, the
band structure is quite complicated, because bands associated with multiple potentials begin to overlap and mix. Considering the lowest five tightly bound bands
in Fig. 1(b), it is noted that the separation between adjacent bands decreases with
increasing energy. This trend reflects the anharmonicity of the lowest adiabatic potential. The potential anharmonicity causes wave-packet dispersion, as has been
observed in breathing-mode (Raithel et al., 1997) and sloshing-type wave-packets
(Raithel et al., 1998).
198
G. Raithel and N. Morrow
[3
F IG . 1. Simulation of laser cooling in a red-detuned optical lattice of rubidium (detailed parame2 /(2M) vs position.
ters provided in text). (a) Adiabatic potentials Vα in recoil energies ERec = h¯ 2 kL
(b) Band structure Ek vs quasimomentum q. (c) Kinetic energy of cooled atoms vs time t. (d) Degree of atom localization in the wells, as defined in text, vs time t. (e) Single-atom fluorescence rate
vs time t. The exponential fits (dashed lines) in panels (c)–(e) yield the rates at which the respective
quantities approach a steady-state.
Under the absence of decoherence, atoms prepared in the tightly bound bands
would tunnel between neighboring wells at rates given by the width of the bands,
which increases with increasing excitation energy in the lattice wells. For the lowest band in Fig. 1(b), the width and thus the well-to-well tunneling rate νT amount
to only 3 s−1 . (This bandwidth cannot be resolved on the scale of Fig. 1(b), but
is evident from the numerical data used for the figure.) This tunneling rate is in
qualitative agreement with a basic estimate
Ωosc
exp(−2G),
2π
!
b 2M
G=
Vα=1 (Z) − E1 dZ,
h¯ 2
νT ≈
a
(10)
3]
ATOM MANIPULATION IN OPTICAL LATTICES
199
where Ω2πosc is the center-of-mass oscillation frequency of the atoms in the wells
(≈140 kHz in Fig. 1), G is the Gamow factor, E1 is the average energy of the lowest lattice band (≈−390 ERec in Fig. 1), Vα=1 (Z) is the lowest adiabatic potential,
and the locations a and b denote the left and right boundaries of the tunneling barrier (i.e. a pair of locations where Vα=1 (Z) = E1 ). While the integral in Eq. (10)
could be easily calculated numerically, a brief survey of Fig. 1(a) allows us to
quickly estimate the integral by ∼100 ERec × 0.1λ, leading to νT ∼ 1 s−1 . This
qualitative value agrees well with the exact value obtained from the band structure
calculation.
The laser-cooling performance and other properties of the lattice are studied using QMCWF simulations. Atoms are entered into the simulation with a Gaussian
velocity distribution that corresponds to an average kinetic energy of 100 ERec .
Due to Sisyphus cooling, the atoms become cooled into the lowest few oscillatory
levels of the wells. As seen in Fig. 1(c), most of the cooling occurs within about
100 µs, and a steady-state energy of about 30 ERec is reached. Comparing this
value with the band structure in Fig. 1(b), and noting that in view of the virial
theorem the total energy relative to the potential minimum is about twice the kinetic energy, it is seen that in steady-state the laser-cooled atoms mostly reside
in the lowest two or three tightly-bound vibrational states of the lowest adiabatic
potential.
The degree of localization of the atoms in the lattice reaches Z =
Ψ |(Z − Z0 )2 |Ψ ≈ λ/18, where Z0 is the location of the nearest potential
minimum (see Fig. 1(d)). The laser-cooling and localization dynamics in reddetuned optical lattices has been studied in detail by Raithel et al. (1997). There,
it has been found that the laser-cooling and localization rates are about the same,
as is evident in Figs. 1(c) and (d), and are proportional to the fluorescence rate of
the atoms in the lattice field. For the parameters of Fig. 1, the cooling rate equals
about 1/30 of the single-atom fluorescence rate. This ratio is fairly typical for
laser-cooling in one-dimensional red-detuned lattices.
While red-detuned lattices exhibit good initial laser-cooling performance and
a high degree of atom localization in the lattice wells, the steady-state kinetic
energy does not reach very low values because the atoms settle at locations of
maximal light scattering rate, as can be seen in Fig. 1(e). Comparing Figs. 1(c)
and (e) it is actually noticed that the degree to which the photon scattering has
approached its steady-state value reflects on the progress in cooling. The photon
scattering rate reaches a steady-state value of about 1.4 × 106 s−1 . This value is
close to the value one can calculate for atoms at the maxima of the two circular
2I1 /Isat
= 1.6×106 s−1 . This
standing waves the lattice is composed of, γ = Γ2 1+4(δ/Γ
)2
fluorescence rate exceeds the tunneling rate on the lowest band, which is 3 s−1 ,
by about six orders of magnitude. The disparity between fluorescence-induced
coherence decay rate and tunneling rate renders red-detuned lattices unsuitable
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G. Raithel and N. Morrow
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for experiments on well-to-well tunneling (and similar experiments that would
require long wave-packet coherence times).
We add that red-detuned optical lattices are still suitable to study wavepackets that are confined to single lattice wells, which have length scales smaller
than λ. Single-well wave-packets include sloshing-type wave-packets, observed
by Kozuma et al. (1996) and Raithel et al. (1998), and breathing-mode wavepackets (Raithel et al., 1997). In such cases, the decay rate of coherences between
the lowest few oscillatory states can be considerably less than the photon scattering rate due to the Lamb–Dicke effect (Dicke, 1953). A reasonable estimate is that
the coherence decay rate is reduced relative to the photon scattering rate by a fac= kL2 Z 2 , where Ωosc /2π is
tor of order of the Lamb–Dicke factor, η2 = h¯EΩRec
osc
the sloshing frequency of the atoms in the wells. This reduction factor is of order
0.1 for the parameters in Fig. 1.
3.2. “G RAY ” L ATTICES
Efficient laser cooling occurs on the blue-detuned side of F ↔ F = F resonances, as predicted by Guo and Berman (1993) and Grynberg and Courtois
(1994), and observed by Hemmerich et al. (1995). To evaluate the suitability
of blue-detuned lattices for tunneling and other wave-packet experiments, we
first consider the case of a lin-perp-lin lattice with single-beam intensity I1 =
10 mW/cm2 that is blue-detuned by 6Γ with respect to the F = 2 ↔ F = 2
component of the 87 Rb D2-line. The potential diagram of this lattice, shown in
Fig. 2(a), exhibits a lowest adiabatic potential that is quite shallow and fairly well
separated from the higher-lying potentials. The lowest adiabatic potential would
be identical zero if the only relevant transition were the F = 2 ↔ F = 2 transition, because an F ↔ F = F transition always has one dark state (i.e. a state
with zero light shift and fluorescence rate) regardless of the light polarization. The
negative light shift and the small modulation of the lowest adiabatic potential seen
in Fig. 2(a) result from residual interactions on the F = 2 ↔ F = 3 transition,
which is blue-shifted by 45Γ relative to the F = 2 ↔ F = 2 transition, and
the F = 2 ↔ F = 1 transition, which is red-shifted by 26Γ relative to the
F = 2 ↔ F = 2 transition.
As in the case discussed in Section 3.1, in the present case the atoms are also
rapidly optically pumped onto the lowest adiabatic potential and subsequently
cooled via Sisyphus cooling. The cooling and localization performance are displayed in Figs. 2(d) and (e), respectively. The achieved steady-state temperature is
considerably lower than that of the example considered in Section 3.1, while the
achieved degree of localization is somewhat less (λ/14 here vs λ/18 above). Also,
both the cooling and localization rates are about half of those in Section 3.1. The
diminished degree of steady-state localization obviously is a result of the larger
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ATOM MANIPULATION IN OPTICAL LATTICES
201
F IG . 2. Simulation of laser cooling in a blue-detuned optical lattice of 87 Rb on the D2-line (detailed parameters provided in text). (a) Adiabatic potentials Vα vs position. (b) Band structure Ek vs
quasimomentum q. (c) Fluorescence-induced decay rates γk of the lattice bands. (d) Kinetic energy of
cooled atoms vs time t. (e) Degree of atom localization in the wells vs time t. (f) Single-atom fluorescence rate vs time t. The exponential fits (dashed lines) in panels (d)–(f) yield the rates at which the
respective quantities approach a steady-state.
size of the localized quantum states supported by the lowest potential in Fig. 2(a),
which is shallower than the one in Fig. 1(a). The reduced temperature and cooling
rates are both a consequence of the low photon scattering rate (compare Fig. 1(e)
with Fig. 2(f)). In the present case, the fluorescence rate strongly decreases vs
time, because in the process of laser cooling atoms accumulate in the lowest few
bands supported by the lowest adiabatic potential. These bands have very low
fluorescence-induced decay rates because they inherit the “almost” dark character of the internal adiabatic states |Ψα=1 (Z) associated with the lowest adiabatic
potential. The very low steady-state fluorescence rate, evident from Fig. 2(f), has
given rise to the term “gray optical lattice” used for lattices that are blue-detuned
with respect to F ↔ F = F transitions. The low temperatures and fluorescence
rates afforded by the gray lattice are achieved on the expense of reduced cooling
and localization rates. The gray lattice in Fig. 2 cools about half as fast as the
red-detuned lattice in Fig. 1.
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The band structure, displayed in Fig. 2(b), reveals two tightly bound bands.
The lack of a larger number of tightly bound states renders this type of lattice
unsuitable for single-well wave-packet experiments. The lowest band has a width
of about 103 s−1 , meaning that the well-to-well tunneling rate of atoms prepared
in that band is about 103 s−1 . This value is in qualitative agreement with estimates
that can be made based on Eq. (10). The decay rates of the Bloch states, shown
in Fig. 2(c), are of order forty times as large as the tunneling rate associated with
the lowest band. While this factor is much smaller than the corresponding factor
for the red-detuned lattice studied in Section 3.1, which was of order 106 , it is not
low enough for experiments on well-to-well tunneling.
An improved version of a gray lattice can be realized by elimination of the main
source of fluorescence of atoms localized on the lowest adiabatic potential. These
atoms are in internal states |m = +F near the maxima of the σ + -polarized
standing-wave component of the lattice field, and |m = −F near the maxima
of the σ − -component. Therefore, the residual fluorescence of the Bloch states in
the lowest few bands of Fig. 2(c) mostly occurs via off-resonant excitation into
sub-states of the 5P3/2 F = 3 hyperfine level. This level can be eliminated by
using the D1-line—which only has F = 1 and F = 2 hyperfine components—
instead of the D2-line. Further, the F = 1 and F = 2 components of the D1-line
are separated by 138Γ , which is much larger than the corresponding separation
in the D2-line. Thus, in D1-lattices there is less perturbation due to the F = 1
component. These advantages lead to lower residual fluorescence rates and lower
temperatures.
In Fig. 3 we show simulation results for a lin-perp-lin lattice of 87 Rb with
single-beam intensity I1 = 10 mW/cm2 that is blue-detuned by 6Γ with respect to the F = 2 ↔ F = 2 component of the D1-line. To allow for a direct
comparison of Fig. 3 with Fig. 2, in the respective simulations we have used the
same wavelength (λ = 780 nm). The lowest adiabatic potential of the D1 optical
lattice (see Fig. 3(a)) is considerably shallower than that of the D2-lattice. Also,
in the D1-lattice all potentials are positive. These differences reflect the absence
of the F = 3 hyperfine component in the D1-lattice. The small residual modulation of the lowest adiabatic potential in the D1 lattice is caused by off-resonant
interaction with the F = 1 hyperfine component. The lower temperatures and
fluorescence rates afforded by the D1 gray lattice, evident from Figs. 3(d) and (f),
are achieved on the expense of reduced cooling and localization rates. The D1
gray lattice cools about half as fast as the D2 gray lattice. Nevertheless, the cooling rate of the D1 gray lattice is still high enough to allow for comfortable cooling
under typical experimental conditions.
The band structure of the D1 gray lattice (Fig. 3(b)) exhibits only one tightly
bound band. Due to the shallow potential barrier on the lowest adiabatic potential, the width of the lowest band and the well-to-well tunneling rate are fairly
high (7 × 103 s−1 ). Further, the fluorescence rate of the Bloch states in the lowest
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ATOM MANIPULATION IN OPTICAL LATTICES
203
F IG . 3. Simulation of laser cooling in a blue-detuned optical lattice of 87 Rb on the D1-line.
(a) Adiabatic potentials Vα vs position. (b) Band structure Ek vs quasimomentum q. (c) Fluorescence-induced decay rates γk of the lattice bands. (d) Kinetic energy of cooled atoms vs time t.
(e) Degree of atom localization in the wells vs time t. (f) Single-atom fluorescence rate vs time t.
The exponential fits (dashed lines) in panels (d)–(f) yield the rates at which the respective quantities
approach a steady-state.
band, shown in Fig. 3(c), is only of order 103 s−1 , entailing two important conclusions. First, most of the residual steady-state scattering seen in Fig. 3(f) is caused
by a small percentage of atoms that are not laser-cooled into the lowest band of
the lattice. Second, for the atoms that are cooled into the lowest band the tunneling rate is of order seven times higher than the fluorescence rate. Thus, the D1
gray optical lattice is suited to perform both efficient laser cooling and to observe
coherent well-to-well tunneling over multiple periods. No other type of lattices
we have studied offers this combination of possibilities.
Gray lattices can also be realized on the lower ground-state hyperfine component of alkali atoms (F = 1 for 87 Rb). We have observed cooling (experimentally and in simulations) on lin-perp-lin lattices of 87 Rb that are blue-detuned
by amounts of order 5Γ with respect to the F = 1 ↔ F = 1 component of the D1-line. Since those lattices require a re-pumper laser tuned to the
F = 2 ↔ F = 1 or 2 transition, some care needs to be taken to avoid the trap-
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G. Raithel and N. Morrow
[3
ping of atoms in F = 2 states that are dark with regard to the re-pumping laser.
While this issue is solvable, the observed cooling performance of gray lattices on
the lower ground-state hyperfine component is not as good as on the upper one.
3.3. M AGNETIC -F IELD -I NDUCED L ATTICES
There exist some less known types of laser cooling in lattices that are quite robust.
Some of these are reviewed in this subsection.
We consider a standing wave of well-defined helicity driving an atom bluedetuned from an F ↔ F = F transition. The lattice potentials produced by this
field are curves ∝ cos2 (2kL Z), with proportionality constants given by squares
of Clebsch–Gordan coefficients. For an isolated F ↔ F = F transition, i.e. if
there are no other coupled hyperfine levels, one of the potentials that correspond
to the outmost m-sublevels is identical zero, reflecting the presence of an exact
dark state. If there is a perturbing excited-state hyperfine level F = F + 1, the
lowest potential exhibits some added spatially modulated light shift. The internalstate wave-functions associated with the lattice potentials are equivalent to the
|m-states, since the circularly polarized standing wave does not couple different |m-states. If a weak transverse magnetic field B pointing in x-direction is
added, the degeneracy of the lattice potentials near the nodes of the light field
becomes lifted, as seen in Fig. 4(a), and the internal states associated with the
potentials become coherent mixtures of different |m-states. Obviously, at the exact nodes of the field the adiabatic states are given by the eigenstates |mx of the
x-component of angular momentum, Fˆx , rotated by π/2 about the y-axes into
the z-direction. Laser-cooling results from a Sisyphus-type mechanism that involves optical pumping from the higher onto the lowest adiabatic potential near
the maxima of the adiabatic potentials, where the field intensity is maximum,
and non-adiabatic transitions of atoms from the lowest potential back onto higher
adiabatic potentials in the node region of the laser field. Since the non-adiabatic
mixing near the field nodes is instrumental in closing the Sisyphus-type cooling
cycle, the cooling only works with a transverse magnetic field present. Therefore,
this type of cooling is known as magnetic-field-induced laser cooling (MILC, see
(Sheehy et al., 1990)).
The efficiency of this cooling can be optimized by tuning the strength of the
transverse magnetic field, and it is fairly easy to achieve steady-state temperatures and laser-cooling rates that closely rival those achieved with the previously discussed cooling methods (see Figs. 4(d) and (f)). For the case of the
D2-MILC-lattice studied in Fig. 4, the off-resonant interaction with the excitedstate hyperfine level F = 3, which is 45Γ above the F = 2 level, causes the
lowest adiabatic potential to exhibit a moderately deep potential well. This well
leads to considerable accumulation of atoms in the region near the field maxima,
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ATOM MANIPULATION IN OPTICAL LATTICES
205
F IG . 4. Simulation of magnetic-field-induced laser cooling in a blue-detuned optical lattice of
87 Rb on the D2-line. The single-beam lattice intensity is I = 10 mW/cm2 and the laser detuning
1
4Γ relative to the F = 2 ↔ F = 2 transition. A transverse magnetic field of 0.1 Gauss is applied.
(a) Adiabatic potentials Vα vs position. (b) Band structure Ek vs quasimomentum q. (c) Fluorescence-induced decay rates γk of the lowest ten lattice bands. (d) Kinetic energy of cooled atoms vs
time t. (e) Steady-state spatial distribution of the atoms in the five m-levels. (f) Single-atom fluorescence rate vs time t. The exponential fits (dashed lines) in panels (d) and (f) yield the rates at which
the respective quantities approach a steady-state.
as can be seen in Fig. 4(e). MILC-type lattices are somewhat reminiscent of gray
lattices, because the fluorescence rate drops as the cooling progresses (compare
Figs. 4(d) and (f)). The residual photon scattering rate is limited by the magneticfield-induced mixing into non-dark m-states and off-resonant excitation into the
F = 3 hyperfine level. For the lowest few bands, the fluorescence rate has values
around 50 × 103 s−1 (see Fig. 4(c)), which exceeds the width of the lowest band
by many orders of magnitude. In fact, since in the MILC-type lattice the wellto-well separation is λ/2, as opposed to λ/4 in Figs. 1–3, the tunneling rates in
MILC-type lattices are extremely small (only about 0.5 s−1 in the case of Fig. 4).
Therefore, MILC-type lattices are not suited for tunneling experiments. As an
aside, it is noted that the issue of how to measure coherent tunneling in MILC-type
lattices, even if it were present, would impose further problems, because atoms in
neighboring wells cannot be distinguished by their magnetic moments.
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The number of tightly bound levels in Fig. 4(b) is only about two, rendering
MILC-type lattices rather unsuitable for wave-packet experiments in single wells.
In calculations and experiments not presented, we have also investigated cases of
MILC-type lattices on the D1-transition of rubidium. We have found no significant difference between MILC-type lattices on the D1- and the D2-transitions.
3.4. R ELATED L ASER -C OOLING M ETHODS
3.4.1. “Corkscrew” Cooling
The last configuration we consider consists of two counter-propagating laser
beams of opposite helicity (in a fixed frame), which produce a light-field of
position-independent intensity and spatially rotating and temporally fixed linear
polarization. Due to this polarization geometry, laser-cooling in this configuration
(Dalibard and Cohen-Tannoudji, 1989) has been dubbed “corkscrew cooling”.
Since in the corkscrew field geometry there are no spatially dependent light-shift
potentials, this geometry is not suited for wave-packet and tunneling experiments.
As shown in Fig. 5(a) for the case I1 = 10 mW/cm2 and laser detuning δ =
−6Γ relative to the F = 2 ↔ F = 3-component of the 87 Rb D2-line, corkscrew
laser-cooling performs not quite as well as cooling in comparable red-detuned
lin-perp-lin lattices (see Fig. 1). For the case in Fig. 5(a), the fluorescence rate is
about 400 × 103 s−1 and does not significantly depend on time.
3.4.2. Velocity-Selective Coherent Population Trapping
We have found that optical lattices that exhibit dark or nearly dark states generally
exhibit some degree of velocity-selective coherent population trapping (VSCPT),
which was first observed by Aspect et al. (1988) in the cooling of metastable
helium. VSCPT refers to optical pumping into coherent superpositions of entangled states of the internal and external degrees of freedom of the trapped atoms
that exhibit particularly low fluorescence rates. Atoms tend to accumulate in such
states. In our one-dimensional lattice geometry, coherently coupled states have
center-of-mass momenta that differ by integer multiples of 2kL .
Even the lin-perp-lin D1-gray lattice studied in detail in Section 3.2 and Fig. 3,
the cooling is a combination of Sisyphus cooling and VSCPT, as evidenced by
the presence of separated peaks of the momentum distribution at integer multiples
of 2kL (see Fig. 5(b)). These separated peaks indicate that atoms tend accumulate
in certain Bloch states that are particularly long-lived. The accumulation of atoms
in selected Bloch states implies some amount of well-to-well coherence, as has
been observed elsewhere (Teo et al., 2002). Signs of VSCPT are further observed
on the blue side of F ↔ F = F transitions in corkscrew field configurations
produced by counter-propagating circularly polarized beams with opposite helicity (in a fixed frame). An example is shown in Fig. 5(c), where I1 = 4 mW/cm2
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ATOM MANIPULATION IN OPTICAL LATTICES
207
F IG . 5. (a) “Corkscrew cooling” for a field that is red-detuned relative to the F = 2 ↔ F = 3
component of the 87 Rb D2-line. The plot shows average energy vs time (solid) and an exponential
fit (dashed). (b)–(f) Indications of velocity-selective coherent population trapping in the momentum
distributions for various field configurations explained in the text.
and the field is detuned by 3Γ relative to the F = 2 ↔ F = 2 component of
the 87 Rb D1-line. VSCPT in fairly clean form occurs if the field has a corkscrew
polarization configuration, is weak, and is on-resonant with an F = 1 ↔ F = 1
or F = 0 transition so that other types of laser cooling are absent. The case
in Fig. 5(d), which is for I1 = 0.2 mW/cm2 and zero detuning relative to the
F = 1 ↔ F = 1 component of the 87 Rb D1-line, exhibits very clear VSCPT
momentum peaks at ±h¯ kL , as is typical for VSCPT (Aspect et al., 1988). Measurements on VSCPT of rubidium on the F = 1 ↔ F = 1 component of the
D1-line have been performed by Esslinger et al. (1996). On the same transition
of the D2-line, the VSCPT is less pronounced due to off-resonant fluorescence
on the F = 2 ↔ F = 3 transition (see Fig. 5(e); lattice intensity and detuning
same as in Fig. 5(d)). The additional fluorescence causes increased coherence loss
and therefore washes out the VSCPT momentum peaks. Finally, three-component
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G. Raithel and N. Morrow
[4
VSCPT occurs on F = 2 ↔ F = 2-transitions, as evident in Fig. 5(f) (lattice
intensity and detuning same as in Fig. 5(d)). There, the three peaks at momentum values 0 and ±2kL reflect the accumulation of atoms in a gray, coherent
superposition of the three states |m = −2, k = −2kL , |m = 0, k = 0 and
|m = 2, k = 2kL . Various related configurations that lead to VSCPT have been
discussed in detail by Aspect et al. (1989) and Papoff et al. (1992).
4. Periodic Well-to-Well Tunneling in Gray Lattices
In the following, we discuss experiments on periodic well-to-well tunneling of
87 Rb atoms in optical lattices that exhibit efficient sub-Doppler laser cooling.
From Section 3 it is concluded that within this class of lattices only gray optical
lattices on the D1-line are a reasonable choice to conduct tunneling experiments,
because these are the only lattice type that combines a relatively large tunneling
frequency (of order 104 s−1 ) with a decoherence rate low enough for the tunneling
to become observable.
4.1. E XPERIMENTAL AND S IMULATION R ESULTS
In each cycle of our experiment (Dutta et al., 1999), 87 Rb atoms are collected for
14 ms in a standard vapor-cell magneto-optic trap (MOT). After switching off the
MOT magnetic field, the atomic cloud is further cooled for about 1 ms in a threedimensional corkscrew optical molasses. The atoms are then loaded into the onedimensional gray optical lattice formed by two counter-propagating laser beams
with orthogonal linear polarizations (lin-perp-lin lattice). As in Fig. 3, the lattice is
blue-detuned by δ = 6Γ from the D1 F = 2 to F = 2 hyperfine component (λ =
795 nm). A re-pumping laser tuned to the transition F = 1 → F = 2 is required
to re-pump atoms scattered into the F = 1 ground-state hyperfine level back into
the F = 2-level the lattice is operating on. It takes ∼1 ms to cool most atoms into
the lowest potential Vα=1 (Z). We have seen in our QMCWF simulations that at
single-beam intensities around I1 = 5 mW/cm2 60% of the atoms are prepared
in the lowest band of the lattice. Atoms in wells with predominantly σ + - (σ − -)
polarized light are predominantly in the |m = 2 (|m = −2) state. In order to
ensure that the σ + - and σ − -wells of the lattice are equally deep, great care has
been taken to reduce environmental magnetic fields to values of order 1 mG or
less.
To initiate observable tunneling between the σ + - and σ − -wells of the lattice,
the atom distribution needs to be initialized such that atoms are present in only
one type of wells. The initialization is accomplished as follows. The re-pumping
laser is turned off, and a σ + -polarized laser, which is co-linear with the lattice
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ATOM MANIPULATION IN OPTICAL LATTICES
209
beams and resonant with the D2 F = 2 → F = 2 transition, is turned on
for 15 µs (intensity ∼0.1 mW/cm2 ). This laser pulse removes most atoms from
the σ − -lattice wells by optical pumping into the F = 1 ground state, whereas
the atoms in the σ + -wells mostly survive. The intensity and duration of this initialization laser is adjusted such that about half the atoms are removed from the
lattice. The removed atoms remain inactive for the remainder of the tunneling
experiment. Coherent well-to-well tunneling of the atoms left over in the lattice
commences at the end of the initialization pulse.
A tunneling event from a σ + -well into a σ − -well is associated with an exchange of 4h¯ angular momentum between the atom and the lattice field, amounting to an exchange of two photon pairs between the σ + - and σ − -components of
the lattice beams. All ∼106 atoms in the lattice tunnel in phase, because their
wave-functions were prepared identically by the initialization laser. While the
photon exchange rate caused by a single tunneling atom would not be detectable,
the exchange rate caused by the whole atomic sample is substantial. We can measure the tunneling current by separating the lattice beams after their interaction
with the atoms into their σ -polarized components, and measuring the difference
of the powers of the σ + - and σ − -components. The power exchange between the
components can be described as
Pσ = N
hc d
m,
λ dt
(11)
where N is the number of atoms, and dtd m is the rate of change of the average magnetic quantum number. Based on the measured tunneling period and the
known atom number, the maximum tunneling-induced power exchange can be estimated to be of order 10−3 to 10−4 of the incident power. The tunneling-induced
power exchanges observed in the experiment are of that order of magnitude.
It is noted that the periodic tunneling current is measured in real-time and that
the measurement is non-destructive. A single experimental cycle yields the timedependence of the tunneling current over the whole time interval of interest, which
is of order one millisecond. Since it is possible to observe the tunneling signal on
an oscilloscope without data averaging, experimental parameters such as background magnetic fields and lattice-beam alignments can be optimized quite easily.
Averages over of order 1000 realizations, which take less than a minute to accumulate, yield noise-free measurements of the tunneling current.
In Fig. 6, left panel, a set of experimental data taken at different lattice intensities is shown. The curves exhibit oscillations with a period of 150 to 200 µs;
these oscillations reflect the periodic tunneling current of atoms trapped in the
lowest band of the optical lattice. As estimated in Section 3.2, the coherence of
the tunneling lasts quite long: the tunneling signal is noticeable over at least five
tunneling periods. At intensities above ≈5 mW/cm2 , a higher-frequency periodic
contribution to the tunneling current is observed. To show this more clearly, the
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[4
F IG . 6. Experimental (left) and simulated (right) periodic well-to-well tunneling currents in a gray
lin-perp-lin optical lattice. The curves are shifted vertically by amounts proportional to the respective
single-beam intensities I1 . In the experiment, I1 is varied up to 15.5 mW/cm2 , and in the QMCWF
simulations up to 15 mW/cm2 . (Reprinted with permission from (Dutta et al., 1999).)
F IG . 7. Theoretical (a) and experimental (b) result for the tunneling current at I1 = 10 mW/cm2 .
The derivative of the magnetic-dipole autocorrelation function (c), d/dτ ψ|Fˆz (t + τ )Fˆz (t)|ψ, exhibits a time-dependence similar to that of curves (a) and (b). (Reprinted with permission from (Dutta
et al., 1999).)
data taken at 10 mW/cm2 are displayed in more detail in Fig. 7, curve (b). The
higher-frequency oscillations, the maxima of which are highlighted in Fig. 7 by
vertical lines, are due to tunneling on the first excited lattice band. This band has
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ATOM MANIPULATION IN OPTICAL LATTICES
211
a larger width than the lowest band, resulting in faster tunneling oscillations. We
simulated the experiment using QMCWF simulations, shown in the right panel of
Fig. 6 and in Fig. 7, curve (a). The agreement between theory and experiment in
both the long-period and short-period oscillations is excellent.
The tunneling should also be visible in the autocorrelation function of the
z-component of the magnetic-dipole, Fˆz . At integer multiples of the tunneling
period, atoms should mostly reside in the type of wells they resided in initially,
and the magnetic-dipole correlation should be positive. Conversely, at half-integer
multiples of the tunneling time the atoms should mostly reside in the opposite
type of wells, leading to negative values of the magnetic-dipole correlation function. Very importantly, this correlation should even exist if the atoms were not
initialized in one type of wells. The QMCWF method allows the computation of
two-time correlation functions, as explained by Marte et al. (1993). Our result for
the parameters of Fig. 7, shown in curve (c) of the figure, confirms our expectation: the time derivative dτd ψ|Fˆz (t + τ )Fˆz (t)|ψ is very similar to the actual
tunneling current. Note, however, an important difference: the tunneling current,
shown in curves (a) and (b) of Fig. 7, reveals tunneling oscillations only after suitable initialization of the atoms in one type of wells, while curve (c) is obtained
by simulating the evolution of quantum trajectories under steady-state conditions,
i.e. without any initialization procedure.
4.2. A NALYSIS BASED ON BAND -S TRUCTURE
In the following, we analyze the described experiment using the band structure of
the system. In the initialization process, most atoms are removed from one type
of wells. Without loss of generality we may assume that the atoms remaining in
the lattice are located in the σ + -wells (see Fig. 8). Also, most of the observed
signal is due to atoms in the lowest pair of bands. Inspecting the eigenfunctions
associated with the band structure plotted in Fig. 8, it is found that the Bloch states
for q = 0 in the lowest two bands approximately are
|1 ≈ −1 − sin(2kL Z) |m = −2 + 1 − sin(2kL Z) |m = 2,
|2 ≈ 1 + sin(2kL Z) |m = −2 + 1 − sin(2kL Z) |m = 2
(12)
with respective energies of E1 = 3.76 ERec and E2 = 5.54 ERec . The initialization process amounts to the generation of a symmetric superposition of Bloch
states, √1 (|1 + |2) ∝ (1 − sin(2kL Z))|m = 2, corresponding to the localiza2
tion of most atoms in the σ + -wells (dashed curve in Fig. 8(b)). The time-evolved
state, √1 (exp(− iEh¯1 t )|1 + exp(− iEh¯2 t )|2), is identical with the initial state—up
2
h
. This
to an irrelevant global phase—for times that are integer multiples of E2 −E
1
value is the well-to-well tunneling period. At half-integer multiples of the tunneling period, the wave-function is ∝(|1 − |2) = (1 + sin(2kL Z))|m = −2,
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G. Raithel and N. Morrow
[4
F IG . 8. (a) Lowest four lattice bands for I1 = 10 mW/cm2 and detuning 6Γ . The initialization
procedure applied in the experiment amounts to the preparation of symmetric superpositions of Bloch
states from the lowest band pair, as indicated by the black circles for the case q = 0. (b) Lowest
adiabatic potential Vα=1 (Z) (solid line, left axis) and approximate spatial probability distributions of
the atoms in state |m = 2 immediately after the initialization (dashed line, right axis) and in state
|m = −2 half a tunneling period later (dotted line).
corresponding to a localization of most atoms in the σ − -wells (dotted curve in
Fig. 8(b)).
The described situation obviously parallels that of periodic tunneling in a
double-well potential, but there are some differences. The above description in
terms of Bloch functions properly accounts for the periodicity of the superposition state: in the Bloch-state description, a delocalized atom does not tunnel
between individual wells, but tunnels back and forth between all σ + -wells of the
lattice and all σ − -wells of the lattice. Consequently, there is no directionality in
the tunneling process (as opposed to the case of a double-well potential). Further,
since the quasimomentum q is quite randomly distributed, the tunneling period
h
E2 −E1 is not fixed but follows a quite random probability distribution. Inspecting
the band structure in Fig. 8(a) it becomes obvious that the tunneling period vs q
exhibits a broad minimum for q = 0. Therefore, the tunneling signals produced by
a fairly large group of atoms with q ∼ 0 will add up, while atoms with quasimomenta very different from zero will produce tunneling signals that tend to cancel
each other. The experimentally observed tunneling signal should therefore equal
h
E2 −E1 evaluated at q = 0. Detailed numerical modeling confirms this assessment
for all lattice intensities we have studied (Dutta et al., 1999). For the specific case
h
= 150 µs.
of Fig. 8, the tunneling period is expected to be TT = ERec (5.54−3.76)
This value is experimentally observed (see Fig. 7, where the lattice parameters are
the same as in Fig. 8).
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ATOM MANIPULATION IN OPTICAL LATTICES
213
In the following, we comment on the dependence of the tunneling period on
the lattice intensity. As observed in Fig. 6, the tunneling period exhibits a broad
minimum around I1 ∼ 5 mW/cm2 . While this behavior accords with predictions
based on the band structure and QMCWF simulations, it evades an immediate
explanation based on the adiabatic potentials of the system. The lowest adiabatic light-shift potential Vα=1 (Z) continuously decreases with decreasing intensity. As a result, one would expect the tunneling time to continuously decrease
with decreasing intensity. Such a trend is observed experimentally in the range
I1 > 5 mW/cm2 . However, in the range I1 < 5 mW/cm2 the experimental observation contradicts the expected trend. The discrepancy can be partially attributed
to the presence of a gauge potential that needs to be added to the lowest adiabatic
potential in order to improve the description of the tunneling behavior. The gauge
potential, predicted to occur in light-shift potentials by Dum and Olshanii (1996)
and observed by Dutta et al. (1999), is
d2 Gα=1 (Z) = − h¯ 2 /2M Ψα=1 (Z) 2 Ψα=1 (Z) ,
(13)
dZ
where |Ψα=1 (Z) is the position-dependent internal state associated with the
lowest adiabatic potential. The potential Gα=1 (Z) is always positive and is, in
the present case, intensity-independent and peaks at the maxima of the lowest adiabatic potential Vα=1 (Z). Adopting the notion that the sum potential
Vα=1 (Z) + Gα=1 (Z) determines the tunneling time, the gauge potential increases
the tunneling time and improves the agreement between the observed tunneling time and the tunneling time one may estimate based on a simple potential
picture. However, the addition of an intensity-independent potential Gα=1 (Z) to
Vα=1 (Z) cannot explain why at the lowest intensities in Fig. 6 the tunneling time
increases with decreasing intensity. It turns out that in this low-intensity domain
the notion that the wave-function adiabatically evolves (and tunnels) on a single potential breaks down. The underlying reason for this breakdown is that the
Born–Oppenheimer approximation separating the dynamics of internal and external degrees of freedom fails. A more detailed discussion of this effect can be
found in (Dutta and Raithel, 2000).
5. Influence of Magnetic Fields on Tunneling
5.1. M OTIVATION AND E XPERIMENTAL O BSERVATIONS
In Section 4 we have studied well-to-well tunneling for the case of zero applied
magnetic field. In that case, the σ + - and σ − -wells of the lattice are identical, and
the tightly bound states in them are degenerate with each other. This symmetry
can be lifted by the application of a magnetic field parallel to the lattice-beam
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G. Raithel and N. Morrow
[5
F IG . 9. (a) Lowest adiabatic potential (left) and band structure (right) of a gray lin-perp-lin optical lattice with single-beam intensity I1 = 11 mW/cm2 and detuning 6.3Γ with respect to the
5S1/2 , F = 2 → 5P1/2 , F = 2 component of the 87 Rb D1-line for B = 0 mG. (b) Same as (a),
except that a longitudinal magnetic field B = 12 mG is applied. The arrows in the band structures
indicate the coherences that dominate the experimentally observed well-to-well tunneling signals,
shown below in Fig. 10.
directions, B . In this section, we study the effect of such longitudinal magnetic
fields on the tunneling dynamics.
In Fig. 9 we show the influence of a weak longitudinal magnetic field on the
lowest adiabatic potential and the band structure of a typical gray lattice. The
single-beam lattice intensity is I1 = 11 mW/cm2 and the detuning 6.3Γ with
respect to the 5S1/2 , F = 2 → 5P1/2 , F = 2 component of the 87 Rb D1-line.
Panel (a) shows a situation similar to that in Fig. 8. In panel (b), a magnetic field
B = 12 mG parallel to the quantization axis of the lattice is added. This field
can be accounted for
in Eq. (8) and in QMCWF simulations by adding a positionindependent term m gF μB B m|mm| to the atom-field interaction, where the
g-factor gF = 1/2 for 5S1/2 , F = 2 of 87 Rb. According to the predominant
magnetic states in the σ + - and σ − -wells of the lattice, indicated in the left panels
of Fig. 9, the longitudinal magnetic field lowers the σ − -wells (left well) by an
energy of h × 1.4 kHz = 0.386 ERec per mG and raises the σ + -wells (right
well) by the same amount. Due to these shifts, clearly seen in the left panel in
Fig. 9(b), the localized states in the two types of wells are tuned out of resonance
with each other. Using selected magnetic-field values, higher-lying states in the
5]
ATOM MANIPULATION IN OPTICAL LATTICES
215
down-shifting type of wells can be brought into resonance with lower-lying states
in the up-shifting wells. In Fig. 9(b), for instance, the lowest (and only) state in
the up-shifted well is resonant with the first excited state in the down-shifted well.
These resonances manifest themselves in the tunneling behavior of the system.
In order to measure the dependence of the tunneling frequency on B , we
apply a variable, well-defined (to within 1 mG) magnetic field parallel to the lattice laser beams using a set of Helmholtz coils. Atoms are laser-cooled into the
slightly magnetized lattice using the same procedure as described in Section 4.
In order to observe the well-to-well tunneling, the atoms are initialized into one
set of wells via application of a 10 µs long σ -polarized pulse resonant with the
5S1/2 , F = 2 → 5P3/2 , F = 2 transition. This pulse selectively removes atoms
from one type of lattice wells into the 5S1/2 , F = 1 level and thereby initializes the remaining atoms in the other type of wells. At the end of the initialization
pulse the remaining atoms begin to tunnel in-phase between the wells. The tunneling current is measured non-destructively and in real-time by detecting the power
exchange between the σ + - and σ − -polarized components of the lattice beams
after their interaction with the atom cloud.
Figure 10(a) shows the tunneling current measured for the indicated values of
the applied magnetic field B . The signals are approximately symmetric about the
value B = 0, and the tunneling frequency generally increases with |B |. The experimental data are compared with corresponding results of QMCWF simulations,
shown in Fig. 10(b). We observe satisfactory agreement between experiment and
simulations. The simulations show more high-frequency modulations than the experimental data; this difference may be attributed to the limited bandwidth of the
photodiode detector used in the experiment. Also, the simulated data are more
asymmetric about B = 0 than the experiment. In this regard, it is noted that a
certain degree of asymmetry is to be expected, because the polarization of the
initialization pulse is kept fixed. Consequently, for one field polarity the deeper
wells are depleted of atoms during the initialization, while for the other polarity
the less deep wells are depleted. This asymmetry in the initialization sequence
causes some differences between tunneling signals observed for magnetic fields
B of equal magnitude but opposite polarity.
The tunneling frequencies νT of the experimental data in Fig. 10 have been
determined graphically and are represented as a function of B in Fig. 11. As
expected from the picture presented in Fig. 9, the tunneling frequency is symmetric in B (even though the underlying measured tunneling-current curves are not
entirely symmetric in B ). The tunneling frequency generally increases with the
magnitude of B . However, at values B = ±12 mG secondary minima are observed. This behavior can be qualitatively explained by considering a double-well
potential with two bound states in each of the wells. In the following, we first
discuss such a simplified double-well model that largely reproduces Fig. 11. We
then turn to a more rigorous analysis based on the band structure of the lattice.
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G. Raithel and N. Morrow
[5
F IG . 10. (a) Tunneling currents measured for the indicated values of the longitudinal magnetic
field B and lattice parameters as in Fig. 9. (b) Corresponding results obtained from QMCWF simulations.
F IG . 11. Tunneling frequencies νT vs applied longitudinal magnetic field B obtained from the
experimental data in Fig. 10(a).
5]
ATOM MANIPULATION IN OPTICAL LATTICES
217
5.2. I NTERPRETATION OF THE R ESULTS BASED ON T WO M ODELS
In a simplified double-well picture, the situation of Fig. 9 can be described by
localized ground and excited states |0L and |1L in the left well of a double-well
potential, and two analogous states |0R and |1R in the right well. The tunnelinginduced coupling between the ground states, |0L and |0R , is denoted c00 , and
that between the excited states c11 . The off-resonant couplings between |0L and |1R and |1L and |0R are both c01 . The band structure in Fig. 9 suggests
the use of the following values for the coupling constants: c00 = h × 3 kHz,
c11 = h × 12 kHz, and c01 = h × 7 kHz. The magnetic-field-free energies of the
localized ground (excited) states are estimated as 0.5h × fosc and 1.5h × fosc with
an oscillation frequency fosc = 35 kHz. In analogy with the situation in Fig. 9,
the effect of a magnetic field B is that the states in the left well are down-shifted
by d × B with d = 1.4 kHz per mG, while the states in the right well are upshifted by that same amount. The corresponding Hamiltonian, represented in the
basis {|0L , |0R , |1L , |1R },
⎛ 1f − d × B
⎞
c
0
c
⎜
Hˆ = ⎜
⎝
2 osc
00
1f
+
d × B
osc
2
c01
0
0
c01
3f
2 osc − d × B
c01
0
c11
c11
3f
+ d × B
osc
2
c00
01
⎟
⎟ (14)
⎠
has eigenvalues vs B as shown in Fig. 12(a). The lowest tunneling frequencies
in this system are given by the energy differences between the lowest two of
F IG . 12. (a) Energy levels of the double-well model system discussed in the text vs B . The
two arrows identify the energy differences that correspond to the lowest tunneling frequencies of the
system. (b) Lowest tunneling frequencies as indicated in (a) vs B .
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G. Raithel and N. Morrow
[5
F IG . 13. Frequencies of the coherences between the lowest band pair at q = 0 (squares) and
between the first- and second-excited band at q = ±1 (triangles) vs magnetic field B . The frequencies
are determined as shown by the arrows in Fig. 9.
eigenvalues and the next-higher pair of eigenvalues, as indicated by the two arrows
in Fig. 12(a). Plotting these two tunneling frequencies vs B and assuming that
the lower one will be dominant in an experimental observation of the tunneling,
we obtain a plot that closely resembles the actual experimental observation in the
lattice (compare Fig. 11 with Fig. 12(b)).
For a more rigorous description using the band structure of the system, we first
recall that the experimentally observed tunneling signal is due to coherences between neighboring bands, which are generated by application of the initialization
pulse. Since the atoms in the lattice follow a fairly random distribution in quasimomentum q, the coherences that produce observable effects will come from
regions in the band structure where the energy difference between neighboring
bands is stationary in q. The coherences at q-values as identified by the arrows
in Fig. 9 are likely to produce a signal, because their frequencies exhibit a broad
maximum as a function of quasimomentum q. Plotting the frequency difference
between the lowest band pair at q = 0 and between the first- and second-excited
band at q = ±1 vs B , we obtain the curves shown in Fig. 13. The lower envelope
of these curves, identified by the filled symbols, agrees well with the experimentally obtained result shown in Fig. 11.
The findings obtained in this section lead to the following summarizing assessment. For B = 0, the atoms tunnel resonantly on the lowest band between
the σ + - and σ − -wells of the lattice. In the range 0 < |B | < 6 mG the
tunneling frequency increases, because the tunneling becomes increasingly offresonant with increasing |B |. The observed tunneling frequency has a maximum
at |B | ≈ 6 mG, because at this field value the tunneling between the ground
6]
ATOM MANIPULATION IN OPTICAL LATTICES
219
bands associated with the σ + - and σ − -wells is off-resonant by about the same
amount as tunneling from the ground band of the up-shifted wells into the first
excited band of the down-shifted wells. Around |B | ≈ 12 mG the tunneling
frequency has as shallow secondary minimum because at that field value the tunneling from the ground band of the up-shifted wells into the first excited band
of the down-shifted wells is resonant. The observed dependence of the tunneling
frequency on B therefore reflects the passage of the system through a couple of
tunneling resonance. At each resonance, a minimum of the tunneling frequency
occurs.
6. Sloshing-Type Wave-Packet Motion
6.1. WAVE -PACKETS L OCALIZED IN S INGLE L ATTICE W ELLS
So far, we have studied well-to-well tunneling in gray lattices initiated by an initialization laser pulse that removes atoms from one type of wells. In the following,
we will be interested in wave-packets evolving in single wells of the lattice, with
negligible tunneling-induced well-to-well coupling. An optical lattice must have
at least a couple of tightly bound bands so that meaningful wave-packets in single
wells can be formed. Lattices of the red-detuned type discussed in Section 3.1
usually are deep enough to support a number of tightly bound states that is sufficient for the excitation of single-well wave-packets. In contrast, gray lattices in a
vanishing magnetic field support only one tightly bound band and will therefore
not allow one to form superposition states in single wells (see Sections 3.2 and 4).
This situation changes when a longitudinal magnetic field of order B ∼ 100 mG
is added, because one type of lattice wells is deepened by the magnetic field,
while the other type of wells essentially disappears. For sufficiently large B , the
deepened wells support a sufficient number of tightly bound bands to form wavepackets in single wells.
In this section, we will consider atomic center-of-mass wave-packets that are
excited by a sudden displacement of the lattice, which causes a subsequent
sloshing-type wave-packet motion that takes place in single lattice wells (Raithel
et al., 1998). In particular, we will be interested in the role of the longitudinal
magnetic field B in enabling the formation of these sloshing-type wave-packets.
It is noted that, while we refer to the sloshing-type atomic states as wavepackets, these are not pure quantum states but are more properly described by a
time-dependent density operator. The atoms are initially prepared by laser cooling
in an incoherent, quasithermal density operator ρˆ0 with most population residing
in the lowest oscillatory states of the lattice wells. The wave-packet initialization, which is implemented via a lattice shift, amounts to a certain excitation of
higher-lying oscillatory states. The lattice-shift-induced excitation is an entirely
220
G. Raithel and N. Morrow
[6
coherent process. In a harmonic approximation of the lattice wells, the excitation
is described by the application of the usual shift operator Dα = exp(αa † − α ∗ a)
on ρˆ0 , yielding the density operator after the shift,
ρˆ = Dα ρˆ0 Dα† .
(15)
There, a and a † are the lowering and raising operators, the complex number α
is the shift parameter, and |α|2 is an approximate measure for how high up the
harmonic energy ladder the state is shifted. After the shift is applied to the lattice, fluorescence of the atoms and the associated laser cooling cause damping of
the sinusoidal oscillation, Tr[ρ(t)Z].
ˆ
In the experiment, damping can be directly
observed as a decay of a sinusoidal signal. The experimentally measured signal
decay is not only due to coherence decay but, in large parts, also due to the anharmonicity of the lattice wells (i.e. the harmonic-oscillator formalism can merely
serve to provide a qualitative discussion). In this context, we also re-iterate that
QMCWF simulations one may perform in order to model optical-lattice experiments are entirely accurate in that they yield an approximation to the evolution of
a density operator (as opposed to the evolution of a single wave-function). Keeping these clarifications in mind, we will continue to use the term “wave-packets”
for the states discussed this section.
6.2. E XPERIMENTAL S TUDY OF S LOSHING -T YPE M OTION
IN A M AGNETIZED G RAY L ATTICE
In order to experimentally initiate sloshing-type motion, we apply a voltage
change to a phase modulator positioned in one of the lattice beams after a steadystate of laser-cooling is achieved. The resultant sudden shift in lattice position,
amounting to 0.1λ, initializes the center-of-mass wave-packet motion on the
lowest adiabatic potential. To measure the subsequently occurring sloshing-type
wave-packet motion, the lattice beams are directed onto photodiodes after they
have interacted with the atomic cloud. The power difference signal P (t) between the photodiodes is then measured. The average electric-dipole force acting
on the atoms is related to the measured power exchange P (t) via
P (t) = N cF ,
(16)
where F is the electric-dipole force averaged over the ensemble of N atoms
(Raithel et al., 1998).
In Fig. 14(a), sloshing-type wave-packet oscillations measured for the indicated values of B are represented. The utilized lattice is a gray optical lattice
with single-beam intensity I1 = 11 mW/cm2 and a detuning of +6.3Γ relative
to the F = 2 ↔ F = 2 component of the 87 Rb D1-line (same as in Section 5).
As evident in Fig. 14(a), there are two regimes of the sloshing-type motion. In
6]
ATOM MANIPULATION IN OPTICAL LATTICES
221
F IG . 14. (a) Wave-packet oscillations for the indicated values of B . (b) Wave-packet oscillations
for low B -values varied in small steps of 1.2 mG. (c) Frequency νslosh of the wave-packet oscillations obtained from the experimental data (squares) and from the geometry of the adiabatic potentials
(triangles). The experimental uncertainty νslosh = 15%.
a low-magnetic-field regime, |B | < 10 mG, there is no clear signature of a
sloshing-mode wave-packet oscillation. There are, however, some reproducible
low-amplitude higher-frequency structures that were also observed in QMCWF
simulations (not shown here). The absence of sloshing oscillations in the lowmagnetic-field regime, |B | < 10 mG, can be attributed to two factors. First, the
lattice wells are not deep enough to support more than one tightly bound state.
Therefore, it is not possible to form wave-packet states that oscillate in individual
wells. Second, in shallow lattices such as gray lattices the sloshing-type motion
competes with rapid tunneling between neighboring wells. Tunneling leads to a
rapid spread of wave-packets over multiple lattice wells. The dipole force F averaged over spread-out wave-packets will always be near zero, as the averaging
will extend over regions of both polarities of the force. Therefore, in the lowmagnetic-field regime any net sloshing-type wave-packet signal that might still
222
G. Raithel and N. Morrow
[7
appear will be very weak and it will reflect a complicated superposition of effects
caused by both tunneling and sloshing-type dynamics. This is demonstrated in the
detailed plot in Fig. 14(b).
With increasing magnetic field, the σ + - and σ − -wells of the lattice become increasingly asymmetric, leading to a suppression of well-to-well tunneling on the
lowest tightly bound bands and to the appearance of more and more tightly bound
bands in the deepening wells. Therefore, for large enough |B | we expect to observe the signatures of well-defined sloshing-mode oscillations. Fig. 14(a) shows
that the high-magnetic field regime in which the gray lattice supports sloshingmode oscillations approximately is |B | > 20 mG. As expected in this regime,
the frequency of the sloshing-mode oscillations is dependent on the shape of the
lowest adiabatic potential near its minima. In Fig. 14(c), the solid squares represent the frequencies νslosh of the sloshing oscillations as a function of the magnetic
field obtained from the experimental data. The observed trend reflects the fact that
both the depth and the curvature of the wells increase with increasing magnetic
field. Approximating the lattice wells by harmonic potentials that match the curvatures of the lowest adiabatic potential at the minima, we find estimated oscillation
frequencies shown by the triangles in Fig. 14(c). The estimated frequencies are in
quite good agreement with the experimental values, but show a systematic trend
of being ∼20% larger. The deviation may be attributed to the anharmonicity of the
lattice wells. The anharmonicity of the wells also is the main cause of the signal
decay that is observed to take about five wave-packet oscillations (see Fig. 14(a)).
7. Conclusion
In this chapter we have compared different types of one-dimensional optical
lattices with regard to their laser-cooling performance and their suitability for
experiments on well-to-well tunneling and sloshing-type wave-packet motion.
The theoretical models used have been explained in some detail. While numerical results are provided for rubidium, the results and conclusions are expected
to be representative for optical lattices of many atomic species (alkaline atoms,
metastable noble gases, etc.). Only one type of lattice has been identified that
provides reasonably fast and efficient laser cooling, high tunneling rates, and
steady-state coherence decay rates that are significantly lower than the tunneling
rate.
In the second half of the chapter, we have presented typical results on wavepacket motion in a gray optical lattice. The main findings can be summarized as
follows. In a regime of very low magnetic fields parallel to the lattice beams, the
predominant dynamics of atoms is due to well-to-well tunneling. We have explained the magnetic-field dependence of the tunneling using a simplified doublewell potential model as well as the exact band structure of the system. We found
8]
ATOM MANIPULATION IN OPTICAL LATTICES
223
that the tunneling current vs applied magnetic field exhibits signatures of a couple of tunneling resonances. In a domain of higher magnetic fields, the tunneling
rate between the lowest localized center-of-mass states of the lattice generally
decreases, and the number of localized states in the lattice wells increases. Consequently, in the domain of higher magnetic field the gray lattices are found to
support sloshing-type wave-packet oscillations.
In future research, we intend to study non-linearities in the discussed types of
wave-packet motion. We have observed a significant dependence of sloshing-type
wave-packet oscillations on the average atom density in the lattice. This dependence is due to the back-action of the wave-packet oscillation on the refractive
index which the oscillating atomic ensemble presents to the lattice beams. The resultant position- and time-dependence of the lattice phase and its coupling to the
atomic motion amounts to a non-linear atom-field coupling and to the presence
of long-range atom-atom interactions in the lattice. Interesting avenues for further
research include the study of wave-packet motion of Bose–Einstein condensates
(BECs) in lattices. In this case, additional non-linearities will arise from positionand time-dependent mean-field potentials. The natural continuation of the discussed work on tunneling will be to investigate spinor-BECs in spin-dependent
optical lattices.
8. Acknowledgement
This work was supported by the National Science Foundation (PHY-0245532).
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ADVANCES IN ATOMIC, MOLECULAR AND OPTICAL PHYSICS, VOL. 53
FEMTOSECOND LASER INTERACTION
WITH SOLID SURFACES: EXPLOSIVE
ABLATION AND SELF-ASSEMBLY OF
ORDERED NANOSTRUCTURES*
JUERGEN REIF† AND FLORENTA COSTACHE
Brandenburgische Technische Universität Cottbus, Konrad-Wachsmann-Allee 1,
03046 Cottbus, Germany
BTU/IHP JointLab, Erich-Weinert-Strasse 1, 03046 Cottbus, Germany
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2. Energy Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1. Absorption by Ionization of Valence Band Electrons: Multiphoton and Tunneling Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2. Impact Heating/Free Carrier Absorption . . . . . . . . . . . . . . . . . . . . . . . . .
3. Secondary Processes: Dissipation and Desorption/Ablation . . . . . . . . . . . . . . . . .
3.1. Desorption/Ablation Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2. Femtosecond Laser Ablation from Silicon . . . . . . . . . . . . . . . . . . . . . . . .
3.3. Recoil Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4. Surface Morphology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Abstract
The fundamentals of interaction between intensive laser pulses and solid surfaces
are reviewed. In order to distinguish the relevant phenomena from secondary effects, e.g., laser heating of the plasma plume formed upon ablation, emphasis is
placed on the action of ultrashort pulses. The present picture of energy absorption
and dissipation dynamics is discussed, and transient and permanent modification of
the surface, in particular its morphology, are considered.
* It is a great pleasure and honor to dedicate this contribution to Prof. Herbert Walther on the occasion of his 70th birthday. Not only did his innumerable contributions to the basic understanding of
and deep insight into quantum and optical physics open the way for tackling the work presented here,
but also he was and still is an outstandingly inspiring teacher with an ever continuing impact. Happy
birthday!
† E-mail: [email protected].
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© 2006 Elsevier Inc. All rights reserved
ISSN 1049-250X
DOI 10.1016/S1049-250X(06)53008-3
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J. Reif and F. Costache
[1
1. Introduction
Among the many specific types of laser interaction with matter, studied at present,
those exploiting the very high optical fields attainable play a peculiar role [1].
At intensities in the range of 1012 . . . 1018 W/cm2 , easily accessible with present
femtosecond laser systems, the corresponding electric field of ≈109 . . . 1012 V/m
is no longer small compared to the intra-atomic Coulomb field (≈1011 V/m for
the hydrogen 1s-electron). Consequently, the interaction cannot be considered as
a weak perturbation of the irradiated matter anymore, with a linear response. Instead, a transient state is created, where the electrons “feel” a combination of
the nuclear Coulomb field with the electric field of the incident radiation. In
atomic physics this led to the theoretical “dressed atom” approach [2], initially
developed for a strong monochromatic cw-field driving a two level system [3].
A typical observation under these conditions is that the cross-section for multiphoton processes approaches or even exceeds that of linear interaction [4], such
as for high-harmonics generation [5–7], above threshold ionization [8–11], or the
emergence of relativistic effects [12].
In condensed matter, in particular in solids with quasi-localized electrons, e.g.,
dielectrics or semiconductors, this transient high-field state can have even more
dramatic effects than in free atoms. Since the crystalline structure is a consequence of equilibrium binding conditions for the atoms’ outer electrons, any
change of electronic configuration will strongly influence the crystalline stability [13,14]. Thus, an excitation faster than any electron-phonon collision time,
i.e. on a sub-picosecond time scale, tends to almost immediately “soften” the
material [15–17], long before any thermodynamic melting sets in [18,19] via an
equilibration between electron and lattice temperature [20]. A consequence of this
breakdown of crystal stability is the desorption or ablation of particles from the
material surface [21].
For technical applications, this type of interaction plays a very important role as
the basis for most materials processing techniques, e.g., drilling, cutting, shaping,
and for medical use, e.g., (eye) surgery, dermatology, etc. However, the nature and
dynamics of energy coupling and dissipation as well as subsequent processes on
a microscopic scale are still subject of ongoing research.
In most practical cases, for long-pulse excitation (> several ps), the basic
processes are masked by secondary effects. Typically, already during the pulse
duration material removal takes place. Thus, strong interaction of the laser with
the ablation (plasma) plume is expected, consuming a substantial part of incident
energy for heating the plasma via inverse bremsstrahlung [22,23]. Then, plasma
erosion of the surface is no longer negligible. Also, thermodynamic processes,
like mere target heating, must be taken into account. These secondary effects
make a study of fundamental mechanisms and dynamics rather complex and complicated. Fortunately, today’s ultrafast laser sources open the way of separating
2]
FEMTOSECOND LASER ABLATION
229
first and second order effects, thus getting closer to the “atomic” laboratory, so
successfully studied for free particles. In fact, transient instabilities, typical for
high field interaction, can show up easily under ultra-short pulse irradiation [15,
24–26] where, for certain conditions, Coulomb explosion of an electrostatically
unstable surface is the basic ablation channel.
In the following, we review the present picture of laser ablation dynamics
from transparent dielectric crystals and semiconductors (e.g., silicon). We will
start with current models and their experimental equivalent for the energy coupling, continue with sketching follow-up processes of energy dissipation and
particle emission, and, finally, discuss the consequences occurring after the ablation/desorption process. There, i.e. well after the laser pulse termination, we
will show, relaxation dynamics are far from (thermodynamic) equilibrium for a
free evolving system.
2. Energy Coupling
We will start with some considerations about the basic mechanisms of energy
coupling between laser and target, in particular for dielectric materials. First, however, a very peculiar feature of experiments on laser ablation/desorption1 from a
solid crystal must be considered. Unlike in experiments with free atoms, where the
microscopic “laboratory” is always well-characterized and, in most cases, does
not change its intrinsic properties during repetitive interaction, the crystalline surface changes with each particle removal. Each loss of a particle results in the
generation of a microscopic defect at the surface, associated with a change in the
energy band structure, e.g., the introduction of defect states within the bandgap.
Further, increasing surface erosion may result in a surface roughening sufficient
to give rise to a local enhancement of the optical electric field. These effects can
substantially change the coupling efficiency for sub-bandgap photons from pulse
to pulse. Only after a certain number of desorption events, the surface “decomposition” reaches a kind of steady state, and additional desorption does not change
the average number of defects in the irradiated area any more. The irradiation
phase before reaching this steady state is usually termed “incubation” [27,28] and
is shown, typically, in Fig. 1 [29].
The incubation does, however, not only serve to generate a “stable” defect density at the surface. It also helps to overcome another problem usually encountered:
a typical surface of a solid target does, usually, not consist of the actual target material. Instead, it is often covered with thin films of contaminants, e.g., water,
1 Here, we apply the following convention: “desorption” denotes the taking away of individual particles from the crystal surface whereas “ablation” refers to more massive material removal.
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J. Reif and F. Costache
[2
F IG . 1. Effect of incubation for the ablation from BaF2 (emission of Ba+ ions) at different laser
intensities [29].
oxides, CO2 , etc., both chemisorbed and physisorbed. During incubation, these
films are removed and, if the experiments are conducted under sufficiently good
vacuum conditions, do not re-grow during the course of subsequent investigations.
In the following, we will only consider nonmetallic targets with an incubated
surface, i.e. with a stable average defect density and practically free from contaminations.
2.1. A BSORPTION BY I ONIZATION OF VALENCE BAND E LECTRONS :
M ULTIPHOTON AND T UNNELING I ONIZATION
The fundamental understanding of the energy coupling between valence band
electrons in a solid and a strong electromagnetic wave at sub-bandgap frequency
2]
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231
F IG . 2. Ionization yield for high laser intensity according to the Keldysh model.
has been developed already in 1965 by Keldysh [30,31]. He discusses two principal mechanisms, multiphoton ionization and tunneling ionization, the latter
prevailing at very high light intensity as shown in Fig. 2.
2.1.1. Multiphoton Ionization
The possibility to bridge energy gaps larger than the photon energy by the simultaneous interaction/absorption of several photons with a sufficient sum energy
has been termed “multiphoton” interaction and has been studied extensively during the last four decades, mainly in atomic and molecular systems but as well in
solids. The interaction becomes possible if the photon density is sufficiently high
for a reasonable probability for several photons being at the same spatio-temporal
interaction site simultaneously.
In a semiclassical description this situation is equivalent to a reasonably high
electric field of the incident wave, which still can be introduced into the target’s
Hamiltonian via a perturbational ansatz. There, in principle, the light electric
field induces a periodic deformation of the—initially symmetric—Coulomb field
binding the electrons (cf. Fig. 3).
This can be considered as the generation of an oscillating dipole. In the Hamiltonian, the increased anharmonicity leads to a coupling of two Eigenstates of the
unperturbed Hamiltonian. The induced dipoles are equivalent to a polarization of
the medium, P, which, for conventional light intensities, is just proportional to the
light electric field, E, where α is the polarizability of the medium:
P = αE.
(1)
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J. Reif and F. Costache
[2
F IG . 3. Influence of a strong light field on the binding potential of an electron (perturbation
model): (a) unperturbed near-harmonic potential, two “Eigenstates” are indicated by highlighted disks;
(b) perturbed potential. The spherical symmetry is perturbed in one direction, leading to the induced
dipole μ. The two Eigenstates are coupled (one in x- one in y-direction).
For higher intensity, the anharmonicity becomes increasingly larger. In Perturbation Theory, this is accounted for by developing the polarization in a power series
of the electric field:
P = α + α (2) E + α (3) EE + α (4) EEE + · · · E
(2)
which involves contributions of higher harmonics of the incident field, as can
easily be seen when taking the electric field as E = E0 exp[i(ωt − kr)]. In the
Hamiltonian, this is equivalent to the coupling of more and more Eigenstates in
the resulting wave function.
For the case of absorption, it must be considered that the electromagnetic field
energy is given by the square of the field, i.e. the nonlinear contributions are proportional to the square of the respective term in the polarization (2), with the
coupling given by the imaginary part of the polarizability. Correspondingly, the
probability for an n-photon transition is found to be proportional to the nth power
of the incident intensity:
2
2 2 P(n) ∝ P(n) E ∝ Im α (n) En−1 E = Im α (n) E2n ∝ I n .
(3)
As a consequence, in typical experiments multiphoton transitions can be identified
by plotting the absorbed energy (e.g., the ionization rate) as a function of incident
intensity in a log-log plot: straight lines are obtained with the slope indicating the
number of photons involved [32] (cf. Fig. 2).
In a real system, the situation may be complicated by the detailed energy structure of the material, which is contained in the explicit shape of α. According to
Fermi’s Golden Rule [33], resonance denominators can enhance the polarizability
[34] and even reduce the nonlinearity, if an intermediate (m-photon) resonance is
directly met.
3]
FEMTOSECOND LASER ABLATION
233
F IG . 4. Tunneling ionization: (a) unperturbed near-harmonic potential with corresponding Eigenstates (b). Influence of a strong electric field (indicated by the dash-dotted line): The potential well
is—on one side—decreased so much that several of the unperturbed energy levels are coupled to the
vacuum and thus directly ionized.
2.1.2. Tunneling Ionization
When the field of the incident light increases further, it might approach the
Coulomb field binding the electrons (for the hydrogen atom this is at the order
of 1011 V/m, corresponding to an intensity of 1016 W/cm2 ). Already at about
10% of that intensity, the perturbational treatment appears no longer justified. In
this the regime, the potential may be considered to be so strongly changed, that direct (above-barrier) or tunneling ionization appears to be the dominant ionization
process (Fig. 4) [30,35–37].
2.2. I MPACT H EATING /F REE C ARRIER A BSORPTION
Different from free particles, the ionization does not simply result in an escape
of the excited electrons. Only those close to the surface, i.e. within the average
inelastic mean free path [38] according to the “universal curve” (Fig. 5), can really
leave the sample (if they are not held back by space charge effects, see below). All
other electrons will be free carriers in the conduction band where they can absorb
additional energy (“free carrier absorption”) [39]. In fact, the energy gained can
be larger than the bandgap, and the electrons can generate further conduction band
electrons by impact ionization [30,35,36,40–43]. This process is characterized by
the absorption of very substantial amounts of energy in an avalanche process.
3. Secondary Processes: Dissipation and Desorption/Ablation
In the following, we will only consider only processes leading to the removal of
particles from the surface of the irradiated target, i.e. ablation or, at a low parti-
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J. Reif and F. Costache
[3
F IG . 5. The “Universal Curve” of electrons’ inelastic mean free path (IMFP) in a solid in dependence on their kinetic energy.
cle emission rate, desorption. Whatever happens within the bulk of the material
is beyond the scope of this contribution. For these considerations, we refer to
experiments with laser pulses of 100 fs duration at a wavelength of 800 nm.
At moderate incident intensity, i.e., below ≈1013 W/cm2 , the desorbed particles are mostly positive ions, even of highly electronegative species typically
forming anions. This ion emission is strongly coupled to the ejection of electrons
from the surface [44]. In this regime (desorption regime), well below the so-called
“ablation threshold” [45], a very large number of incident pulses is required for
observable surface damage to occur. Only at higher fluence, a more massive material removal sets in [46,47] (ablation regime), characterized by a considerable and
even prevailing contribution of neutrals, and even negative ions can be detected
[48,49].
In the following section, some detailed results from these two regimes will be
resumed.
3.1. D ESORPTION /A BLATION DYNAMICS
First, we consider the desorption regime, i.e. at moderate particle emission well
below the classical damage threshold. It is characterized by a strongly nonlinear
coupling of the incident laser energy to the irradiated material, as it is displayed
in Fig. 6.
The relevant absorption process can be identified as multiphoton surface ionization. This is obvious from the close connection to the observed emission of
electrons, shown in Fig. 6(b). There, above ≈0.6×1012 W/cm2 , the dominant ionization process corresponds to a band-to-band transition. Below, the lower slope
indicates ionization of an occupied surface defect state [50–52].
In Fig. 7, the dramatic enhancement of the laser-surface coupling via defect
states within the bandgap is shown, exemplarily, for the ionization of an Al2 O3
3]
FEMTOSECOND LASER ABLATION
235
F IG . 6. Yield of electrons and Ba+ ions from BaF2 irradiated with intense fs laser pulses [51].
F IG . 7. Electron emission from Al2 O3 for excitation with 1.5 eV photons. (a) Yield in dependence on the incident intensity. Obviously, the nonlinearity of 4 is lower than expected from the 9 eV
bandgap, corresponding to a 6-photon transition [51]. (b) Energy structure of Al2 O3 , indicating defect
states within the bandgap [52].
surface. As indicated in the schematic in Fig. 7(b), the observed 4-photon nonlinearity corresponds to a transition in the F-center defect, i.e. an oxygen vacancy.
The defect can serve as a relay for the ionization and thus reduces the order of
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J. Reif and F. Costache
[3
F IG . 8. Positive ion yield from BaF2 : cluster emission [25]. The signal for masses above 250 amu
(right of the dashed line) are magnified by a factor of 17.
nonlinearity by two. This interpretation is corroborated by the fact, that blue fluorescence at ≈3 eV was detected in the experiment, corresponding to an internal
relaxation of the F-center excitation via the 3 P–1 S transition [52].
This significant role of defects, such as missing anions, for the enhancement of
the coupling efficiency confirms the observed incubation effect (Fig. 1). Indeed,
an increasing density of induced defects leads to an increased coupling efficiency,
correspondingly.
An analysis of the desorbed positive ions by Time-of-Flight (ToF) spectroscopy
reveals that not only monoatomic ions are emitted but also larger clusters. This
observation is, as the previous ones, independent on the specific materials under
study, as shown in Fig. 8 for a dielectric (BaF2 ) and in Section 3.2 for a semiconductor (Si).
Closer inspection shows that all these clusters have the same kinetic energy
(Fig. 9(c)). This means that their velocities are different, thus excluding gas phase
interaction as the origin of cluster formation. Consequently, they are emitted intact
from the surface, indicating massive surface breakdown. In fact, the ions’ velocity,
as derived from retarding field measurements (Fig. 9(a), (b)), does not correspond
to a thermal Maxwellian distribution. More likely, we find a narrow distribution
superimposed on a large drift velocity, similar to a seeded molecular beam and
3]
FEMTOSECOND LASER ABLATION
237
F IG . 9. Kinetic energy of ions emitted from Al2 O3 . (a) Retarding voltage transmission;
(b) corresponding velocity distribution (solid line), compared to a Maxwell–Boltzmann distribution
(dash-dotted) and a shifted Maxwellian (dotted line, cf. Eq. (4)); (c) kinetic energy of different desorbed clusters [28].
described by a modified Maxwellian:
"
#
m · ( Dt − u)2
A
f (t) = 4 · N · exp −
2kTu
t
(4)
with the drift time t and the “Maxwellian” velocity u.
The resulting kinetic energies are rather high, at the order of 100 eV, with a
narrow distribution of only ≈1 eV, indicating a fast, monochromatic ion beam!
The strong coupling between ion and electron emission, the fact that the desorbed particles are almost exclusively positive ions, and the emitted ion dynamics
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J. Reif and F. Costache
[3
F IG . 10. General behavior of desorbed ion yield as a function of incident intensity, for two different targets (normalized to the transition between multiphoton-ionization/Coulomb-explosion to
hyperthermal particle emission [47]. (Note that the total particle emission, including neutrals increases
substantially!)
suggest the following desorption scenario [24–26]: the main action of the laser is
rapid (surface) ionization, with the excited electrons in the surface region rapidly
leaving the sample. This results in a fast positive surface charging, inducing an
electrostatic instability. Much faster than any charge equilibration can take place
(in a dielectric or semiconductor), the instable surface decomposes via Coulomb
explosion (this explains the identical kinetic energies for all singly charged surface
fragments).
At increasing incident fluence, the increase of ion yield with intensity appears
to saturate (Fig. 10). At the same time, the total ablation rate increases dramatically, indicating that other than ionic species start to make up most of the ablated
material, pointing more towards a different ablation mechanism than to a change
in energy coupling, e.g., avalanche processes. This general behavior does not depend on the material investigated, as shown in Fig. 10 where normalized data from
two different materials are superimposed.
At the same threshold, also the distribution of the ions’ kinetic energy changes
(cf. also [46,47]). This can be seen in the drift-mode2 ToF spectra of positive
2 In drift-mode, the ToF spectrometer is operated without an extraction field between sample and
the spectrometer which, usually, is applied to compensate an initial kinetic energy distribution by a
larger drift velocity. Note, that only a moderate mass resolution is obtained, which makes it difficult
to distinguish between the different species.
3]
FEMTOSECOND LASER ABLATION
239
F IG . 11. Drift-mode ToF spectra from BaF2 , at intensity (a) close to, (b) well above saturation
threshold (cf. Fig. 10). The solid lines in (a), (b) are a fits assuming identical kinetic energies (fast
peaks), respectively, temperatures (slow peaks) for both species, Ba+ and F+ [47].
ions from BaF2 in Fig. 11: Above the threshold a second peak of slower ions
appears which, with increasing intensity becomes more and more important. In
this regime, the excitation density in the irradiated volume becomes so high, that
a sufficient density of hot electrons is created in the conduction band, which can
be further heated by free carrier absorption [39] and then transfer their energy by
electron–phonon collisions to the crystal lattice [20,39,53]. The associated rapid
heating results in new ablation mechanisms to occur, such as phase explosion [54,
55]. Consequently, the ablation plume does not only consist of positive ions but
also, and particularly, of neutrals and even negative ions (Fig. 12), which may be
the result of electron capture within the plume [48].
3.2. F EMTOSECOND L ASER A BLATION FROM S ILICON
Below the single-shot ablation threshold of silicon the high electronic excitation
leads to a nonthermal, ultrafast phase change within less than 1 ps. Here, the
percentage of fast ions ejected from the silicon surface increases [56]. Indeed, the
resulting mass spectrum reveals positive atomic ions and clusters (Fig. 13, left).
The ion kinetic energies distribution shows fast (several tens of eV) and slow
(down to few eV) contributions suggesting a superposition between a nonthermal
mechanism (such as Coulomb explosion [57,58]) and a thermal-mechanism such
as phase explosion [56].
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J. Reif and F. Costache
[3
F IG . 12. Negative ions observed during ablation from CaF2 (panel (a): mass spectrum). As can be
seen panel (b), the negative ions’ distribution is much broader and slower than for the positive ions,
indicating a different ablation mechanism. In fact, the negatives’ arrival time cannot only be explained
by a lower drift velocity but also a later generation time, for instance, in the ablation plume [49].
F IG . 13. Positive ion mass spectrum from a silicon surface irradiated by ∼100 fs laser pulses
(left); nonmass resolved spectrum (drift-mode): peaks attributed to Si+ fast ions and Si+ slow ions.
3.3. R ECOIL P RESSURE
The emission of many particles at substantial kinetic energies is associated, in
turn, with a considerable recoil pressure onto the sample. This results in a nonnegligible pressure load on the interaction region. For silicon, it is known that
localized high pressure in the GPa range results in phase transformations in the
crystal lattice [59,60]. In simple words, some of the atoms are pushed out of their
usual position and squeezed into the surrounding part of the lattice, thus changing coordination and distances. The resulting new phases, e.g., hexagonal, bcc,
3]
FEMTOSECOND LASER ABLATION
241
F IG . 14. Raman spectra from the ablated area on p-doped Si(100), taken at different areas of a
spot of a few µm depth [57]: From bottom to top, the traces are taken 1 at a virgin area (reference),
2 in the center of a flat crater, 3 in the ripples area, and 4 at the steep wall of a deep (several 10 µm)
crater.
or rhombohedral silicon, can be detected by micro-Raman spectroscopy of the
corresponding, new phonon frequencies [61–63] as shown, exemplary, in Fig. 14.
Note that, due to the penetration depth of the 532-nm Raman laser of ≈1 µm in
silicon, all spectra are dominated by the TO-phonon peak of crystalline silicon at
520.7 cm−1 .
Interestingly, the different spectra show distinctly different behavior in the region close to the TO-phonon peak (cf. Fig. 15). In Fig. 16, a more detailed analysis
of this situation is presented, fitting the experimental curves to a sum of known
contributions from different silicon structures [63], namely a broad peak due
to amorphous silicon [a-Si (TO)] at 475 cm−1 , a peak attributed to zincblende
(Wurtzite) structure (Si-IV) at 516 cm−1 , and a contribution from polycrystallites, resulting in a broadening and red-shift of the c-silicon TO peak. Obviously,
in the ripples zone, a significant amount of the Si-IV, polymorph is generated,
whereas at the crater wall the presence of micro- and nanocrystallites is indicated.
In fact, molecular-dynamics calculations for a Coulomb explosion of silicon upon highly-charged ion impact [64] have demonstrated, that the massive
positive-ion ejection at high kinetic energies results, indeed, in considerable recoil pressures of up to 103 GPa, initially, and falling down to about 10 GPa after
360 fs.
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[3
F IG . 15. Micro Raman spectra of reference (1 in Fig. 14), ripples area (3 in Fig. 14), and crater
wall (4 in Fig. 14) in the vicinity of the TO-peak of crystalline silicon (Si-I) at 520.7 cm−1 . Note the
different asymmetries, at the low frequency side, between spectra 3 and 4.
3.4. S URFACE M ORPHOLOGY
Both ablation mechanisms, Coulomb and phase explosion, are associated with a
strong transient perturbation of the target in the interaction volume. The interaction volume, determined by the average phonon mean path, is by far not in
thermal equilibrium with the surrounding matrix, with a strong gradient between
both regimes. Consequently, the subsequent relaxation is very unlikely to occur
via thermodynamic processes like crystallization or glass formation. Instead, nonlinear dynamics models offer possible relaxation pathways.
As can be seen in Fig. 17, regular, aligned periodic structures have developed
at the bottom of the ablation crater after several thousand shots, with no obvious relation to the underlying crystal structure [65]. Instead, the laser polarization
appears to play an important role for the orientation. Similar features, termed
“ripples” have been known in laser ablation for more than three decades [66,67],
classically attributed to an inhomogeneous energy input due to an interference of
the incident light with a surface scattered wave from the same pulse [68]. Closer
inspection of the ripples structures as in Fig. 17 reveals, however, that they are
not compatible with such model: the periodicity can be substantially smaller (at
the order of 100. . . 300 nm) than the wavelength of 800 nm; the regularity is multiply interrupted and interconnected, no dependence on angle of incidence and
wavelength can be established.
More likely, the local intensity or irradiation density has a marked influence
on the structure width. This is shown, impressively, in Fig. 18 [69]: even at one
ablated spot, the periodicity changes between the center (high intensity, wide
3]
FEMTOSECOND LASER ABLATION
243
F IG . 16. Fit of Micro Raman spectra (solid gray lines) of (a) ripples area (3 in Fig. 14) and
(b) crater wall (4 in Fig. 14) to a combination (-+- lines) of c-Si (TO; 520.7 cm−1 /HWHM 2 cm−1 ),
respectively, c-Si+crystallites (520.0 cm−1 /HWHM 5.15 cm−1 , a-Si (475 cm−1 /HWHM 70 cm−1 ),
and Si-IV (Wurtzite; 516 cm−1 /HWHM 4 cm−1 ), the only fit parameters being the relative abundance. The different contributions, divided by a factor of 10 for visibility, are indicated in the lower
part of (a), (b).
spacing) and the edge region (low intensity, narrow spacing). Interestingly, the
transition between both features is abrupt and does not follow the intensity distribution.
For really high irradiation density, the feature shape changes dramatically, exhibiting wide, flat crests and very narrow, very deep valleys instead of the almost
sinusoidal variation at lower intensity. Also, the alignment changes from long,
parallel lines to a more meandrous appearance (Fig. 19). Though in the right two
panels, at first sight, the surface appears almost like refrozen from a liquid melt,
the deep valleys in between the broad flat crests indicate, that this seems to be
very unlikely. The very narrow trenches between the large flat areas are about
244
J. Reif and F. Costache
[3
F IG . 17. Typical “ripples” patterns at the bottom of the ablation crater after several thousand
pulses (normal incidence) at low ablation rate: regular ordered structures of sub-micron feature size.
The double arrows indicate the laser polarization [65].
F IG . 18. Change in ripple spacing across one ablation spot in CaF2 (The double arrow denotes
the direction of laser polarization). Indicated below is a schematic of a corresponding beam profile.
Note the abrupt transition between the narrow (≈200 nm) and the coarse (≈450 nm) spacing despite
the smooth intensity profile [69].
1 µm deep, thus exhibiting an aspect ratio of about 10 or more, which is not expected from a refreezing liquid.
On the other hand, at low irradiation dose, i.e. comparably low irradiance or,
at very high fluence, very few pulses, only arrays of aligned nanoparticles are
observed, very similar to what is observed for the debris outside the actual ablation
crater (Fig. 20). This suggests a formation of the ripples via an agglomeration of
these nanoparticles in a scenario similar to a percolation process.
3]
FEMTOSECOND LASER ABLATION
245
F IG . 19. Change in ripple spacing, shape and orientation on Si(001) with irradiation density (left
panel: 60,000 pulses, 0.4 × 1012 W/cm2 ; middle and right panels: 20,000 pulses, 1.6 × 1012 W/cm2 ).
The laser polarization is vertical [57].
F IG . 20. Agglomeration of nanoparticles in the ablation area (CaF2 : left and middle panels; AFM
pictures) and in the debris outside the crater (Si: right panel; SEM picture). (Left panel: 3 pulses,
middle panel 5 pulses at 8 × 1013 W/cm2 ). The dotted line in the middle panel indicates the trace
analyzed by atomic force microscopy, shown in the lower panel and yielding an average particle size
of ≈200 nm [65].
A very interesting feature, important for an interpretation of the origin of the
observed structures, is shown in Fig. 21: the ripples are not simple, parallel lines
but exhibit very many bifurcations, as are typical for self-organization phenomena.
246
J. Reif and F. Costache
[4
F IG . 21. Bifurcations at the bottom of an ablation crater on CaF2 . The double arrow indicates the
direction of the laser polarization [69].
4. Discussion
Bringing all observed features together, the following picture of the laser-material
interaction evolves: the main action of the incident laser energy is a massive
ionization at the irradiated surface. The corresponding positive surface charging
results in a Coulomb explosion of the positive ions, at a femtosecond time scale,
much faster than any intrinsic charge transfer times. This situation is very similar
to that induced at the surface by the impact of highly charged positive ions. Theoretical model calculations [64] demonstrate that the ion ejection starts during the
first 40 femtoseconds after ionization and continues for several 100 femtoseconds.
The target surface is left behind in a state of extreme thermal nonequilibrium and
instability. Also at higher etch rate, when sufficient electrons are created via free
carrier absorption and their energy is transferred to the lattice [39], i.e. substantial amounts of also neutral particles are taken away (cf. Fig. 10), such instability
should be expected [13–17].
Similar results are found in completely different experiments, namely in typical ion etching configurations [70], where it was shown that this instability tends
to relax to self-assembled structures with a typical feature size in a few-100-nm
range, very similar to those shown above (Figs. 17–20). A particular clue to assume a self-organization at the origin of the observed morphology is given in
Fig. 21, where many bifurcations are shown, typical for such nonlinear-dynamics.
Further, the experiments at lower irradiation dose (Fig. 20) indicate a possible way
for the development of the long, parallel structures: it appears that, first, nanoparticles form with a typical size below 200 nm. These particles do not only occur
inside the illuminated spot but are also observed in the debris precipitated around,
which may have two reasons: either, such particles are already contained in the
4]
FEMTOSECOND LASER ABLATION
247
F IG . 22. Model of a corrugated thin liquid film, homogeneously charged. The arrows at two highlighted ions represent their ejection probability.
ejected material, or they coagulate from the pre-formed clusters. As shown in
Fig. 20, the nanoparticles tend to arrange in long, parallel arrays.
Also, the dependence of the ripples width on the incident intensity and dose
suggests the role of self-organization processes. Assuming the excitation density
to correspond to a “perturbation depth”, this thickness of the instability might
control the order parameters for the determination of the feature size, similar as,
e.g., the structure width in a Benard–Marangoni instability [71]: the thicker the
instability layer, the larger is the structure. However, the stepwise variation in
Fig. 18 cannot be fully explained this way.
In first numerical simulations [72], an attempt is made to simulate the unstable
surface by a thin liquid-like layer (Fig. 22). Then, the first laser pulses are necessary to produce a randomly corrugated surface (cf. the incubation). Subsequently,
assuming, e.g., surface ionization and Coulomb explosion as the possible ablation
mechanism, each laser pulse results in a homogeneous charging of the surface.
For positive ions sitting in a valley of the corrugation, the number of next neighbor positive charges (holes) is much larger than for an ion sitting on the hill of
the corrugation. Thus, the desorption probability is much higher in the valleys,
resulting in an increased erosion of the valleys and a growing surface roughening. On the other hand, at the hill of the film the surface is particularly stretched.
The resulting surface tension acts to minimize the surface by refilling the valleys.
Thus, we have a competition between surface roughening (erosion of the valleys)
and surface smoothening (diffusion from the hills), well reflecting the postulated
unstable surface.
This situation is similar to that in ion beam erosion, postulating a surface instability after massive erosion of ions [73,74] relaxing by the formation of regular
patterns like those observed in this contribution. It can be studied using wellknown formalisms of nonlinear hydrodynamics of thin films [75–77]. An equation
of the Cahn–Hilliard or Kuramoto–Sivashinsky type [78,79] can describe the
linear growth of periodic structures (stripes, squares) which turns into a typical
coarsening upon increased dose of interaction. This equation is of the KPZ type
248
J. Reif and F. Costache
[4
F IG . 23. Surface structure as a result of a numerical solution of Eq. (6).
(Kadar et al. [80]):
∂
h = −V [h] 1 + (∇h)2 − D2 h,
∂t
(5)
where the variation of the corrugation, V (h) explicitly contains erosion and
smoothening parameters in integral form. Considering that the interaction only
is rather local, (5) can be reduced to a partial differential equation
∂ 2h
∂ 2h
∂h
∂
h = − V0 + γ
+ vx 2 + vy 2 − D2 h
∂t
∂x
∂x
∂y
2
2
λy ∂h
λx ∂h
+
+ higher orders.
+
2 ∂x
2 ∂y
(6)
Such equations are well known from nonequilibrium physics and may be studied
by analytical (stability and bifurcation analysis, spectral analysis) and numerical methods, showing as transient solutions similar structures as those observed
experimentally (Fig. 23).
An open question concerns the orientation of the ripples structures. It appears
that, at least for moderate intensities, the laser polarization plays an important role
whereas the underlying crystal structure seems to have no influence. Experiments
with circularly polarized light [81], however, show a similar structure of ordered
ripples (Fig. 24), without the possibility of the laser electric field as a control
parameter. Further, the meandering structures at high irradiation dose cannot yet
be understood. Up to now, no reliable model has been found to account for the
structures’ orientation.
Ongoing work concentrates on a more detailed analysis of the instabilities involved and the mechanisms responsible for the orientation of the self-organized
nanostructures. Further, investigations are aimed at the possibility to control the
structures for possible applications.
5]
FEMTOSECOND LASER ABLATION
249
F IG . 24. Ripples structures on CaF2 after ablation with circularly polarized light [81].
5. Acknowledgements
We gratefully acknowledge fruitful collaboration and discussions with and valuable support by Tz. Arguirov, J. Bertram, M. Bestehorn, S. Eckert, M.E. Garcia, I. Georgescu, M. Henyk, W. Kautek, M. Ratzke, R.P. Schmid, W. Seifert,
O. Varlamova, D. Wolfframm, and L. Zhu. The BTU/IHP JointLab is supported
by an HWP grant, a joint initiative of the German Federal Government and
the Land of Brandenburg. We also gratefully acknowledge support from the
Land-Brandenburg-Schwerpunktprogramm “Qualitätsforschung” and the European Funds for Regional Development (EFRE).
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This page intentionally left blank
ADVANCES IN ATOMIC, MOLECULAR AND OPTICAL PHYSICS, VOL. 53
CHARACTERIZATION OF SINGLE
PHOTONS USING TWO-PHOTON
INTERFERENCE∗
T. LEGERO† , T. WILK, A. KUHN‡ and G. REMPE§
Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, 85748 Garching, Germany
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
2. Single-Photon Light Fields . . . . . . . . . . . . . . . .
2.1. Frequency Modes . . . . . . . . . . . . . . . . . . .
2.2. Spatiotemporal Modes . . . . . . . . . . . . . . . .
2.3. Single-Photon Detection . . . . . . . . . . . . . . .
3. Two-Photon Interference . . . . . . . . . . . . . . . . .
3.1. Quantum Description of the Beam Splitter . . . . .
3.2. Principle of the Two-Photon Interference . . . . . .
3.3. Temporal Aspects of the Two-Photon Interference .
3.4. Correlation Function . . . . . . . . . . . . . . . . .
3.5. Two-Photon Interference without Time Resolution
3.6. Time-Resolved Two-Photon Interference . . . . . .
4. Jitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1. Frequency Jitter . . . . . . . . . . . . . . . . . . . .
4.2. Emission-Time Jitter . . . . . . . . . . . . . . . . .
4.3. Autocorrelation Function of the Photon’s Shape . .
5. Experiment and Results . . . . . . . . . . . . . . . . . .
5.1. Single-Photon Source and Experimental Setup . . .
5.2. Average Detection Probability . . . . . . . . . . . .
5.3. Time-Resolved Two-Photon Interference . . . . . .
5.4. Interpretation of the Results . . . . . . . . . . . . .
6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . .
7. Acknowledgements . . . . . . . . . . . . . . . . . . . .
8. References . . . . . . . . . . . . . . . . . . . . . . . . .
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254
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288
∗ It is a pleasure for us to dedicate this paper to Prof. Herbert Walther, a pioneer in quantum optics
from the very beginning. The investigation of the amazing properties of single photons both in the
microwave and the optical domain has always been a central theme in his research. We wish him all
the best in the years to come!
† Now at Physikalisch-Technische Bundesanstalt, 38116 Braunschweig, Germany.
‡ Now at Clarendon Laboratory, Oxford University, Parks Road, Oxford OX1 3PU, United Kingdom.
§ Corresponding author. E-mail: [email protected].
253
© 2006 Elsevier Inc. All rights reserved
ISSN 1049-250X
DOI 10.1016/S1049-250X(06)53009-5
254
T. Legero et al.
[1
1. Introduction
Four decades after the pioneering work on optical coherence and photon statistics
by Glauber (1965), the controlled generation of single photons with well-defined
coherence properties is now of fundamental interest for many applications in
quantum information science. First, single photons are an important ingredient
for quantum cryptography and secure quantum key distribution (Gisin et al.,
2002). Second, the realization of quantum computing with linear optics (LOQC),
which was first proposed by Knill et al. (2001), relies on the availability of
deterministic single-photon sources. And third, various schemes have been proposed to entangle and teleport the spin of distant atoms, acting as emitters of
single photons, by means of correlation measurements performed on the singlephoton light fields (Cabrillo et al., 1999; Bose et al., 1999; Browne et al., 2003;
Duan and Kimble, 2003). Therefore, in recent years, a lot of effort has been made
to realize single-photon sources. As a result, the controlled generation of single
photons has been demonstrated in various systems, as summarized in a review
article of Oxborrow and Sinclair (2005).
Using the process of spontaneous emission from a single quantum system is
the simplest way to realize a single-photon source. In this case, the quantum system is excited by a short laser pulse and the subsequent spontaneous decay of the
system leads to the emission of only one single photon. This has been successfully demonstrated many times, e.g., using single molecules (Brunel et al., 1999;
Lounis and Moerner, 2000; Moerner, 2004), single atoms (Darquié et al., 2005),
single ions (Blinov et al., 2004), single color centers (Kurtsiefer et al., 2000;
Brouri et al., 2000; Gaebel et al., 2004) or single semiconductor quantum dots
(Santori et al., 2001; Yuan et al., 2002; Pelton et al., 2002; Aichele et al.,
2004). If the quantum system radiates into a free-space environment, the direction of the emitted photon is unknown. This limits the efficiency of the source.
To overcome this problem, the enhanced spontaneous emission into a cavity
has been used. The system is coupled to a high-finesse cavity and the photon is preferably emitted into the cavity mode, which defines the direction of
the photons. Although a cavity is used, most properties of the photons, like
the frequency, the duration and the bandwidth, are given by the specific quantum system. Only if the generation of single photons is driven by an adiabatic
passage, these spectral parameters can be controlled. This technique uses the
atom-cavity coupling and a laser pulse to perform a vacuum stimulated Ramantransition (STIRAP), which leads to the generation of one single photon. Up to
now, this has been demonstrated with single Rubidium atoms (Kuhn et al., 2002;
Hennrich et al., 2004), single Caesium atoms (McKeever et al., 2004) and single
Calcium ions (Keller et al., 2004) placed in high-finesse optical cavities.
The characterization of a single-photon source usually starts with the investigation of the photons statistics, which is done by a g (2) correlation measurement
1]
SINGLE PHOTONS USING TWO-PHOTON INTERFERENCE
255
using a Hanbury Brown and Twiss (1957) setup. The observation of antibunching
shows that the source emits only single photons. However, the requirements on
a single-photon source for LOQC and for the entanglement of two distant atoms
go far beyond the simple fact of antibunching. The realization of these proposals
relies on the indistinguishability of the photons, so that even photons from different sources need to be identical with respect to their frequency, duration and
shape. Therefore it is desirable to investigate the spectral and temporal properties of single photons emitted from a given source. We emphasize that properties
like bandwidth or duration always deal with an ensemble of photons and cannot
be determined from a measurement on just a single photon. Therefore any measurement of these properties requires a large ensemble of successively emitted
photons. Several methods have been employed to characterize these.
The first measurement of the duration of single photons has been performed
by Hong et al. (1987). In this experiment, the fourth-order interference of two
photons from a parametric down-conversion source was investigated by superimposing the signal and the idler photon on a 50/50 beam splitter. The coincidence
rate of photodetections at the two output ports of the beam splitter was measured
in dependence of a relative arrival-time delay between the two photons. Indistinguishable photons always leave the beam splitter together, so that no coincidence
counts can be observed. If the photons are slightly different, e.g., because they
impinge on the beam splitter at slightly different times, the coincidence rate increases. Therefore, as a function of the photon delay, the coincidence rate shows
a minimum if the photons impinge simultaneously on the beam splitter, and for
otherwise identical photons the width of this dip is the photon duration. The minimum in the coincidence rate goes down to zero if the photons are identical. Any
difference between the two interfering photons reduces the depth of this dip. The
first demonstration of such a two-photon interference of two independently emitted photons from a quantum-dot device has been shown by Santori et al. (2002).
In addition to the two-photon coincidence experiments, a correlation measurement between the trigger event and the detection time of the generated photon can
be used to determine the temporal envelope of the photon ensemble (Kuhn et al.,
2002; Keller et al., 2004; McKeever et al., 2004). This latter method is insensitive
to the spectral properties of the photons. In general, it does not reveal the shape
of the single-photon wavepackets, unless all photons are identical. In case of variations in the photon duration or a jitter in the emission time, only the temporal
envelope of the photon ensemble is observed. No conclusions can be drawn on
the envelope of the individual photons.
The standard way to determine the coherence time of a given light source is the
measurement of the second-order interference using a Mach–Zehnder or Michelson interferometer. This measurement can also be done with single photons, so
that each single photon follows both paths of the interferometer and interferes
with itself. The detection probability of the photons at both outputs of the inter-
256
T. Legero et al.
[2
ferometer shows a fringe pattern if the length of one arm is varied. The visibility
of this pattern depends on the length difference of both arms and determines the
coherence length (or the coherence time) of the photons. This method has been
used by Santori et al. (2002) and Jelezko et al. (2003) to measure the coherence
time of their single-photon sources. However, this method is hardly feasible for
photons of long duration, because the length of one arm of the interferometer
must be varied over large distances. Furthermore, the measurement depends on
the mechanical stability of the whole setup, i.e. the interferometer must be stable
within a few per cent of the wavelength of the photons, which might not be easy.
Only recently, adiabatic passage techniques have allowed the generation of
photons which are very long compared to the detector time resolution. Therefore the detection time of a photon can be measured within the duration of the
single-photon wavepacket. As a consequence, the two-photon interference can be
investigated in a time-resolved manner, i.e. the coincidence rate can be measured
as a function of the time between photodetections (Legero et al., 2003, 2004). The
theoretical analysis shows that this method not only gives information about the
duration of single photons, but also about their coherence time. Here we discuss
how to use this method for a spectral or temporal characterization of a singlephoton source.
The article is organized as follows: After a brief summary of the nature of
single-photon light fields (Section 2), we discuss the interference of two photons
on a beam splitter and introduce the time-resolved two-photon interference (Section 3). Thereafter, we show how a frequency and an emission-time jitter affects
the results of a time-resolved two-photon interference experiment (Section 4). On
this basis, the experimental characterization of a single-photon source, based on
an adiabatic passage technique, is discussed (Section 5).
2. Single-Photon Light Fields
The quantum theoretical description of light within an optical cavity is well understood (Meystre and Sargent III, 1998). The electromagnetic field between the two
mirrors is subject to boundary conditions which lead to a discrete mode structure
of the field. Each mode can be labeled by a number l and is characterized by its
individual frequency, ω. These eigenfrequencies are separated by ω = 2πc/L,
where L is the round-trip length of the cavity. The quantization results in a discrete
set of energies, En = hω(n
+ 1/2), and the appropriate eigenstates are defined
¯
†
by means of creation, aˆ l , and annihilation, aˆ l , operators. The energy eigenstates
(aˆ † )n
|n = √l |0
n!
(1)
2]
SINGLE PHOTONS USING TWO-PHOTON INTERFERENCE
257
are states with a fixed photon number, n. In this context, photons are the quanta
of energy in the modes of the cavity.
In the limit of L → ∞ and ω → 0, the mode spectrum becomes continuous. In this case it is convenient to introduce continuous-mode operators aˆ † (ω)
and a(ω)
ˆ
according to
aˆ l† → (ω)1/2 aˆ † (ω).
(2)
These operators create and annihilate photons as quanta of monochromatic waves
in free space. These waves of infinite spatial extension do not have any beginning or any end. However, photons generated in a laboratory are characterized by
a certain frequency bandwidth or a finite spatial extension. Therefore it is desirable to define operators which create or annihilate photons in modes of a given
bandwidth, or, in other words, of a well-defined spatiotemporal spread.
2.1. F REQUENCY M ODES
In contrast to modes describing monochromatic waves, it is possible to define
field modes of a given frequency distribution. These modes represent wavepackets travelling with the speed of light c through the vacuum, and the bandwidth κ
of such a mode determines the duration δt of the wavepacket. A frequency distribution is described by a normalized complex function χ(ω) which is called the
mode function of the field. The operators aˆ † (ω) and a(ω)
ˆ
can be used to define
a new set of operators for the creation and annihilation of photons in these new
modes (Blow et al., 1990). The creation operator, e.g., is given by
bˆχ† = dω χ(ω)aˆ † (ω).
(3)
Note that the mode function χ(ω) can be written as the product of a real amplitude, ε(ω), and a complex phase, exp (−iΦ(ω)). The phase term includes the
emission time τ0 and the propagation of the wavepacket. In the following, we restrict our discussion to Gaussian wavepackets centered at the frequency ω0 . Their
mode functions read
$
2
(ω − ω0 )2
4
χ(ω) =
(4)
exp
−
exp −iω(τ0 + z/c) .
2
2
πκ
κ
For an ideal single-photon source which always produces identical photons, the
light field is always described by the same quantum mechanical state vector. In
other words, the state vector is given by the creation operator bˆ † (χ) acting on the
vacuum state |0 for every single photon:
|1χ = bˆχ† |0.
(5)
258
T. Legero et al.
[2
We emphasize that such an ideal source is hardly feasible. Usually the generation
process cannot be controlled perfectly and therefore the mode function is subject
to small variations. To take this into account the light field must be described by
a quantum mechanical density operator
ˆ = dϑ f (ϑ)|1χ(ϑ) 1χ(ϑ) |.
(6)
Here we assume that the source produces single photons with a Gaussian frequency distribution and the parameters of this distribution are subject to small
variations, according to a distribution function f (ϑ). The parameter ϑ stands for
the center frequency, ω0 , the bandwidth, κ, or the emission time, τ0 , of the photon,
or a combination of these.
2.2. S PATIOTEMPORAL M ODES
Due to the Fourier theorem, each mode with a certain frequency distribution χ(ω)
can be assigned to a temporal wavepacket which is travelling through space.
A mode with the Gaussian frequency distribution given by Eq. (4) therefore belongs to a spatiotemporal mode ξ(t −z/c) of Gaussian shape. With the substitution
q := t − z/c this mode is given by the function
$
2
q2
4
exp − 2 exp iω0 (τ0 − q)
ξ(q) =
2
πδt
δt
≡ (q) exp iω0 (τ0 − q) .
(7)
The duration δt of this Gaussian wavepacket is given by the reciprocal bandwidth
of the frequency distribution, δt = 2/κ. Blow et al. (1990) have shown that creation and annihilation operators can also be assigned to spatiotemporal modes. In
order to do that, one has to define the Fourier-transformed operators
aˆ † (q) = (2π)−1/2 dω aˆ † (ω)e−iωq ,
(8)
iωq
ˆ
a(q)
ˆ
= (2π)−1/2 dω a(ω)e
(9)
.
ˆ
Its expectation
By means of these operators we define a flux operator aˆ † (q)a(q).
value has the unit of photons per unit time. We need this operator in the next
subsection to describe the detection of single photons.
Equations (8) and (9) are only valid if the bandwidth of the modes is much
smaller than the frequency of the light, κ ω0 . This also limits the localization
of a single photon in such a spatiotemporal mode. In case of optical frequencies,
this condition is usually fulfilled.
2]
SINGLE PHOTONS USING TWO-PHOTON INTERFERENCE
259
In analogy to Eq. (3) the Fourier-transformed operators can be used to define
creation and annihilation operators for photons of spatiotemporal modes ξ(q):
cˆξ† = dq ξ(q)aˆ † (q).
(10)
To take fluctuations into account, one can again write the density operator of the
light field as in Eq. (6), but using spatiotemporal modes. In this case, ϑ stands for
any combination of ω0 , δt, and τ0 .
2.3. S INGLE -P HOTON D ETECTION
Choosing spatiotemporal modes for describing the state of a single-photon light
field simplifies the formal description of the detection of a photon. We assume a
detector with quantum efficiency η placed at the position z = 0. The response
of the detector within a time interval [t0 − dt0 /2, t0 + dt0 /2] is given by the
expectation value of the flux operator:
t0 +dt
0 /2
P
(1)
(t0 ) = η
dt tr ˆ aˆ † (t)a(t)
ˆ
.
(11)
t0 −dt0 /2
In case of single-photon wavepackets, the function P (1) (t0 ) gives the probability
to detect this photon within the considered time interval. In practice, the lower
limit of the duration dt0 is given by the detector time resolution T , i.e. dt0 T .
If the photon duration is much longer than the detector time resolution, δt T
and dt0 = T , Eq. (11) can be simplified to
P (1) (t0 ) = ηT tr ˆ aˆ † (t0 )a(t
(12)
ˆ 0) .
The measurement of the detection probability requires a large ensemble of single
photons. In the following, we therefore assume a periodic stream of single photons
emitted one-after-the-other, so that the photons always hit the detector one by one.
If all photons of this stream are identical, the light field can simply be described
by a state vector |1ξ and the density operator is given by ˆ = |1ξ 1ξ |, with
|1ξ = cˆξ† |0. In this case, the average detection probability of the ensemble of
photons is given by the square of the absolute value of the mode function, ξ(q),
and is therefore identical to the shape of each individual photonic wavepacket,
2
P (1) (t0 ) = ηT ξ(t0 ) = ηT 2 (t0 ).
(13)
As already discussed, the photons may differ from one another, and the density
operator is given according to Eq. (6). The average detection probability is then
given by
260
T. Legero et al.
[3
P (1) (t0 ) = ηT
dϑ f (ϑ) 2 (t0 , ϑ).
(14)
To obtain this equation, we assume that trace and integration can be exchanged.
Obviously the average detection probability for the photon ensemble differs from
that for individual photons. The average detection probability is, in general, affected by the variation, f (ϑ), of the parameters of the mode function, ξ(t).
Therefore it shows only a temporal envelope of the photon ensemble. However,
the effect of each parameter onto P (1) (t0 ) can be very different. A variation of
the frequency, e.g., does not affect the real amplitude of the mode function, so
that the average detection probability, P (1) (t0 ), is simply given by Eq. (13). This
is not the case for variations of the other parameters, as will be shown in Section 4.
3. Two-Photon Interference
We now consider two independent streams of Gaussian-shaped single photons
that impinge on a 50/50 beam splitter such that always two photons are superimposed. As we show in Section 5, these two streams can originate from one
single-photon source by directing each photon randomly into two different paths
of suitable length, so that successively generated photons hit the beam splitter
at the same time. Here we ask for the probability to detect the photons of each
pair in different output ports of the beam splitter. In case of identical photons,
the joint detection probability is zero. With polarization-entangled photon pairs
emitted from a down-conversion source, this effect has first been used by Alley
and Shih (1986) to test the violation of Bell’s inequality by joint photodetections,
and one year later, Hong et al. (1987) have used it to measure the delay between
two photons with sub-picosecond precision. Recently, two-photon interference
phenomena have successfully been employed to test the indistinguishability of
independently generated single photons (Santori et al., 2002). To illustrate this
interference effect, we first assume that each photon of a given stream can be described by the same quantum mechanical state vector, |1ξ , but allow the state
vectors of the two considered streams to differ from one another. In Section 4
we generalize this discussion to streams of photons which show a variation in the
parameters of the mode functions, e.g., a variation in the photon frequency. Finally we show that the interference of photon pairs reveals information about the
variations of the mode functions.
In Sections 3.1 and 3.2 we start with a brief discussion of the beam splitter
and the principle of the two-photon interference. Afterwards we analyze the joint
detection probability for photons in the limits of a photon that is either very short
or very long compared to the detector time resolution.
3]
SINGLE PHOTONS USING TWO-PHOTON INTERFERENCE
261
3.1. Q UANTUM D ESCRIPTION OF THE B EAM S PLITTER
The beam splitter is an optical four-port device with two inputs and two outputs.
The principle of the beam splitter is shown in Fig. 1. As discussed by Leonhardt
(1997), each port has its own creation and annihilation operators, and the output
operators can be expressed by the input operators using a unitary transformation
matrix B. This relation is valid for creation and annihilation operators a(ω)
ˆ
of
monochromatic waves as well as for operators of spatiotemporal modes, bˆχ or cˆξ .
It reads:
aˆ 3
aˆ 1
(15)
=B
and aˆ 3† , aˆ 4† = aˆ 1† , aˆ 2† B∗ .
aˆ 4
aˆ 2
In the following discussion, we assume an ideal lossless
and polarization inde√
pendent beam splitter with transmission coefficient σ . The matrix of this beam
splitter is given by
√
√
σ
1−σ
√
.
B=
(16)
√
σ
− 1−σ
The opposite signs of the off-diagonal terms reflect the phase jump of π for the
reflection at one side of the beam splitter.
The transmission of photons from the input side to the output side of the beam
splitter can be understood as a quantum mechanical evolution of the system. This
F IG . 1. The ideal lossless beam splitter√
is fully characterized by its transmission coefficient σ .
The reflection coefficient is then given by 1 − σ . Light can enter the beam splitter through two
different input ports 1 and 2. According to the transmission and the reflection coefficient, it is divided
into the output ports 3 and 4. For one of the reflections, the light is subject to a phase jump of π
which is indicated by the minus sign. In the Heisenberg picture (a), one accounts for this process by
transforming the creation and annihilation operators of the two input modes (1 and 2) into suitable
operators of the output modes (3 and 4), whereas in the Schrödinger picture (b), the action of the
%† , which acts on the wavevector and couples the
beamsplitter is expressed by the unitary operator B
two through-going modes (1 and 2).
262
T. Legero et al.
[3
evolution can be described in two equivalent pictures corresponding to the Heisenberg and the Schrödinger picture in quantum mechanics (Campos et al., 1989;
Leonhardt, 2003). In the Heisenberg picture, the evolution is described by the
creation and annihilation operators. The output operators are considered as the
evolved input operators whereas the state vector of the field remains unchanged
% and B
%† , this evolution can also be
(see Fig. 1(a)). Using the unitary operators B
expressed by
aˆ 1
aˆ 1 %†
aˆ 3
%
and
=B
=: B
B
aˆ 4
aˆ 2
aˆ 2
aˆ 1
%† aˆ 1 B.
%
B∗
(17)
=: B
aˆ 2
aˆ 2
Alternatively, in the Schrödinger picture, the evolution can be calculated using the
state vector of the light field. In this case, the state vector of the input side |Ψin %† |Ψin , while the modes
evolves to a state vector at the output side, |Ψout = B
themselves do not change, that is modes 1 and 2 are defined as the through-going
modes (see Fig. 1(b)). In the next subsection the Schrödinger picture is used to
illustrate the principle of the two-photon interference.
3.2. P RINCIPLE OF THE T WO -P HOTON I NTERFERENCE
We consider two identical photons that impinge on a 50/50 beam splitter. The
input state of the light field is given by |Ψin = |11H |12H , where the indices
label the two input ports and the polarization of the photons. Here, we assume
two photons of horizontal polarization. In the Schrödinger picture we describe
%† as follows:
the evolution of the state using the unitary operator B
%† aˆ † aˆ † |0.
%† |11H |12H = B
B
1H 2H
(18)
%B
%† is equal to the identity operator 1 and B
%† |0 = |0 we can write
Since B
%† aˆ † B
%%† ˆ † B|0,
%
%† aˆ † aˆ † |0 = B
B
1H 2H
1H B a
2H
(19)
which according to Eq. (17) gives
√ √ %%† ˆ † B|0
% = 1/ 2 aˆ † − aˆ † 1/ 2 aˆ † + aˆ † |0
%† aˆ † B
B
1H B a
2H
1H
2H
1H
2H
†2
†2
†
†
†
† |0.
− aˆ 2H
+ aˆ 1H
aˆ 2H
− aˆ 2H
aˆ 1H
= 1/2 aˆ 1H
Each term in this sum of creation operators corresponds to one of four possible
photon distributions in the beam splitter output ports, shown in Fig. 2. In the first
two cases, (a) and (b), both photons are found in either one or the other output,
whereas in the cases (c) and (d), the photons go to different ports. The last two
cases are indistinguishable, but the two expressions leading to cases (c) and (d)
3]
SINGLE PHOTONS USING TWO-PHOTON INTERFERENCE
263
F IG . 2. Two impinging photons lead to four possible photon distributions at the beam-splitter
output. In the first two cases (a) and (b) the photons would be found together. In the remaining two
cases (c) and (d) the photons would leave the beam splitter through different ports. Since the quantum
states of the cases (c) and (d) show different signs, they interfere destructively.
have opposite sign. Therefore the two possibilities interfere destructively. As a
consequence, the two photons always leave the beam splitter as a pair and the
output state is given by the superposition
%† |11H |12H = √1 |21H |02H − |01H |22H .
B
(20)
2
This quantum interference occurs only if the photons are identical. If the photons
were distinguishable, no interference takes place. For example, two photons of
orthogonal polarization, |Ψin = |11H |12V , give rise to four different output
states which are distinguishable by the photon polarization. In this case the overall
output state can be written as a product state, e.g.,
%† |11H |12V = √1 |11H |02 − |01 |12H
B
2
1 ⊗ √ |11V |02 + |01 |12V ,
2
which describes the state of two independently distributed photons.
Note that all temporal aspects of the light field are neglected in the above
discussion. In the next sections, the two-photon interference is discussed under
consideration of the photon duration and the time resolution of the detectors.
3.3. T EMPORAL A SPECTS OF THE T WO -P HOTON I NTERFERENCE
We now take into account that the photodetections in the output ports of the beam
splitter might occur at different times, t1 and t2 . We use the Heisenberg picture to
calculate the probability of a joint photodetection from the second-order correlation function,
tr ˆ 1,2 Aˆ 3s,4s (t1 , t2 ) ,
G(2) (t1 , t2 ) =
(21)
s,s 264
T. Legero et al.
[3
where ˆ 1,2 describes the two-photon input state and the operator Aˆ 3s,4s (t1 , t2 ) is
given by
†
†
(t1 )aˆ 4s
ˆ 4s (t2 )aˆ 3s (t1 )
Aˆ 3s,4s (t1 , t2 ) := aˆ 3s
(t2 )a
and s, s ∈ {H, V }. (22)
The probability for a photodetection at the first detector within the time interval
[t0 − dt0 /2, t0 + dt0 /2] and at the second detector within a time interval shifted
by τ , [t0 + τ − dτ/2, t0 + τ + dτ/2], is then given in analogy to Eq. (11),
t0 +dt
0 /2
P
(2)
(t0 , τ ) = η3 η4
t0 +τ+dτ/2
dt1
t0 −dt0 /2
dt2 G(2) (t1 , t2 ).
(23)
t0 +τ −dτ/2
Here we assume that the detectors have different efficiencies, η3 and η4 . In analogy to Section 2.3 the smallest duration of the detection intervals is given by the
detector time resolution, T , so that dt0 T and dτ T . In the following, we
calculate the joint detection probability in the limit of very short and very long
photons.
If the photons are very short compared to the time resolution of the detectors,
δt T , the limits of the integration in Eq. (23) can be extended to infinity, so
that
dt1 dt2 G(2) (t1 , t2 )
P (2) = η3 η4
(24)
gives the probability of a coincidence of photodetections.
For very long photons with δt T , the integration in Eq. (23) leads to
P (2) (t0 , τ ) = η3 η4 G(2) (t0 , t0 + τ ) dt0 dτ.
(25)
Therefore the probability of a joint photodetection can be studied as a function
of the two detection times, t0 and t0 + τ . In practice, only the time difference, τ ,
between two photodetections is relevant. Therefore we integrate P (2) (t0 , τ ) over
the time t0 of the first photodetection. This gives
(2)
P (τ ) = η3 η4 T
(26)
dt0 G(2) (t0 , t0 + τ ),
where dτ is substituted by the detector time resolution T . The second-order correlation function, G(2) , plays a central role in the calculation of the joint detection
probability. It is now analyzed taking the polarization and the spatiotemporal
modes of the photons into account.
3.4. C ORRELATION F UNCTION
We calculate the correlation function G(2) for two photons characterized by two
mode functions, ξ1 and ξ2 . Without loss of generality, we assume that both photons are linearly polarized with an angle ϕ between the two polarization directions. The state of the photons is then given by |1ξ1 1H and cos ϕ|1ξ2 2H |02V +
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SINGLE PHOTONS USING TWO-PHOTON INTERFERENCE
265
sin ϕ|02H |1ξ2 2V , respectively. The density operator, ˆ 1,2 = |Ψin Ψin |, is given
by the input state
|Ψin = cos ϕ|1ξ1 1H |1ξ2 2H + sin ϕ|1ξ1 1H |1ξ2 2V ,
(27)
which is a superposition of the cases in which the impinging photons are parallel
and perpendicular polarized to each other. The correlation function can then be
(2)
(2)
written as a sum of two expressions GH H and GH V ,
(2)
(2)
G(2) = cos2 ϕGH H + sin2 ϕGH V ,
(28)
(2)
where the first function GH H accounts for the input state in which both pho(2)
tons have the same polarization and the second function GH V accounts for the
perpendicular polarized case. Taking the mode functions into account, these two
expressions read
|ξ1 (t1 )ξ2 (t2 ) − ξ2 (t1 )ξ1 (t2 )|2
,
(29)
4
|ξ1 (t1 )ξ2 (t2 )|2 + |ξ1 (t2 )ξ2 (t1 )|2
G(2)
(30)
(t
,
t
)
=
.
1
2
HV
4
We emphasize that the correlation function for parallel polarized photons is always zero for t1 = t2 , even if the mode functions ξ1 (t) and ξ2 (t) are not identical.
As a consequence, the probability of a joint photodetection, Eq. (26), is always
zero for τ = t2 − t1 = 0, i.e. no simultaneous photodetections are expected even
if the photons are distinguishable with respect to their mode functions.
As already mentioned in Section 2.1, the mode function can be written as the
product of a real amplitude and a complex phase, ξj (t) = j (t) exp (−iΦj (t))
with j ∈ {1, 2}. Since the correlation function G(2)
H V for perpendicular polarized
photons is independent of the phase, it can be written as
(2)
GH H (t1 , t2 ) =
(1 (t1 )2 (t2 ))2 + (1 (t2 )2 (t1 ))2
.
(31)
4
This is not the case for the correlation function of parallel polarized photons which
(2)
carries a phase-dependent interference term. It can be expressed as GH H (t1 , t2 ) =
G(2)
H V (t1 , t2 ) − F (t1 , t2 ), with
(2)
GH V (t1 , t2 ) =
F (t1 , t2 ) :=
1 (t1 )2 (t2 )1 (t2 )2 (t1 )
2
× cos Φ1 (t1 ) − Φ1 (t2 ) + Φ2 (t2 ) − Φ2 (t1 ) .
However, this phase-dependency is only relevant, if Φ1 (t) and Φ2 (t) display a
different time evolution. Otherwise the sum over the phases is always zero. Such
a difference in the time evolution is given if, e.g., the frequencies of the photons are different. In that case, the interference term oscillates with the frequency
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difference, which gives rise to an oscillation in the joint photodetection probability, P (2) (τ ). This will be discussed further in Section 3.6.
Taking the interference term into account, the overall correlation function,
Eq. (28), can be summarized to
2
G(2) (t1 , t2 ) = G(2)
H V (t1 , t2 ) − cos ϕF (t1 , t2 ),
(32)
where the effect of the interference term depends on the angle ϕ between the two
photon polarizations.
In the next two subsections, the joint detection probability, Eq. (23), is analyzed
for very long and very short photons.
3.5. T WO -P HOTON I NTERFERENCE WITHOUT T IME R ESOLUTION
First we assume Gaussian-shaped photons which are very short compared to the
time resolution of the photodetectors, δt T . In this case one can only decide
whether there is a coincidence of detections within the time interval T or not,
and the coincidence probability is given by Eq. (24). With a possible frequency
difference := ω02 − ω01 and an arrival-time delay δτ := τ02 − τ01 of the
photons, the coincidence probability is given by
1
δt 2
δτ 2
(2)
2
P =
(33)
1 − cos ϕ exp −
exp − 2
,
2
4/2
δt
where we assume that the photons hit perfect photodetectors with η3 = η4 = 1.
We analyze the coincidence probability as a function of the photon delay δτ for
different photon polarizations and frequency differences. This is shown in Fig. 3.
As already discussed in Section 3.2, perpendicular polarized photon pairs, ϕ =
π/2, show no interference at all. Therefore, the probability for detecting photons
at different output ports of the beam splitter is always 1/2, independent of the
photon delay, δτ .
If the photons have identical polarizations, ϕ = 0, and identical frequencies,
= 0, the coincidence probability shows a Gaussian-shaped dip centered at
δτ = 0. The minimum of P (2) is zero, indicating that the photons never leave the
beam splitter through different output ports. If the photon pairs show any difference in their polarization, shape or frequency, there is no perfect interference and
the minimum of the dip is no longer zero. Therefore the two-photon interference
can be used to test the indistinguishability of photons.
The first measurement of the coincidence rate as a function of the relative
photon delay was performed by Hong et al. (1987) using photon pairs from a parametric downconversion source. They controlled the relative delay of the photons
by shifting the position of the 50/50 beam splitter. The frequencies and bandwidths of the photons were adjusted by using two identical optical filters, so that
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267
F IG . 3. Coincidence probability as a function of the arrival time delay, δτ , of two linear polarized
photons. In case of perpendicular polarized photons (ϕ = π/2) there is no interference at all and the
coincidence probability shows the constant value 1/2. If the photons are parallel polarized (ϕ = 0)
and have identical frequency ( = 0), there is a Gaussian-shaped dip which goes down to zero for
simultaneous impinging photons, δτ = 0. Any difference in polarization or frequency leads to a
reduced depth of this dip.
the coincidence rate dropped nearly to zero for simultaneously impinging photons. As one can see from Eq. (33), the width of the dip is identical to the photon
duration, δt. Therefore this experiment was used to measure the duration and
bandwidth of the photons.
So far, most two-photon interference experiments were performed with very
short photons. Therefore the joint detection probability was only considered as a
function of the photon delay, δτ . However, if the photon duration is much larger
than the detector time resolution, the time τ between the photodetections in the
two output ports can be measured and the joint detection probability can additionally be analyzed in dependence of the detection-time difference.
3.6. T IME -R ESOLVED T WO -P HOTON I NTERFERENCE
We now assume, that the photon duration is much larger than the detection time
resolution, δt T . Since the time, τ , between the photodetections can be measured within the photon duration, the joint detection probability can be analyzed
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[3
as a function of this detection-time difference. Using Eq. (26) and assuming
Gaussian-shaped photons of identical duration, δt, the joint detection probability is given by
1 − cos2 ϕ cos(τ )
T
(2)
2 τ δτ
+ sinh
P (τ, δτ ) = √
2
δt 2
π δt
2
2
δτ + τ
× exp −
(34)
.
δt 2
In Fig. 4 the joint detection probability is shown as a function of the photon
arrival-time delay, δτ , and the detection time difference, τ .
The sign of τ indicates which detector clicks first. Similarly, the sign of δτ
determines which photon arrives first at the beam splitter. Note that the joint detection probability can only be different from zero if |τ | ≈ |δτ |. This leads to the
cross-like structure in Fig. 4(a–c). Since the photons only interfere if the relative
delay is smaller than the photon duration, we focus our attention to the center of
Fig. 4(a–c).
Again, we start our analysis with perpendicular polarized photon pairs. Obviously, no interference takes place, and as one can see in Fig. 4(a), even simultaneously impinging photons (with δτ = 0) can be detected in different output ports
of the beam splitter. The joint detection probability shows therefore a Gaussianshaped peak. According to Eq. (34), the width of this peak is identical to the
photon duration, δt. Since the photons are distinguishable by their polarization, an
additional frequency difference, , does not affect this result. Assuming photon
pairs with identical mode functions, the joint detection probability of perpendicular polarized photons can be used to determine the photon duration.
Figure 4(b) shows the joint detection probability for parallel polarized photons
of identical frequency, = 0. For simultaneously impinging photons the joint
detection probability is always zero, which indicates that the photons coalesce
and leave the beam splitter always together.
If the parallel polarized photons show a frequency difference, the joint detection
probability oscillates as a function of the detection time difference, τ . This is
shown in Fig. 4(c). As one can see from Eq. (34), the frequency difference, ,
determines the periodicity of this oscillation. We emphasize that the oscillation
always leads to a minimum at τ = 0, independent of , so that even photons of
different frequencies are never detected simultaneously in different output ports.
Furthermore, the joint detection probability at the maxima is always larger than
the joint detection probability for perpendicular polarized photons.
Without time resolution, the detection-time difference cannot be measured and
the joint detection probability, P (2) (τ, δτ ), has to be integrated over τ . This links
the results of a time-resolved two-photon interference to the discussion of Section 3.5. In case of perpendicular polarized photons, the τ -integrated function
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269
F IG . 4. Joint detection probability, P (2) , as a function of the relative delay between the photons, δτ , and the time difference between photodetections, τ , for perpendicular polarized photons (a)
and parallel polarized photons (b). In (c) the parallel polarized photon pairs have a frequency difference , which leads to an oscillation in the joint detection probability. All times and frequencies are
normalized by the photon duration, δt.
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[4
P (2) (δτ ) shows the constant value 1/2. If the photons are identical, the integration leads to a Gaussian-shaped dip, which was already discussed in Section 3.5.
However, the oscillation in the joint detection probability for photon pairs with a
frequency difference is no longer visible. The integration leads, in accordance to
Eq. (33), only to a reduced depth of the dip in P (2) (δτ ).
4. Jitter
Up to now, we assumed that all photons of a given stream can be described by
the same state vector |1ξ . However, this requires a perfect single-photon source,
which is able to generate a stream of photons without any variation in the parameters of the Gaussian mode functions. Here, we consider a more realistic scenario,
in which a stream of single photons shows a jitter in the parameters, ϑ. The
quantum mechanical state of the photons is then given by the density operator
of Eq. (6).
Such a jitter in the mode functions has important consequences on the results
of measurements which can be performed on the single-photon stream. On the
one hand, as already discussed in Section 2.3, it affects the average detection
probability of the photons in a way that its measurement does in general not reveal
information about the duration or shape of each single photon. On the other hand,
variations in the mode functions of photon pairs have an influence on the joint
detection probability in two-photon interference experiments. This is discussed in
some detail in the following two subsections.
To analyze the effect of jitters on the two-photon interference, we consider
two streams of Gaussian-shaped photons with a variation in the parameters of
their mode functions. In analogy to Eq. (6) the density operator for photon pairs
impinging on the beam splitter is given by
ˆ 1,2 =
(35)
dϑ1 dϑ2 f1 (ϑ1 )f2 (ϑ2 )|1ξ1 |1ξ2 1ξ1 |1ξ2 |,
so that, using Eq. (21), the correlation function reads
ˆ 0 , t0 + τ ) .
G(2) (t0 , t0 + τ ) =
dϑ1 dϑ2 f1 (ϑ1 )f2 (ϑ2 ) tr (ξ
ˆ 1 , ξ2 )A(t
(36)
Here the expression (ξ
ˆ 1 , ξ2 ) substitutes |1ξ1 |1ξ2 1ξ1 |1ξ2 |. In Eq. (35) we assumed that all photons have identical polarization and that they are completely
independent from each other. Therefore the density operator has only diagonal
elements. The parameters of the mode functions ξ1 and ξ2 of the two streams are
summarized by ϑ1 and ϑ2 , respectively. In general, all parameters of the mode
functions could be subject to a variation, and all the variations could in principle
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271
depend on each other. However, in the following two subsections, we focus our
attention only on two examples of jitters and analyze the detection probability of
photons, P (1) , for a single photon stream as well as the joint detection probability, P (2) , of two streams superimposed on a beam splitter.
First, in Section 4.1, we consider streams of photons which are characterized
by a variation of the center frequency, ω0j , so that each photon pair exhibits a
variation of the frequency difference, = ω02 −ω01 . All remaining parameters of
the mode functions, e.g., the duration of the photons, are assumed to be identical.
In Section 4.2, we consider photons, which show only a variation in their emission
time, so that photon pairs are characterized by a variation in their arrival-time
delay δτ = τ02 − τ01 .
4.1. F REQUENCY J ITTER
We start our discussion of a frequency jitter by analyzing its effect on the average detection probability, P (1) (t0 ), for a perfect photodetector with the detection
efficiency η = 1. If the frequency variation in the stream of single photons is
described by a normalized distribution function, f (ω), the average detection probability is given, according to Eq. (14), by the integral
2
P (1) (t0 ) = T
(37)
dω f (ω)ξ(t0 , ω) .
Since only the phase of the Gaussian mode functions depends on the frequency,
the absolute value, |ξ(t0 , ω)|2 = 2 (t0 ), is independent of ω. Thus, the average
detection probability is not affected by any frequency jitter and is entirely determined by the spatiotemporal mode function of each single photon.
However, a frequency jitter affects the joint detection probability of photon
pairs superimposed on a beam splitter. To illustrate this, we assume two independent streams of photons, each fluctuating around a common center frequency ω0
according to a normalized Gaussian frequency distributions, f1 (ω01 ) and f2 (ω02 ).
Hence, the frequency difference of the photon pairs, = ω02 − ω01 , shows also
a normalized Gaussian variation,
1
exp −2 /δω2 ,
f () = √
πδω
(38)
with width δω depending on the widths of the frequency distributions of both
2 + δω2 . The density operator of the photon pairs can then
streams, δω = δω01
02
be expressed in terms of the distribution function of the frequency difference,
ˆ 1 , ξ2 ).
ˆ 1,2 = d f ()(ξ
(39)
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[4
F IG . 5. Joint detection probability as a function of the detection-time difference, τ , for simultaneously impinging photons, δτ = 0, of identical polarization. The photons are subject to a frequency
jitter of width δω.
As the operations of trace and integration are exchangeable, the correlation function, according to Eq. (36), can be written as
ˆ 0 , t0 + τ ) .
ˆ 1 , ξ2 )A(t
G(2) (t0 , τ ) = d f () tr (ξ
(40)
For photons which are very long compared to the detector time-resolution, the
joint detection probability is given by Eq. (26). In case of simultaneously impinging photons, δτ = 0, this leads to
T
τ2
τ2
1 − cos2 ϕ exp −
exp
−
.
P (2) (τ ) = √
(41)
4/δω2
δt 2
2 π δt
For photons of parallel polarization, the result is shown in Fig. 5. In the limit
of δω → ∞ the joint detection probability shows a Gaussian-shaped peak of
width T1 = δt, which is the photon duration. As one can see from Eq. (41), the
joint detection probability is always zero for τ = 0 as long as the width of the
frequency distribution, δω, is finite. In fact, as one can deduce from Eq. (41), this
leads to a dip in the joint detection probability around τ = 0 that is T2 = 2/δω
wide. Note that T2 represents a coherence time which must not be mixed up with
the duration of each single photon, δt.
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To determine the amount of a frequency jitter from a time-resolved two-photon
interference experiment, one has to perform two measurements. First, the joint
detection probability of perpendicular polarized photons, ϕ = π/2, reveals the
photon-duration, δt. Afterwards, the joint detection probability of parallel polarized photons, ϕ = 0, is used to measure T2 and derive the width of the frequency
jitter, δω. This is shown in detail in Section 5.
If the photons are very short compared to the detector time-resolution, the
coincidence probability must be calculated using Eq. (24). The coincidence probability is then a function of the relative photon delay, δτ , and is given by
1
δτ 2
2 cos2 ϕ
P (2) (δτ ) =
(42)
exp − 2
1− √
.
2
δt
4 + δt 2 δω2
In analogy to Section 3.5, a frequency jitter, δω, now leads to a decreased depth
of the Gaussian-shaped dip, while the width of this dip is not affected and always
identical to the photon duration.
In principle, it is possible to derive the frequency jitter also from a two-photon
interference experiment without time-resolution, but there are some major disadvantages. First, the depth of the dip depends not only on a frequency jitter, but
also on the mode matching of the transversal modes of both beams. A nonperfect
mode matching leads to a factor comparable to cos2 ϕ in Eq. (42). Therefore, in
contrast to the time-resolved measurement, one cannot distinguish between a nonperfect mode matching or a frequency jitter. Second, in case of two independent
streams of photons from two different single-photon sources, it is impossible to
decide whether a constant frequency difference or a frequency jitter is the reason
for a decreased dip depth. And third, if the frequency jitter is large, the depth of
the dip is very small, whereas in a time-resolved measurement, the dip-depth remains unchanged. As it is much more reliable to determine a small width rather
than a small depth, the time-resolved method is much more powerful.
4.2. E MISSION -T IME J ITTER
Now we assume a stream of single photons which shows a jitter in the emission
time of each photon. This variation of the emission time is assumed to be given by
a normalized Gaussian distribution function, f (τ0 ). The average detection probability of the photons for an ideal photodetector with η = 1 is again given by
Eq. (14),
2
P (1) (t0 ) = T
(43)
dτ0 f (τ0 )ξ(τ0 − t0 ) .
This is a convolution of the detection probability, |ξ(t0 )|2 , of each single photon and the emission-time distribution, f (τ0 ), of the photon stream. Therefore
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[4
the average detection probability is always broader than the detection probability
which would arise solely from the duration of each single photon. This shows that
a variation in the parameters of the spatiotemporal mode functions can alter the
detection probability of the photons. Therefore, in general, the average detection
probability is not identical to the detection probability of individual photons.
To investigate the influence of an emission-time jitter on the joint detection
probability in a two-photon interference experiment, we now assume two streams
of photons with a Gaussian emission-time distribution of identical width, τ . In
this case, the photon pairs are characterized by a jitter in the arrival-time delay of
the photons, which is again given by a Gaussian distribution,
1
f (δτ ) = √
exp −δτ 2 /τ 2 .
π τ
(44)
The correlation function G(2) (t0 , t0 + τ ) can be written in analogy to Eq. (40),
using only the variation of the relative photon delay,
(2)
ˆ 0 , t0 + τ ) ,
G (t0 , τ ) = d(δτ ) f (δτ ) tr (ξ
(45)
ˆ 1 , ξ2 )A(t
and the joint detection probability has to be calculated according to Eq. (26). In
case of simultaneously impinging photons, this leads to
T
τ2
(2)
2
P (τ ) = √ √
1 − cos ϕ exp − 2
δt + δt 4 /τ 2
2 π δt 2 + τ 2
τ2
.
× exp − 2
(46)
δt + τ 2
In contrast to the previous case, now the width of the Gaussian-shaped peak in the
joint detection probability of perpendicular polarized photons is no more identical to the photon duration. The variation in the emission time affects also the
amplitude of the spatiotemporal mode functions and affects the joint detection
probability even without interference. This can be seen in Fig. 6, which shows
the joint detection probability for photon pairs characterized by a distribution of
the photon arrival-time delay. The width, T1 , of the Gaussian-shaped peak is no
longer identical to the photon duration, δt. As one can derive from Eq. (46), it is
now broadened by the width of the variation in the photon delay, τ :
T1 = δt 2 + τ 2 .
(47)
Furthermore, the width of the dip in case of parallel polarized photons is not
independent of the width T1 , but is given by
δt
T2 = δt 2 + δt 4 /τ 2 =
(48)
T1 .
τ
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SINGLE PHOTONS USING TWO-PHOTON INTERFERENCE
275
F IG . 6. Joint detection probability as a function of the detection time difference, τ , for photon pairs
with a variation in their relative photon delay, δτ . The photons are assumed to be (a) perpendicular and
(b) parallel polarized to each other. Curve (c) is a Gaussian mode function of width δt. The width, T1 ,
of the Gaussian peak in (a) is broadened by the variation of the relative photon delay, τ , which is
here assumed to be 2δt.
A time-resolved two-photon interference experiment can again be used to determine the variation in the emission time of the photon streams. However, since
the shapes of the joint detection probabilities for a frequency and an emissiontime jitter are identical, it is in general not possible to distinguish between the
two. Nonetheless, one can determine the maximum values of both jitters, as well
as all pairs of frequency and an emission-time jitters matching the data. This is
discussed in the next section.
Note that in case of very short photons, the coincidence probability has again
to be calculated using Eq. (24). The width of the Gaussian-shaped dip in the coincidence probability
1
δτ 2
cos2 ϕ
exp − 2
1− P (2) (δτ ) =
(49)
2
δt + τ 2
1 + τ 2 /δt 2
is broadened by the emission-time jitter, τ , and the depth of the dip is also decreased by τ . Again it is possible to determine the emission-time jitter from such
a two-photon interference experiment without time resolution. The disadvantages
of such a procedure have already been discussed in Section 4.1.
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4.3. AUTOCORRELATION F UNCTION OF THE P HOTON ’ S S HAPE
We now consider only one source of single photons that we want to characterize
using a two-photon interference experiment. We assume that the stream of photons generated by this source is split up in such a way that each single photon
is randomly directed along two different paths. These two paths are of different
length and the repetition rate of the source is chosen in such a way that only successively generated photons impinge on the beam splitter at the same time. The
details of such an experiment are discussed in Section 5.
In general, we must distinguish between the photon ensemble of the whole
stream and the subensemble of photon pairs superimposed on the beam splitter.
The latter consists only of successively generated photons and the characterization of a single-photon source by a two-photon interference experiment takes into
account only this subensemble. Since the jitter in the subensemble of successive photon pairs does not have to be identical to the jitter in the whole photon
stream, we need a method to decide whether the results of a two-photon interference experiment can be generalized to all photons generated by the single-photon
source. In case of very long photons, this can be done by comparing the joint
detection probability, P (2) (τ ), for perpendicular polarized photons and the autocorrelation function, A(2) (τ ), of the average detection probability of the whole
photon stream.
We start our discussion with a stream of identical single photons, so that the
average detection probability is simply given by the square of the amplitude of
the spatiotemporal mode function, P (1) (t0 ) = T 2 (t0 ). On the one hand, the autocorrelation function of P (1) (t0 ) reads
2
A(2) (τ ) = dt0 P (1) (t0 )P (1) (t0 + τ ) = T 2 dt0 (t0 )(t0 + τ ) . (50)
On the other hand, the joint detection probability for perpendicular polarized photon pairs of this stream is given by Eq. (31), which leads to
2
P (2) (τ ) ∝ T 2 dt0 (t0 )(t0 + τ ) .
(51)
Therefore the joint detection probability for perpendicular polarized photons and
the autocorrelation function have the same shape.
However, if the photons show a variation in their spatiotemporal modes, the
two functions are no longer equal. In the following, we assume that the variations in the whole photon stream are described by a normalized distribution
function f (ϑ), whereas the variations in the subensemble of successively emitted
photons is given by f˜(ϑ). In general, both functions do not have to be identical,
i.e. the jitter in the subensemble can be smaller than the jitter in the whole photon
stream. The autocorrelation function of the average detection probability P (1) (t0 ),
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SINGLE PHOTONS USING TWO-PHOTON INTERFERENCE
277
see Eq. (14), is given by
2
A(2) (τ ) = T 2 dt0
dϑ1 dϑ2 f (ϑ1 )f (ϑ2 ) (t0 , ϑ1 )(t0 + τ, ϑ2 ) .
(52)
The joint detection probability,
2
dϑ1 dϑ2 f˜(ϑ1 )f˜(ϑ2 ) (t0 , ϑ1 )(t0 + τ, ϑ2 )
P (2) (τ ) ∝ T 2 dt0
(53)
is either independent
0 , ϑ1 )(t0 +τ, ϑ2
of ϑ1 and ϑ2 , or if the distribution function f (ϑ) of the whole photon stream is
identical to the distribution function f˜(ϑ) of successively emitted photons.
The comparison of the joint detection probability of perpendicular polarized
photon pairs and the autocorrelation function of the average detection probability
therefore answers the question whether the results of a two-photon interference
experiment can be generalized to the whole photon stream.
is therefore only equal to A(2) (τ ), if ((t
))2
5. Experiment and Results
In the previous three sections we discussed the theoretical background for characterizing single photons using two-photon interference. Now, we show how to
use this method to experimentally characterize single photons that are emitted
from only one source. This single-photon source has been realized using vacuumstimulated Raman transitions in a single Rb atom located inside a high-finesse
optical cavity. In Section 5.1 we briefly review the principle of this source and
discuss the experimental setup, which was used to investigate the two-photon interference. For further details concerning the single-photon generation, we refer
to Kuhn et al. (2002) and references therein. The measurement of the average detection probability of a stream of photons emitted from this source is discussed
in Section 5.2. As already discussed, we use the autocorrelation function of the
average detection probability to determine whether the results of a two-photon
interference experiment can be generalized to the whole photon stream. Since the
duration of the photons is much longer than the time resolution of the detectors,
the interference of successively emitted photons is measured in a time-resolved
manner. The results and the interpretation of these measurements are discussed in
detail in Sections 5.3 and 5.4.
5.1. S INGLE -P HOTON S OURCE AND E XPERIMENTAL S ETUP
A sketch of the single-photon source and the experimental setup that we use to
superimpose successively generated photons on a 50/50 beam splitter is shown
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F IG . 7. Single-photon source and experimental setup used to investigate the two-photon interference of successively emitted photons. The photons are generated in an atom-cavity system by an
adiabatically driven stimulated Raman transition. A polarizing beam splitter directs the photons randomly into two optical fibers. The delay from photon to photon matches the travel-time difference in
the two fibers, so that successively emitted photons can impinge simultaneously on the 50/50 beam
splitter. Using a half-wave plate, the polarization of the photons can be chosen parallel or perpendicular to each other. The photons are detected using avalanche photodiodes with a detection efficiency
of 50% and a dark-count rate of 150 Hz.
in Fig. 7. The single-photon generation starts with 85 Rb atoms released from a
magneto-optical trap. The atoms enter the cavity mostly one-at-a-time (the probability of having more than one atom is negligible). Each atom is initially prepared
in |e ≡ |5S1/2 , F = 3, while the cavity is resonant with the transition between
|g ≡ |5S1/2 , F = 2 and |x ≡ |5P3/2 , F = 3. On its way through the cavity,
the atom experiences a sequence of laser pulses that alternate between triggering single-photon emissions and repumping the atom to state |e: The 2 µs-long
trigger pulses are resonant with the |e ↔ |x transition and drive an adiabatic
passage (STIRAP) to |g by linearly increasing the Rabi frequency. This transition goes hand-in-hand with a photon emission. In the ideal case, the duration
and pulse shape of each photon depend in a characteristic manner on the temporal
shape and intensity of the triggering laser pulses (Keller et al., 2004). As we will
discuss in Section 5.3, the photon-frequency can be chosen by an appropriate fre-
5]
SINGLE PHOTONS USING TWO-PHOTON INTERFERENCE
279
quency of the trigger laser (Legero et al., 2004). Between two photon emissions,
another laser pumps the atom from |g to |x, from where it decays back to |e.
While a single atom interacts with the cavity, the source generates a stream of single photons one-after-the-other. The efficiency of the photon generation is 25%.
As described in detail in Legero (2005), the source has been optimized with
respect to jitters by compensating the Earth’s magnetic field inside the cavity and
by adding to the recycling scheme an additional π-polarized laser driving the
transition |5S1/2 , F = 3 ↔ |5P3/2 , F = 2 to produce a high degree of spinpolarization in 5S1/2 , F = 3, mF = ±3. This results in an increased coupling of
the atom to the cavity. We have characterized the emitted photons by two-photon
interference measurements before and after this optimization.
To superimpose two successively emitted photons on the 50/50 beam splitter,
they are directed along two optical paths of different length. These paths are realized using two polarization maintaining optical fibers with a length of 10 m and
1086 m, respectively. Since the photon polarization is a priori undefined, a polarizing beam splitter is used to direct the photons randomly into the long or short
fiber. The time between two trigger pulses is adjusted to match the travel-time
difference of the photons in the two fibers, which is t = 5.28 µs. With a probability of 25%, two successively emitted photons therefore impinge on the beam
splitter simultaneously. In addition, we use a half-wave plate to adjust the mutual
polarization of the two paths.
5.2. AVERAGE D ETECTION P ROBABILITY
First, we investigate the average detection probability of the photon stream. For
this measurement, the long fiber is closed and the detection times of about 103
photons are recorded with respect to their trigger pulses. From these photons,
we calculate the probability density for a photodetection, shown in Fig. 8. The
measurement has been done (a) before and (b) after optimizing the single-photon
source. The autocorrelation functions of both curves, shown in the inset of Fig. 8,
were calculated using Eq. (50). The width of these curves is (a) 1.07 µs and
(b) 0.81 µs. In Section 5.4 we compare both results with the joint detection probability of perpendicular polarized photon pairs.
Note that one obtains no information on the shape or duration of individual
photons from a detection probability that is averaged over many photodetections.
As already stated in Section 4.2, such a measurement does not exclude that the
photons are very short so that the average probability distribution reflects only an
emission-time jitter. Only from a time-resolved two-photon interference experiment, one obtains information on the duration of the photons. This is discussed in
the next subsection.
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F IG . 8. Probability density for photodetections averaged over 103 photons (a) before and (b) after
optimizing the single-photon source. The data are corrected for the detector dark-count rate. The inset
shows the corresponding autocorrelation functions, A(2) (τ ).
5.3. T IME -R ESOLVED T WO -P HOTON I NTERFERENCE
The detection times of about 105 photons are registered by the two detectors in
the output ports of the beam splitter, while the photons in each pair impinge simultaneously, i.e. with δτ = 0. The number of joint photodetections, N (2) , is then
determined from the recorded detection times as a function of the detection-time
difference, τ (using 48 ns to 120 ns long time bins). To do that, the photon duration must exceed the time resolution of the detectors. Otherwise, joint detection
probabilities could only be examined as a function of the arrival-time delay, δτ ,
like in most other experiments.
We have performed these two-photon interference experiments before and after optimizing the single-photon source. The results are shown in Figs. 9 and 10,
respectively. Each experiment is first performed with photons of (a) perpendicular
and then with photons of (b) parallel polarization, until about 105 photons are detected. Note that the number of joint photodetections is corrected for the constant
background contribution stemming from detector dark counts.
In case of perpendicular polarization, no interference takes place. In accordance
with Section 3.6, the number of joint photodetections shows a Gaussian peak centered at τ = 0. This signal is used as a reference, since any interference leads to
5]
SINGLE PHOTONS USING TWO-PHOTON INTERFERENCE
281
F IG . 9. Number of joint photodetections in 120 ns-long time bins as a function of the detection-time difference, τ , before optimizing the single-photon source. The results are shown for photons
of (a) perpendicular polarization and (b) parallel polarization. In both cases, the data is accumulated
for a total number of 73,000 photodetections. The solid lines are numerical fits of the theoretical
expectations to the data. T1 is the width of the Gaussian peak in (a), and the width of the dip for
parallel polarized photons (b) is given by T2 . The dotted curve shows the T3 -wide autocorrelation
function, A(2) (τ ), of the average detection probability.
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F IG . 10. Number of joint photodetections in 48 ns long time bins after optimizing the single-photon source. The data is shown in (a) for perpendicular and in (b) for parallel polarized photons.
In both cases, the data is accumulated for a total number of 139,000 photodetections. The dotted curve
shows the T3 -wide autocorrelation function of the average detection probability. Compared to the results of Fig. 9, the width of the Gaussian peak in (a) is decreased to T1 = 0.64 µs and it is clearly
smaller than T3 = 0.81 µs. With parallel polarizations, a dip of increased width, T2 = 0.44 µs, is
observed. All results are discussed in Section 5.4.
5]
SINGLE PHOTONS USING TWO-PHOTON INTERFERENCE
283
a significant deviation. If we now switch to parallel polarization, identical photons are expected to leave the beam splitter as a pair, so that their joint detection
probability should be zero for all values of τ . In the experiment, however, the
signal does not vanish completely. Instead, we observe a pronounced minimum
around τ = 0, which complies well with the behavior one expects for varying
spatio-temporal modes, as shown in Figs. 5 and 6. In analogy to Eqs. (41) and
(46), respectively, the number of joint photodetections is then given by
τ2
τ2
(2)
(2)
2
N (τ ) = N0 exp − 2 1 − cos ϕ exp − 2 .
(54)
T1
T2
(2)
N0 is the peak value at τ = 0 that we measure for perpendicular polarized photons, ϕ = π/2. The time T1 is the width of this Gaussian peak, whereas T2 gives
the width of the dip for photons of parallel polarization, ϕ = 0. In Section 5.4,
both numbers are used to deduce the frequency and the emission-time jitter. The
two widths, T1 and T2 , are obtained from a fit of Eq. (54) to the measured data.
(2)
This is done in two steps. First, we obtain T1 and N0 from a fit to the data taken
with perpendicular polarized photons. We then keep these two values and obtain
the dip width T2 from a subsequent fit to the data with parallel polarized photons. In this second step, we use a polarization term of cos2 ϕ = 0.92 to take into
account that we have a small geometric mode mismatch. This is well justified
since mode mismatch and non-parallel polarizations affect the signal in the same
manner. The value of ϕ has been obtained from an independent second-order interference measurement.
As one can see by comparing Figs. 9 and 10, the compensation of the Earth’s
magnetic field and the improved recycling scheme lead to a decreased width T1
of the Gaussian peak in (a) and a broader dip T2 in (b). As we discuss in the
following, these results show that this optimization has successfully reduced the
jitter in the mode function of the photons.
Moreover, as shown in Fig. 11(a) and (b), we resolve a pronounced oscillation
in the number of joint photodetections when a frequency difference, , is deliberately introduced between the interfering photons (Legero et al., 2004). This is
achieved by driving the atom-cavity system with a sequence of trigger pulses that
alternate between two frequencies. The frequency difference between consecutive pulses is either (a) 2π × 2.8 MHz or (b) 2π × 3.8 MHz. In accordance with
Section 3.6, the oscillation in the joint detection probability always starts with a
minimum at τ = 0, and the maxima exceed the reference signal that we measure
with perpendicular polarized photons.
The latter experiment has been performed with the optimized single-photon
source. The number of detected photons (corrected for the number of dark counts)
equals the photon number in Fig. 10. Therefore N0(2) , T1 and T2 are well known
from this previous measurement. As shown by Legero (2005), the only remaining
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F IG . 11. Two-photon interference of parallel polarized photons with a frequency difference, ,
of (a) 2π × 2.8 MHz and (b) 2π × 3.8 MHz. The number of joint photodetections is accumulated
over (a) 210,000 and (b) 319,000 detection events. It oscillates as a function of the detection-time
difference. The solid curves represent numerical fits to the data with a frequency-difference of
(a) = 2π × 2.86 MHz and (b) = 2π × 3.66 MHz. The dotted curve shows the reference
signal measured with perpendicular polarized photons.
5]
SINGLE PHOTONS USING TWO-PHOTON INTERFERENCE
parameter one can obtain from a fit of the joint detection probability,
τ2
τ2
N (2) (τ ) = N0(2) exp − 2 1 − cos2 ϕ cos(τ ) exp − 2 ,
T1
T2
285
(55)
to the data is the frequency difference . Its fit value agrees very well with the frequency differences that we imposed on consecutive pulses. We therefore conclude
that the adiabatic Raman transition we use to generate the photons allows us to adjust the single-photon frequency. Moreover, the oscillations in the joint detection
probability impressively demonstrate that time-resolved two-photon interference
experiments are able to reveal small phase variations between the photons, like,
e.g., the frequency difference we have deliberately imposed here.
5.4. I NTERPRETATION OF THE R ESULTS
We start our analysis by comparing the autocorrelation function of the average
detection probability with the result of the two-photon coincidence measurement with perpendicular polarized photons, shown in Fig. 9(a) and Fig. 10(a).
The shape of the autocorrelation function, A(2) (τ ), is commensurable with the
data obtained in the two-photon correlation experiment before the optimization
of the source, but it differs significantly afterwards. As discussed in Section 4.3,
the different widths of both curves, T1 = 0.64 µs and T3 = 0.81 µs, indicate
that the photon stream is subject to variations in the spatiotemporal mode functions that are much less pronounced in the subensemble of consecutive photons.
These variations cannot be attributed to a jitter in the photon frequency, since the
autocorrelation function and the joint detection probability, given by Eqs. (52)
and (53), depend only on the frequency-independent amplitude of the mode functions. Therefore the emission time and/or the duration of the photons must be
subject to a jitter. As a consequence, the average detection probability shown in
Fig. 8 cannot represent the shape of the underlying single-photon wavepackets.
In particular the width of the measured photon detection probability is broadened
due to the emission-time jitter.
Moreover, the discrepancy between A(2) (τ ) (Fig. 8) and the Gaussian peak
(Fig. 9(a) and Fig. 10(a)) shows that the variations in the whole photon stream
are larger than the variations in the subensemble of consecutive photons. Therefore the following analysis of the times T1 and T2 of the two-photon interference
cannot be generalized to the whole photon stream.
To figure out the photon characteristics that can explain the measured quantumbeat signal, we restrict the analysis of T1 and T2 to a frequency and an emissiontime jitter. If we assume that successively emitted photons show only a variation of their frequencies, then T1 is identical to the photon duration, whereas
T2 = 2/δω is solely due to the frequency variation. In this case, T2 is identical
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to the coherence time, which one could also measure using second-order interference (Santori et al., 2002; Jelezko et al., 2003). As shown in Figs. 9 and 10, optimizing the single-photon source reduces the bandwidth of the frequency variation
from δω/2π = 1.03 MHz to δω/2π = 720 kHz. The remaining inhomogeneous
broadening of the photon frequency can be attributed to several technical reasons.
First, static and fluctuating magnetic fields affect the energies of the Zeeman sublevels and spread the photon frequencies over a range of 160 kHz. Second the
trigger laser has a linewidth of 50 kHz, which is also mapped to the photons. And
third, diabatically generated photons lead to an additional broadening.
Another explanation for the measured quantum-beat signal assumes photons
of fixed frequency
and shape, but with an emission-time jitter. In this case, we
√
have T1 = δt 2 + τ 2 and T2 = T1 δt/τ . From these two equations, one can
calculate δt, which is the lower limit of the photon duration, and τ , which is the
maximum emission-time jitter. In our experiment, the optimization of the singlephoton source led to an increase of δt from 0.29 µs to 0.36 µs, and at the same
time to a reduction of the maximum emission-time jitter from τ = 0.82 µs to
τ = 0.53 µs.
However, in general, both the frequency and the emission time are subject to a
jitter. If these fluctuations are uncorrelated, a whole range of (δω, τ )-pairs can
explain the peak and dip widths, T1 and T2 . For our two sets of data (before (a)
and after (b) the optimization of the source), this is illustrated in Fig. 12. All pairs
of frequency and emission-time jitters that are in agreement with the measured
values of T1 and T2 lie on one of the two solid lines. From this figure, it is evident
that the values for δω and τ deduced above represent the upper limits for the
respective fluctuations. Moreover, it is also nicely visible that our optimization
of the source significantly improved the frequency stability and emission-time
accuracy of our single-photon source.
We emphasize again that this information about the photons can be obtained
from time-resolved two-photon interference experiments, but not from a measurement of the average detection probability.
6. Conclusion
We have shown that time-resolved two-photon interference experiments are an
excellent tool to characterize single photons. In these experiments, two photons
are superimposed on a beam splitter and the joint detection probability in the two
output ports of the beam splitter is measured as a function of the detection-time
difference of the photons. This is only possible if the photons are long compared to
the detector time resolution. For identical photons, the joint detection probability
is expected to be zero. Variations of the spatiotemporal modes of the photons lead
to joint photodetections except for zero detection-time difference. Therefore the
7]
SINGLE PHOTONS USING TWO-PHOTON INTERFERENCE
287
F IG . 12. Frequency jitter, δω, and emission-time jitter, τ , before (a) and after (b) optimizing
the single-photon source. The two curves represent all pairs of jitters that match the two widths, T1
and T2 , found in the two-photon interference experiments.
joint detection probability shows a pronounced dip. From the width of this dip,
one can estimate the maximum emission-time jitter and the minimum coherence
time of the photons. In addition, a lower limit of the single-photon duration can be
obtained. This is not possible by just measuring the average detection probability
with respect to the trigger producing the photons. Moreover, we have shown that
a frequency difference between photons leads to a distinct oscillation in the joint
detection probability. This does not only demonstrate that we are able to adjust
the frequencies of the photons emitted from a single-photon source, but also that
one is sensitive to very small frequency differences in time-resolved two-photon
interference measurements.
7. Acknowledgements
This work was supported by the Deutsche Forschungsgemeinschaft (SPP 1078
and SFB 631) and the European Union (IST (QGATES) and IHP (CONQUEST)
programs).
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ADVANCES IN ATOMIC, MOLECULAR AND OPTICAL PHYSICS, VOL. 53
FLUCTUATIONS IN IDEAL AND
INTERACTING BOSE–EINSTEIN
CONDENSATES: FROM THE LASER
PHASE TRANSITION ANALOGY
TO SQUEEZED STATES AND
BOGOLIUBOV QUASIPARTICLES*
VITALY V. KOCHAROVSKY1,2 , VLADIMIR V. KOCHAROVSKY2 ,
MARTIN HOLTHAUS3 , C.H. RAYMOND OOI1 , ANATOLY SVIDZINSKY1 ,
WOLFGANG KETTERLE4 and MARLAN O. SCULLY1,5
1 Institute for Quantum Studies and Department of Physics, Texas A&M University,
TX 77843-4242, USA
2 Institute of Applied Physics, Russian Academy of Science, 600950 Nizhny Novgorod, Russia
3 Institut für Physik, Carl von Ossietzky Universitat, D-2611 Oldenburg, Germany
4 MIT-Harvard Center for Ultracold Atoms, and Department of Physics, MIT, Cambridge,
MA 02139, USA
5 Princeton Institute for Materials Science and Technology, Princeton University,
NJ 08544-1009, USA
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2. History of the Bose–Einstein Distribution . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1. What Bose Did . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2. What Einstein Did . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3. Was Bose–Einstein Statistics Arrived at by Serendipity? . . . . . . . . . . . . . . . .
2.4. Comparison between Bose’s and Einstein’s Counting of the Number of Microstates W
3. Grand Canonical versus Canonical Statistics of BEC Fluctuations . . . . . . . . . . . . .
3.1. Relations between Statistics of BEC Fluctuations in the Grand Canonical, Canonical,
and Microcanonical Ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2. Exact Recursion Relation for the Statistics of the Number of Condensed Atoms in an
Ideal Bose Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
293
298
299
303
307
314
315
316
320
* It is a pleasure to dedicate this review to Prof. Herbert Walther, our guide in so many fields
of physics. His contributions to atomic, molecular and optical physics are enlightened by the deep
insights he has given us into the foundations of quantum mechanics, statistical physics, nonlinear
dynamics and much more.
291
© 2006 Elsevier Inc. All rights reserved
ISSN 1049-250X
DOI 10.1016/S1049-250X(06)53010-1
292
V.V. Kocharovsky et al.
3.3. Grand Canonical Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4. Dynamical Master Equation Approach and Laser Phase-Transition Analogy . . . . . . .
4.1. Quantum Theory of the Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2. Laser Phase-Transition Analogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3. Derivation of the Condensate Master Equation . . . . . . . . . . . . . . . . . . . . .
4.4. Low Temperature Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5. Quasithermal Approximation for Noncondensate Occupations . . . . . . . . . . . .
4.6. Solution of the Condensate Master Equation . . . . . . . . . . . . . . . . . . . . . .
4.7. Results for BEC Statistics in Different Traps . . . . . . . . . . . . . . . . . . . . . .
4.8. Condensate Statistics in the Thermodynamic Limit . . . . . . . . . . . . . . . . . . .
4.9. Mesoscopic and Dynamical Effects in BEC . . . . . . . . . . . . . . . . . . . . . . .
5. Quasiparticle Approach and Maxwell’s Demon Ensemble . . . . . . . . . . . . . . . . . .
5.1. Canonical-Ensemble Quasiparticles in the Reduced Hilbert Space . . . . . . . . . .
5.2. Cumulants of BEC Fluctuations in an Ideal Bose Gas . . . . . . . . . . . . . . . . .
5.3. Ideal Gas BEC Statistics in Arbitrary Power-Law Traps . . . . . . . . . . . . . . . .
5.4. Equivalent Formulation in Terms of the Poles of the Generalized Zeta Function . . .
6. Why Condensate Fluctuations in the Interacting Bose Gas are Anomalously Large, NonGaussian, and Governed by Universal Infrared Singularities? . . . . . . . . . . . . . . . .
6.1. Canonical-Ensemble Quasiparticles in the Atom-Number-Conserving Bogoliubov
Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2. Characteristic Function and all Cumulants of BEC Fluctuations . . . . . . . . . . . .
6.3. Surprises: BEC Fluctuations are Anomalously Large and Non-Gaussian Even in the
Thermodynamic Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4. Crossover between Ideal and Interaction-Dominated BEC: Quasiparticles Squeezing
and Pair Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5. Universal Anomalies and Infrared Singularities of the Order Parameter Fluctuations
in the Systems with a Broken Continuous Symmetry . . . . . . . . . . . . . . . . . .
7. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9. Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A. Bose’s and Einstein’s Way of Counting Microstates . . . . . . . . . . . . . . . . . . . . .
B. Analytical Expression for the Mean Number of Condensed Atoms . . . . . . . . . . . . .
C. Formulas for the Central Moments of Condensate Fluctuations . . . . . . . . . . . . . . .
D. Analytical Expression for the Variance of Condensate Fluctuations . . . . . . . . . . . .
E. Single Mode Coupled to a Reservoir of Oscillators . . . . . . . . . . . . . . . . . . . . .
F. The Saddle-Point Method for Condensed Bose Gases . . . . . . . . . . . . . . . . . . . .
10. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Abstract
We review the phenomenon of equilibrium fluctuations in the number of condensed
atoms n0 in a trap containing N atoms total. We start with a history of the Bose–
Einstein distribution, a similar grand canonical problem with an indefinite total
number of particles, the Einstein–Uhlenbeck debate concerning the rounding of the
mean number of condensed atoms n¯ 0 near a critical temperature Tc , and a discussion of the relations between statistics of BEC fluctuations in the grand canonical,
canonical, and microcanonical ensembles.
First, we study BEC fluctuations in the ideal Bose gas in a trap and explain why
the grand canonical description goes very wrong for all moments (n0 − n¯ 0 )m ,
except of the mean value. We discuss different approaches capable of providing
1]
FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
293
approximate analytical results and physical insight into this very complicated problem. In particular, we describe at length the master equation and canonical-ensemble
quasiparticle approaches which give the most accurate and physically transparent
picture of the BEC fluctuations. The master equation approach, that perfectly describes even the mesoscopic effects due to the finite number N of the atoms in the
trap, is quite similar to the quantum theory of the laser. That is, we calculate a steadystate probability distribution of the number of condensed atoms pn0 (t = ∞) from
a dynamical master equation and thus get the moments of fluctuations. We present
analytical formulas for the moments of the ground-state occupation fluctuations in
the ideal Bose gas in the harmonic trap and arbitrary power-law traps.
In the last part of the review, we include particle interaction via a generalized Bogoliubov formalism and describe condensate fluctuations in the interacting Bose gas.
In particular, we show that the canonical-ensemble quasiparticle approach works
very well for the interacting gases and find analytical formulas for the characteristic
function and all cumulants, i.e., all moments, of the condensate fluctuations. The surprising conclusion is that in most cases the ground-state occupation fluctuations are
anomalously large and are not Gaussian even in the thermodynamic limit. We also
resolve the Giorgini, Pitaevskii and Stringari (GPS) vs. Idziaszek et al. debate on the
variance of the condensate fluctuations in the interacting gas in the thermodynamic
limit in favor of GPS. Furthermore, we clarify a crossover between the ideal-gas and
weakly-interacting-gas statistics which is governed by a pair-correlation, squeezing
mechanism and show how, with an increase of the interaction strength, the fluctuations can now be understood as being essentially 1/2 that of an ideal Bose gas.
We also explain the crucial fact that the condensate fluctuations are governed by a
singular contribution of the lowest energy quasiparticles. This is a sort of infrared
anomaly which is universal for constrained systems below the critical temperature
of a second-order phase transition.
1. Introduction
Professor Herbert Walther has taught us that good physics unifies and unites seemingly different fields. Nowhere is this more apparent than in the current studies of
Bose–Einstein condensation (BEC) and coherent atom optics which draw from
and contribute to the general subject of coherence effects in many-body physics
and quantum optics. It is in this spirit that the present paper presents the recent application of techniques, ideas, and theorems which have been developed
in understanding lasers and squeezed states to the condensation of N bosons.
Highlights of these studies, and related points of BEC history, are described in
the following paragraphs.
(1) Bose [1,2] got the ball rolling by deriving the Planck distribution without
using classical electrodynamics, as Planck [3] and Einstein [4] had done. Instead,
294
V.V. Kocharovsky et al.
[1
he took the extreme photon-as-a-particle point of view, and by regarding these
particles as indistinguishable obtained, among other things, Planck’s result,
1
,
(1)
−1
where n¯ k is the mean number of photons with energy εk and wavevector k, β =
(kB T )−1 , T is the blackbody temperature, and kB is Boltzmann’s constant.
However, his paper was rejected by the Philosophical Magazine and so he sent
it to Einstein, who recognized its value. Einstein translated it into German and got
it published in the Zeitschrift für Physik [1]. He then applied Bose’s method to
atoms and predicted that the atoms would “condense” into the lowest energy level
when the temperature was low enough [5–7].
Time has not dealt as kindly with Bose as did Einstein. As is often the case in
the opening of a new field, things were presented and understood imperfectly at
first. Indeed Bose did his “counting” of photon states in cells of phase space in an
unorthodox fashion. So much so that the famous Max Delbrück wrote an interesting article [8] in which he concluded that Bose made a mistake, and only got
the Planck distribution by serendipity. We here discuss this opinion, and retrace
the steps that led Bose to his result. Sure, he enjoyed a measure of luck, but his
mathematics and his derivation were correct.
(2) Einstein’s treatment of BEC of atoms in a large box showed a cusp in the
number of atoms in the ground state, n¯ 0 , as a function of temperature,
3/2 T
n¯ 0 = N 1 −
(2)
Tc
n¯ k =
eβεk
for T Tc , where N is the total number of atoms, and Tc is the (critical) transition
temperature.
Uhlenbeck [9] criticized this aspect of Einstein’s work, claiming that the cusp
at T = Tc is unphysical. Einstein agreed with the Uhlenbeck criticism but argued
that in the limit of large numbers of atoms (the thermodynamic limit) everything
would be okay. Later, Uhlenbeck and his student Kahn showed [10] that Einstein
was right and put the matter to rest (for a while).
Fast forward to the present era of mesoscopic BEC physics with only thousands (or even hundreds) of atoms in a condensate. What do we now do with this
Uhlenbeck dilemma? As one of us (W.K.) showed some time ago [11], all that is
needed is a better treatment of the problem. Einstein took the chemical potential
to be zero, which is correct for the ideal Bose gas in the thermodynamic limit.
However, when the chemical potential is treated more carefully, the cusp goes
away, as we discuss in detail, see, e.g., Figs. 1a and 3.
(3) So far everything we have been talking about concerns the average number
of particles in the condensate. Now we turn to the central focus of this review:
fluctuations in the condensate particle number. As the reader will recall, Einstein
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FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
295
used the fluctuation properties of waves and particles to great advantage. In particular he noted that in Planck’s problem, there were particle-like fluctuations in
photon number in addition to the wave-like contribution, i.e.,
n2k = n¯ 2k (wave) + n¯ k (particle),
(3)
and in this way he argued for a particle picture of light.
In his studies on Bose–Einstein condensation, he reversed the logic arguing that
the fluctuations in the ideal quantum gas also show both wave-like and particlelike attributes, just as in the case of photons. It is interesting that Einstein was led
to the wave nature of matter by studying fluctuations. We note that he knew of
and credited de Broglie at this point well before wave mechanics was developed.
Another important contribution to the problem of BEC fluctuations came from
Fritz London’s observation [12] that the specific heat is proportional to the variance of a Bose–Einstein condensate and showed a cusp, which he calculated as
being around 3.1 K. It is noteworthy that the so-called lambda point in liquid
Helium, marking the transition from normal to superfluid, takes place at around
2.19 K.
However, Ziff, Uhlenbeck and Kac [13] note several decades later that there is
a problem with the usual treatment of fluctuations. They say:
[When] the grand canonical properties for the ideal Bose gas are derived, it turns out
that some of them differ from the corresponding canonical properties—even in the bulk
limit! . . . The grand canonical ensemble . . . loses its validity for the ideal Bose gas in
the condensed region.
One of us (M.H.) has noted elsewhere [14] that:
This grand canonical fluctuation catastrophe has been discussed by generations of
physicists. . .
Let us sharpen the preceding remarks. Large fluctuations are a feature of the
thermal behavior of systems of bosons. If n¯ is the mean number of noninteracting
particles occupying a particular one particle state, then the mean square occupation fluctuation in the grand canonical picture is n(
¯ n¯ + 1). If, however, the system
has a fixed total number of particles N confined in space by a trapping potential, then at low enough temperature T when a significant fraction of N are in
the ground state, such large fluctuations are impossible. No matter how large N ,
the grand canonical description cannot be even approximately true. This seems
to be one of the most important examples that different statistical ensembles
give agreement or disagreement in different regimes of temperatures. To avoid
the catastrophe, the acclaimed statistical physicist D. ter Haar [15] proposed that
the fluctuations in the condensate particle number in the low temperature regime
(adapted to a harmonic trap) might go as
3
T
n0 ≡ (n0 − n¯ 0 )2 = N − n¯ 0 = N
(4)
.
Tc
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V.V. Kocharovsky et al.
[1
This had the correct zero limit as T → 0, but is not right for higher temperatures
where the leading term actually goes as [N ( TTc )3 ]1/2 . The point is that fluctuations
are subtle; even the ideal Bose gas is full of interesting physics in this regard.
In this paper, we resolve the grand canonical fluctuation catastrophe in several
ways. In particular, recent application of techniques developed in the quantum
theory of the laser [16,17] and in quantum optics [18,19] allow us to formulate
a consistent and physically appealing analytical picture of the condensate fluctuations in the ideal and interacting Bose gases. Our present understanding of the
statistics of the BEC fluctuations goes far beyond the results that were formulated
before the 90s BEC boom, as summarized by Ziff, Uhlenbeck and Kac in their
classical review [13]. Theoretical predictions for the BEC fluctuations, which are
anomalously large and non-Gaussian even in the thermodynamical limit, are derived and explained on the basis of the simple analytical expressions [20,21]. The
results are in excellent agreement with the exact numerical simulations. The existence of the infrared singularities in the moments of fluctuations and the universal
fact that these singularities are responsible for the anomalously large fluctuations
in BEC, are among the recent conceptual discoveries. The quantum theory of
laser threshold behavior constitutes another important advance in the physics of
bosonic systems.
(4) The laser made its appearance in the early 60s and provided us with a new
source of light with a new kind of photon statistics. Before the laser, the statistics
of radiation were either those of black-body photons associated with Planck’s
radiation, which for a single mode of frequency ν takes the form
¯ ,
pn = e−nβ h¯ ν 1 − e−β hν
(5)
or when one considers radiation from a coherent oscillating current such as a radio
transmitter or a microwave klystron the photon distribution becomes Poissonian,
n¯ n −n¯
e ,
(6)
n!
where n¯ is the average photon number.
However, laser photon statistics, as derived from the quantum theory of the
laser, goes from black-body statistics below threshold to Poissonian statistics far
above threshold. In between, when we are in the threshold region (and even above
threshold as in the case, for example, of the helium neon laser), we have a new
distribution. We present a review of the laser photon statistical problem.
It has been said that the Bose–Einstein condensate is to atoms what the laser
is to photons; even the concept of an atom laser has emerged. In such a case, one
naturally asks, “what is the statistical distribution of atoms in the condensate?”
For example, let us first address the issue of an ideal gas of N atoms in contact
with a reservoir at temperature T . The condensate occupation distribution in the
harmonic trap under these conditions at low enough temperatures is given by the
pn =
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FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
297
F IG . 1. (a) Mean value n0 and (b) variance n0 = n20 − n0 2 of the number of condensed
atoms as a function of temperature for N = 200 atoms in a harmonic trap calculated via the solution
of the condensate master equation (solid line). Large dots are the exact numerical results obtained
3
in the canonical ensemble. Dashed line for n0 is a plot of N [1 − (T /T
c ) ] which is valid in the
thermodynamic limit. Dashed line for n0 is the grand canonical answer n¯ 0 (n¯ 0 + 1) which gives
catastrophically large fluctuations below Tc .
BEC master equation analysis as
pn0 =
1 [N (T /Tc )3 ]N−n0
.
ZN
(N − n0 )!
(7)
The mean number and variance obtained from the condensate master equation
are in excellent agreement with computer simulation (computer experiment) as
shown in Fig. 1. We will discuss this aspect of the fluctuation problem in some
detail and indicate how the fluctuations change when we go to the case of the
interacting Bose gas.
(5) The fascinating interface between superfluid He II and BEC in a dilute gas
was mapped out by the experiments of Reppy and coworkers [22]; and finite-size
effects were studied theoretically by M. Fisher and coworkers [23]. They carried
out experiments in which He II was placed in a porous glass medium which serves
to keep the atoms well separated. These experiments are characterized by a dilute
gas BEC of N atoms at temperature T .
Of course, it was the successful experimental demonstration of Bose–Einstein
condensation in the ultracold atomic alkali–metal [24–26], hydrogen [27] and
helium gases [22,28,29] that stimulated the renaissance in the theory of BEC.
In less than a decade, many intriguing problems in the physics of BEC, that were
not studied, or understood before the 90s [30–36], were formulated and resolved.
(6) Finally, we turn on the interaction between atoms in the BEC and find
explicit expressions for the characteristic function and all cumulants of the probability distribution of the number of atoms in the (bare) ground state of a trap
for the weakly interacting dilute Bose gas in equilibrium. The surprising result is
298
V.V. Kocharovsky et al.
[2
that the BEC statistics is not Gaussian, i.e., the ratio of higher cumulants to an
appropriate power of the variance does not vanish, even in the thermodynamic
limit. We calculate explicitly the effect of Bogoliubov coupling between excited
atoms on the suppression of the BEC fluctuations in a box (“homogeneous gas”)
at moderate temperatures and their enhancement at very low temperatures. We
find that there is a strong pair-correlation effect in the occupation of the coupled
atomic modes with the opposite wavevectors k and −k. This explains why the
ground-state occupation fluctuations remain anomalously large to the same extent as in the noninteracting gas, except for a factor of 1/2 suppression. We find
that, roughly speaking, this is so because the atoms are strongly coupled in correlated pairs such that the number of independent stochastic occupation variables
(“degrees of freedom”) contributing to the fluctuations of the total number of excited atoms is only 1/2 the atom number N . This is a particular feature of the
well-studied quantum optics phenomenon of two-mode squeezing (see, e.g., [37]
and [18,19]). The squeezing is due to the quantum correlations that build up in
the bare excited modes via Bogoliubov coupling and is very similar to the noise
squeezing in a nondegenerate parametric amplifier.
Throughout the review, we will check main approximate analytical results
(such as in Eqs. (162), (172), (223), (263), (271)) against the “exact” numerics
based on the recursion relations (79) and (80) which take into account exactly all
mesoscopic effects near the critical temperature Tc . Unfortunately, the recursion
relations are known only for the ideal Bose gas. In the present review we discuss
the BEC fluctuations mainly in the canonical ensemble, which cures misleading
predictions of the grand canonical ensemble and, at the same time, does not have
any essential differences with the microcanonical ensemble for most physically
interesting quantities and situations. Moreover, as we discuss below, the canonical
partition function can be used for an accurate calculation of the microcanonical
partition function via the saddle-point method.
2. History of the Bose–Einstein Distribution
In late 1923, a certain Satyendranath Bose, reader in physics at the University
of Dacca in East Bengal, submitted a paper on Planck’s law of blackbody radiation to the Philosophical Magazine. Six months later he was informed that the
paper had received a negative referee report, and consequently been rejected [38].
While present authors may find consolation in the thought that the rejection
of a truly groundbreaking paper after an irresponsibly long refereeing process
is not an invention of our times, few of their mistreated works will eventually
meet with a recognition comparable to Bose’s. Not without a palpable amount of
self-confidence, Bose sent the rejected manuscript to Albert Einstein in Berlin,
together with a handwritten cover letter dated June 4, 1924, beginning [39]:
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FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
299
Respected Sir:
I have ventured to send you the accompanying article for your perusal and opinion.
I am anxious to know what you think of it. You will see that I have tried to deduce
the coefficient 8π ν 2 /c3 in Planck’s Law independent of the classical electrodynamics,
only assuming that the ultimate elementary regions in the phase-space has the content h3 . I do not know sufficient German to translate the paper. If you think the paper
worth publication I shall be grateful if you arrange its publication in Zeitschrift für
Physik. Though a complete stranger to you, I do not hesitate in making such a request.
Because we are all your pupils though profiting only from your teachings through your
writings. . .
In hindsight, it appears curious that Bose drew Einstein’s attention only to his
derivation of the prefactor in Planck’s law. Wasn’t he aware of the fact that his
truly singular achievement, an insight not even spelled out explicitly in Einstein’s
translation of his paper as it was received by the Zeitschrift für Physik on July 2,
1924 [1,2], but contained implicitly in the mathematics, lay elsewhere?
2.1. W HAT B OSE D ID
In the opening paragraph of his paper [1,2], Bose pounces on an issue which he
considers unsatisfactory: When calculating the energy distribution of blackbody
radiation according to
ν dν =
8πν 2 dν
Eν ,
c3
(8)
that is,
energy per volume of blackbody radiation with frequency between ν
and ν + dν
= number of modes contained in that frequency interval
of the radiation field per volume
× thermal energy Eν of a radiation mode with frequency ν,
the number of modes had previously been derived only with reference to classical
physics. In his opinion, the logical foundation of such a recourse was not sufficiently secure, and he proposed an alternative derivation, based on the hypothesis
of light quanta.
Considering radiation inside some cavity with volume V , he observed that the
squared momentum of such a light quantum is related to its frequency through
h2 ν 2
,
(9)
c2
where h denotes Planck’s constant, and c is the velocity of light. Dividing the
frequency axis into intervals of length dν s , such that the entire axis is covered
p2 =
300
V.V. Kocharovsky et al.
[2
when the label s varies from s = 0 to s = ∞, the phase space volume associated
with frequencies between ν and ν + dν s therefore is
2
hν h dν s
h3 ν 2
dx dy dz dpx dpy dpz = V 4π
(10)
= V 4π 3 dν s .
c
c
c
It does not seem to have bothered Bose that the concept of phase space again
brings classical mechanics into play. Relying on the assumption that a single
quantum state occupies a cell of volume h3 in phase space, a notion which, in
the wake of the Bohr–Sommerfeld quantization rule, may have appeared natural
to a physicist in the early 1920s, and accounting for the two states of polarization, the total number As of quantum cells belonging to frequencies between ν
and ν + dν s , corresponding to the number of radiation modes in that frequency
interval, immediately follows:
8πν 2 s
(11)
dν .
c3
That’s all, as far as the first factor on the r.h.s. of Eq. (8) is concerned. This is
what Bose announced in his letter to Einstein, but this is, most emphatically, not
his main contribution towards the understanding of Planck’s law. The few lines
which granted him immortality follow when he turns to the second factor. Backtranslated from Einstein’s phrasing of his words [1,2]:
As = V
Now it is a simple task to calculate the thermodynamic probability of a (macroscopically defined) state. Let N s be the number of quanta belonging to the frequency
interval dν s . How many ways are there to distribute them over the cells belonging
to dν s ? Let p0s be the number of vacant cells, p1s the number of those containing one
quantum, p2s the number of cells which contain two quanta, and so on. The number of
possible distributions then is
As !
,
p0s !p1s ! . . .
where As = V
8π ν 2 s
dν ,
c3
(12)
and where
N s = 0 · p0s + 1 · p1s + 2 · p2s . . .
is the number of quanta belonging to dν s .
What is happening here? Bose is resorting to a fundamental principle of statistical mechanics, according to which the probability of observing a state with certain
macroscopic properties—in short: a macrostate—is proportional to the number of
its microscopic realizations—microstates—compatible with the macroscopically
given restrictions. Let us, for example, consider a model phase space consisting
of four cells only, and let there be four quanta. Let us then specify the macrostate
by requiring that one cell remain empty, two cells contain one quantum each, and
one cell be doubly occupied, i.e., p0 = 1, p1 = 2, p2 = 1, and pr = 0 for r 3.
2]
FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
301
How many microstates are compatible with this specification? We may place two
quanta in one out of four cells, and then choose one out of the remaining three
cells to be the empty one. After that there is no further choice left, since each of
the two other cells now has to host one quantum. Hence, there are 4 × 3 = 12
possible configurations, or microstates: 12 = 4!/(1! 2! 1!). In general, when there
are As cells belonging to dν s , they can be arranged in As ! ways. However, if a
cell pattern is obtained from another one merely by a rearrangement of those p0s
cells containing no quantum, the configuration remains unchanged. Obviously,
there are p0s ! such “neutral” rearrangements which all correspond to the same
configuration. The same argument then applies, for any r 1, to those prs cells
containing r quanta: Each of the prs ! possibilities of arranging the cells with r
quanta leads to the same configuration. Thus, each configuration is realized by
p0s !p1s ! . . . equivalent arrangements of cells, and the number of different configurations, or microstates, is given by the total number of arrangements divided by
the number of equivalent arrangements, that is, by Bose’s expression (12).
There is one proposition tacitly made in this way of counting microstates which
might even appear self-evident, but which actually constitutes the very core of
Bose’s breakthrough, and which deserves to be spelled out explicitly: When considering equivalent arrangements as representatives of merely one microstate, it is
implied that the quanta are indistinguishable. It does not matter “which quantum
occupies which cell”; all that matters are the occupation numbers prs . Even more,
the “which quantum”-question is rendered meaningless, since there is, as a matter
of principle, no way of attaching some sort of label to individual quanta belonging
to the same dν s , with the purpose of distinguishing them. This “indistinguishability in principle” does not occur in classical physics. Two classical particles may
have the same mass, and identical other properties, but it is nevertheless taken for
granted that one can tell one from the other. Not so, according to Bose, with light
quanta.
The rest of Bose’s paper has become a standard exercise in statistical physics.
Taking into account all frequency intervals dν s , the total number of microstates
corresponding to a pre-specified set {prs } of cell occupation numbers is
W prs =
s
As !
.
p0s ! p1s ! . . .
(13)
The logarithm of this functional yields the entropy associated with the considered
set {prs }. Since, according to the definition of prs ,
As =
(14)
prs for each s,
r
and assuming the statistically relevant prs to be large, Stirling’s approximation
ln n! ≈ n ln n − n gives
302
V.V. Kocharovsky et al.
s
A ln As −
prs ln prs .
ln W prs =
s
s
[2
(15)
r
The most probable macrostate now is the one with the maximum number of microstates, characterized by that set of occupation numbers which maximizes this
expression (15). Stipulating that the radiation field be thermally isolated, so that
its total energy
E=
(16)
N s hν s with N s =
rprs
s
r
is fixed, the maximum is found by variation of the prs , subject to this constraint (16). In addition, the constraints (14) have to be respected. Introducing
Lagrangian multipliers λs for these “number-of-cells” constraints, and a further
Lagrangian multiplier β for the energy constraint, the maximum is singled out by
the condition
s s
δ ln W prs −
(17)
λ
pr − β
hν s
rprs = 0,
s
giving
r
s
r
δprs ln prs + 1 + λs + β
hν s
rδprs = 0.
r,s
s
(18)
r
Since the δprs can now be taken as independent, the maximizing configuration
{pˆ rs } obeys
ln pˆ rs + 1 + λs + rβhν s = 0,
(19)
pˆ rs = B s e−rβhν ,
(20)
or
s
Bs
to be determined from the constraints (14):
with normalization constants
s
B
As =
(21)
pˆ rs =
s .
1
−
e−βhν
r
The total number of quanta for the maximizing configuration then is
As
s
s
r pˆ rs = As 1 − e−βhν
re−rβhν = βhν s
.
Nˆ s =
e
−1
r
r
(22)
Still, the physical meaning of the Lagrangian multiplier β has to be established.
This can be done with the help of the entropy functional, since inserting the maximizing configuration yields the thermodynamical equilibrium entropy:
s
As ln 1 − e−βhν ,
S = kB ln W pˆ rs = kB βE −
(23)
s
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FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
303
where kB denotes Boltzmann’s constant. From the identity ∂S/∂E = 1/T one
then finds β = 1/(kB T ), the inverse energy equivalent of the temperature T .
Hence, from Eqs. (22) and (11) Bose obtains the total energy of the radiation
contained in the volume V in the form
E=
N s hν s =
s
8πhν s 3
s
c3
V
1
s
exp( khν
− 1)
BT
dν s ,
(24)
which is equivalent to Planck’s formula: With the indistinguishability of quanta,
i.e., Bose’s enumeration (12) of microstates as key input, the principles of statistical mechanics immediately yield the thermodynamic properties of radiation.
2.2. W HAT E INSTEIN D ID
Unlike that unfortunate referee of the Philosophical Magazine, Einstein immediately realized the power of Bose’s approach. Estimating that it took the manuscript
three weeks to travel from Dacca to Berlin, Einstein may have received it around
June 25 [8]. Only one week later, on July 2, his translation of the manuscript was
officially received by the Zeitschrift für Physik. The author’s name was lacking its
initials—the byline of the published paper [1] simply reads: By Bose (Dacca University, India)—but otherwise Einstein was doing Bose fair justice: He even sent
Bose a handwritten postcard stating that he regarded his paper as a most important contribution; that postcard seems to have impressed the German Consulate in
Calcutta to the extent that Bose’s visa was issued without requiring payment of
the customary fee [38].
Within just a few days, Einstein then took a further step towards exploring the
implications of the “indistinguishability in principle” of quantum mechanical entities. At the end of the printed, German version of Bose’s paper [1], there appears
the parenthetical remark “Translated by A. Einstein”, followed by an announcement:
Note added by the translator: Bose’s derivation of Planck’s formula constitutes, in my
opinion, an important step forward. The method used here also yields the quantum
theory of the ideal gas, as I will explain in detail elsewhere.
“Elsewhere” in this case meant the Proceedings of the Prussian Academy of
Sciences. In the session of the Academy on July 10, Einstein delivered a paper
entitled “Quantum theory of the monoatomic ideal gas” [5]. In that paper, he
considered nonrelativistic free particles of mass m, so that the energy-momentum
relation simply reads
E=
p2
,
2m
(25)
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V.V. Kocharovsky et al.
[2
and the phase-space volume for a particle with an energy not exceeding E is
4π
(26)
(2mE)3/2 .
3
Again relying on the notion that a single quantum state occupies a cell of volume
h3 in phase space, the number of such cells belonging to the energy interval from
E to E + E is
Φ=V
2πV
(27)
(2m)3/2 E 1/2 E.
h3
Thus, for particles with nonzero rest mass s is the analog of Bose’s As introduced in Eq. (11). Einstein then specified the cell occupation numbers by
requiring that, out of these s cells, prs s cells contain r particles, so that prs
is the probability of finding r particles in any one of these cells,
prs = 1.
(28)
s =
r
Now comes the decisive step. Without attempt of justification or even comment,
Einstein adopts Bose’s way (12) of counting the number of corresponding microstates. This is a far-reaching hypothesis, which implies that, unlike classical
particles, atoms of the same species with energies in the same range E are indistinguishable: Interchanging two such atoms does not yield a new microstate; as
with photons, it does not matter “which atom occupies which cell”. Consequently,
the number of microstates associated with a pre-specified set of occupation probabilities {prs } for the above s cells is
s!
,
s
r=0 (pr s)!
W s = &∞
(29)
giving, with the help of Stirling’s formula,
ln W s = −s
prs ln prs .
(30)
r
Einstein then casts this result into a more attractive form. Stipulating that the
index s does no longer refer jointly to the cells within a certain energy interval,
but rather labels individual cells, the above expression naturally generalizes to
prs ln prs ,
ln W prs = −
(31)
s
r
where the cell index s now runs over all cells, so that prs here is the probability
of finding r particles in the sth cell. It is interesting to observe that this functional (31) has precisely the same form as the Shannon entropy introduced in 1948
in an information-theoretical context [40].
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FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
305
Since r rprs gives the expectation value of the number of particles occupying
the cell labelled s, the total number of particles is
N=
(32)
rprs ,
s
r
while the total energy of the gas reads
E=
Es
rprs ,
s
(33)
r
where E s is the energy of a particle in the sth cell. Since, according to Eq. (26),
a cell’s number s is related to the energy E s through
s=
3/2
V 4π Φs
= 3
,
2mE s
3
h
h 3
(34)
one has
with
E s = cs 2/3
(35)
h2 4πV −2/3
c=
.
2m
3
(36)
Considering an isolated system, with given, fixed particle number N and fixed
energy E, the macrostate realized in nature is characterized by that set {pˆ rs } which
maximizes the entropy functional (31), subject to the constraints (32) and (33),
together with the constraints (28) expressing normalization of the cell occupation
probabilities. Hence,
s s
δ ln W prs −
λ
pr − α
rprs − β
Es
rprs
s
r
= 0,
s
r
s
r
(37)
so that, seen from the conceptual viewpoint, the only difference between Bose’s
variational problem (17) and Einstein’s variational problem (37) is the appearance
of an additional Lagrangian multiplier α in the latter: In the case of radiation, the
total number of light quanta adjusts itself in thermal equilibrium, instead of being
fixed beforehand; in the case of a gas of particles with nonzero rest mass, the
total number of particles is conserved, requiring the introduction of the entailing
multiplier α. One then finds
ln pˆ rs + 1 + λs + αr + βrE s = 0
(38)
pˆ rs = B s e−r(α+βE ) ,
(39)
or
s
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[2
with normalization constants to be determined from the constraints (28):
B s = 1 − e−(α+βE ) .
s
(40)
Here we deviate from the notation in Einstein’s paper [5], in order to be compatible with modern conventions. The expectation value for the occupation number
of the cell with energy E s then follows from an elementary calculation similar to
Bose’s reasoning (22):
1
rprs = α+βE s
.
(41)
e
−1
r
Therefore, the total number of particles and the total energy of the gas can be
expressed as
1
N=
(42)
,
s
α+βE
e
−1
s
Es
E=
(43)
.
s
eα+βE − 1
s
Inserting the maximizing set (39) into the functional (31) yields, after a brief
calculation, the equilibrium entropy of the gas in the form
s −(α+βE s )
ln 1 − e
.
S = kB ln W pˆ r = kB αN + βE −
(44)
s
In order to identify the Lagrangian multiplier β, Einstein considered an infinitesimal heating of the system, assuming its volume and, hence, the cell energies E s
to remain fixed. This gives
d(α + βE s ) dE = T dS = kB T N dα + β dE + E dβ −
s
eα+βE − 1
s
= kB T β dE,
(45)
requiring
1
(46)
.
kB T
As in Bose’s case, the Lagrangian multiplier β accounting for the energy constraint is the inverse energy equivalent of the temperature T . The other multiplier α, guaranteeing particle number conservation, then is determined from the
identity (42).
In the following two sections of his paper [5], Einstein shows how the thermodynamics of the classical ideal gas is recovered if one neglects unity against
s
eα+βE , and derives the virial expansion of the equation of state for the quantum
gas obeying Eqs. (42) and (43).
β=
2]
FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
307
2.3. WAS B OSE –E INSTEIN S TATISTICS A RRIVED AT BY S ERENDIPITY ?
The title of this subsection is a literal quote from the title of a paper by M. Delbrück [8], who contents that Bose made an elementary mistake in statistics in
that he should have bothered “which quantum occupies which cell”, which would
have been the natural approach, and that Einstein first copied that mistake without
paying much attention to it. Indeed, such a suspicion does not seem to be entirely unfounded. In his letter to Einstein, Bose announces only his comparatively
straightforward derivation of the number (11) of radiation modes falling into the
frequency range from ν to ν + dν s , apparently being unaware that his revolutionary deed was the implicit exploitation of the “indistinguishability in principle” of
quanta—a concept so far unheard of. In Einstein’s translation of his paper [1] this
notion of indistinguishability does not appear in words, although it is what underlies the breakthrough. Even more, it does not appear in the first paper [5] on the
ideal Bose gas—until the very last paragraph, where Einstein ponders over
. . .a paradox which I have been unable to resolve. There is no difficulty in treating
also the case of a mixture of two different gases by the method explained here. In this
case, each molecular species has its own “cells”. From this follows the additivity of the
entropies of the mixture’s components. Therefore, with respect to molecular energy,
pressure, and statistical distribution each component behaves as if it were the only one
present. A mixture containing n1 and n2 molecules, with the molecules of the first
kind being distinguishable (in particular with respect to the molecular masses m1 , m2 )
only by an arbitrarily small amount from that of the second, therefore yields, at a given
temperature, a pressure and a distribution of states which differs from that of a uniform
gas with n1 + n2 molecules with practically the same molecular mass, occupying the
same volume. However, this appears to be as good as impossible.
Interestingly, Einstein here considers “distinguishability to some variable degree”, which can be continuously reduced to indistinguishability. But this notion
is flawed: Either the molecules have some feature which allows us to tell one
species from the other, in which case the different species can be distinguished, or
they have none at all, in which case they are indistinguishable in principle. Thus,
at this point, about two weeks after the receipt of Bose’s manuscript and one week
after sending its translation to the Zeitschrift für Physik, even Einstein may not yet
have fully grasped the implications of Bose’s way (12) of counting microstates.
But there was more to come. In December 1924, Einstein submitted a second
manuscript on the quantum theory of the ideal Bose gas [6,7], formally written
as a continuation of the first one. He began that second paper by pointing out a
curiosity implied by his equation of state of the ideal quantum gas: Given a certain number of particles N and a temperature T , and considering a compression
of the volume V , there is a certain volume below which a segregation sets in.
With decreasing volume, an increasing number of particles has to occupy the first
quantum cell, i.e., the state without kinetic energy, while the rest is distributed
over the other cells according to Eq. (41), with eα = 1. Thus, Bose–Einstein condensation was unveiled! But this discovery merely appears as a small addendum
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V.V. Kocharovsky et al.
[2
to the previous paper [5], for Einstein then takes up a different, more fundamental scent. He mentioned that Ehrenfest and other colleagues of his had criticized
that in Bose’s and his own theory the quanta or particles had not been treated as
statistically independent entities, a fact which had not been properly emphasized.
Einstein agrees, and then he sets out to put things straight. He abandons his previous “single-cell” approach and again considers the collection of quantum cells
with energies between Eν and Eν + Eν , the number of which is
zν =
2πV
(2m)3/2 Eν1/2 Eν .
h3
(47)
Then he juxtaposes in detail Bose’s way of counting microstates to what is done
in classical statistics. Assuming that there are nν quantum particles falling into
Eν , Bose’s approach (12) implies that there are
Wν =
(nν + zν − 1)!
nν !(zν − 1)!
(48)
possibilities of distributing the particles over the cells. This expression can easily
be visualized: Drawing the nν particles as a sequence of nν “dots” in a row, they
can be organized into a microstate with specific occupation numbers for zν cells—
again assuming that it does not matter which particle occupies which cell—by
inserting zν − 1 separating “lines” between them. Thus, there are nν + zν − 1
positions carrying a symbol, nν of which are dots. The total number of microstates
then equals the total number of possibilities to select the nν positions carrying a
“dot” out of these nν + zν − 1 positions, which is just the binomial coefficient
stated in Eq. (48). To give an example: Assuming that there are nν = 4 particles
and zν = 4 cells, Eq. (48) states that there are altogether
(4 + 4 − 1)!
7!
7·6·5
=
=
= 35
4! 3!
4! 3!
2·3
microstates. On the other hand, there are several sets of occupation numbers which
allow one to distribute the particles over the cells:
Occupation numbers
Number of microstates
p4 = 1, p0 = 3
4!
1! 3! = 4
4!
1! 1! 2! = 12
4!
2! 2! = 6
4!
1! 2! 1! = 12
4! = 1
4!
p3 = 1, p1 = 1, p0 = 2
p2 = 2, p0 = 2
p2 = 1, p1 = 2, p0 = 1
p1 = 4
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FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
309
The right column of this table gives, for each set, the number of microstates
according to Bose’s formula (12); obviously, these numbers add up to the total
number 35 anticipated above. Thus, the binomial coefficient (48) conveniently
accounts for all possible microstates, without the need to specify the occupation
numbers, according to the combinatorial identity
Z!
N +Z−1
(49)
=
,
N
p0 ! . . . pN !
where the sum is restricted
to those sets
{p0 , p1 , . . . , pN } which comply with
the two conditions r pr = Z and r rpr = N , as in the example above.
In Appendix A we provide a proof of this identity. With this background, let us
return to Einstein’s reasoning: When taking into account all energy
& intervals Eν ,
the total number of microstates is given by the product W = ν Wν , providing
the entropy functional
(nν + zν ) ln(nν + zν ) − nν ln nν − zν ln zν
ln W {nν } =
ν
=
ν
zν
nν
+1 .
nν ln 1 +
+ zν ln
nν
zν
The maximizing set {nˆ ν } now has to obey the two constraints
nν = N,
(50)
(51)
ν
nν Eν = E,
(52)
ν
but there is no more need for the multipliers λs appearing in the previous Eqs. (17)
and (37), since the constraints (14) or (28) are automatically respected when starting from the convenient expression (48). Hence, one has
δ ln W {nν } − α
(53)
nν − β
nν E ν = 0
ν
or
ν
zν
ln 1 +
− α − βEν δnν = 0,
nν
ν
(54)
leading immediately to
zν
,
nˆ ν = α+βE
(55)
ν −1
e
in agreement with the previous result (41).
But what, Einstein asks, would have resulted had one not adopted Bose’s prescription (12) and thus counted equivalent arrangements with equal population
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[2
numbers only once, but rather had treated the particles as classical, statistically
independent entities? Then there obviously are
Wν = (zν )nν
(56)
possibilities of distributing the nν particles belonging to Eν over the zν cells:
Each particle simply is placed in one of the zν cells, regardless of the others.
Now, when considering all intervals Eν , with distinguishable particles it does
Eν are selected from
matter how those nν particles going into the respective
&
all N particles; for this selection, there are N !/ ν nν ! possibilities. Thus, taking
classical statistics seriously, there are
W = N!
ν
(zν )nν
nν !
(57)
possible microstates, yielding
[nν ln zν − nν ln nν + nν ]
ln W {nν } = N ln N − N +
= N ln N +
ν
ν
zν
nν ln
nν
+ nν .
(58)
This is a truly vexing expression, since it gives a thermodynamical entropy which
is not proportional to the total number of particles, i.e., no extensive quantity, because of the first term on the r.h.s. Hence, already in the days before Bose and
Einstein one had got used to ignoring the leading factor N ! in Eq. (57), with
the half-hearted concession that microstates which result from each other by a
mere permutation of the N particles should not be counted as different. Of course,
this is an intrinsic inconsistency of the classical theory: Instead of accepting that,
shouldn’t one abandon Eq. (57) straight away and accept the more systematic
quantum approach, despite the apparently strange consequence of losing the particles’ independence? And Einstein gives a further, strong argument in favor of
the quantum theory: At zero temperature, all particles occupy the lowest cell, giving n1 = N and nν = 0 for ν > 1. With z1 = 1, the quantum way of counting
based on Eq. (48) gives just one single microstate, which means zero entropy in
agreement with Nernst’s theorem, whereas the classical expression (57) yields an
incorrect entropy even if one ignores the disturbing N ! Finally, the variational
calculation based on the classical functional (58) proceeds via
zν ln
− α − βEν δnν = 0,
(59)
nν
ν
furnishing the Boltzmann-like distribution
nˆ ν = zν e−α−βEν
(60)
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FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
311
for the maximizing set {nˆ ν }. In short, the quantum ideal gas of nonzero mass particles with its distribution (55) deviates from the classical ideal gas in the same
manner as does Planck’s law of radiation from Wien’s law. This observation convinced Einstein, even in the lack of any clear experimental evidence, that Bose’s
way of counting microstates had to be taken seriously, since, as he remarks in the
introduction to his second paper [6], “if it is justified to consider radiation as a
quantum gas, the analogy between the quantum gas and the particle gas has to be
a complete one”. This belief also enabled him to accept the sacrifice of the statistical independence of quantum particles implied by the formula (48), which, by
the end of 1924, he had clearly realized:
The formula therefore indirectly expresses a certain hypothesis about a mutual influence of the molecules on each other which is of an entirely mysterious kind. . .
But what might be the physics behind that mysterious influence which noninteracting particles appear to exert on each other? In a further section of his paper [6],
Einstein’s reasoning takes an amazing direction: He considers the density fluctuations of the ideal quantum gas, and from this deduces the necessity to invoke wave
mechanics! Whereas he had previously employed what is nowadays known as the
microcanonical ensemble, formally embodied through the constraints that the total number of particles and the total energy be fixed, he now resorts to a grand
canonical framework and considers a gas within some finite volume V which
communicates with a gas of the same species contained in an infinitely large volume. He then stipulates that both volumes be separated from each other by some
kind of membrane which can be penetrated only by particles with an energy in a
certain infinitesimal range Eν , and quantifies the ensuing fluctuation nν of the
number of particles in V , not admitting energy exchange between particles in different energy intervals. Writing nν = nˆ ν + nν , the entropy of the gas within V
is expanded in the form
Sgas (nν ) = Sˆgas +
∂ Sˆgas
1 ∂ 2 Sˆgas
nν +
(nν )2 ,
∂nν
2 ∂(nν )2
(61)
whereas the entropy of the reservoir changes with the transferred particles according to
∂ Sˆ0
S0 (nν ) = Sˆ0 −
nν .
∂nν
(62)
In view of the assumed infinite size of the reservoir, the quadratic term is negligible here. Since the equilibrium state is characterized by the requirement that the
total entropy S = Sgas + S0 be maximum, one has
∂ Sˆ
= 0,
∂nν
(63)
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V.V. Kocharovsky et al.
[2
so that
1 ∂ 2 Sˆgas
(nν )2 .
S(nν ) = Sˆ +
2 ∂(nν )2
(64)
Hence, the probability distribution for finding a certain fluctuation nν is
Gaussian,
P (nν ) = const · eS(nν )/kB
1 ∂ 2 Sˆgas
2
= const · exp
(n
)
,
ν
2kB ∂(nν )2
(65)
from which one reads off the mean square of the fluctuations,
(nν )2 =
kB
∂ 2 Sˆ
− ∂(ngas)2
ν
.
(66)
Since, according to the previous Eq. (54), one has
1 ∂ Sˆgas
zν
= ln 1 +
,
kB ∂nν
nˆ ν
(67)
one deduces
−zν
1 ∂ 2 Sˆgas
= 2
,
2
kB ∂(nν )
nˆ ν + zν nˆ ν
(68)
resulting in
nˆ 2
(nν )2 = nˆ ν + ν
zν
(69)
or
1
1
(nν /nν )2 =
+ .
nˆ ν
zν
(70)
With zν = 1, this gives the familiar grand canonical expression for the fluctuation
of the occupation number of a single quantum state. With a stroke of genius, Einstein now interprets this formula: Whereas the first term on the r.h.s. of Eq. (70)
would also be present if the particles were statistically independent, the second
term reminds him of interference fluctuations of a radiation field [6]:
One can interpret it even in the case of a gas in a corresponding manner, by associating
with the gas a radiation process in a suitable manner, and calculating its interference
fluctuations. I will explain this in more detail, since I believe that this is more than a
formal analogy.
2]
FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
313
He then refers to de Broglie’s idea of associating a wavelike process with single
material particles and argues that, if one associates a scalar wave field with a
gas of quantum particles, the term 1/zν in Eq. (70) describes the corresponding
mean square fluctuation of the wave field. What an imagination—on the basis of
the fluctuation formula (70) Einstein anticipates many-body matter waves, long
before wave mechanics was officially enthroned! Indeed, it was this paper of his
which led to a decisive turn of events: From these speculations on the relevance
of matter waves Schrödinger learned about de Broglie’s thesis, acquired a copy of
it, and then formulated his wave mechanics.
Having recapitulated this history, let us once again turn to the title of this
subsection: Was Bose–Einstein statistics arrived at by serendipity? Delbrück’s
opinion that it arose out of an elementary mistake in statistics that Bose made
almost certainly is too harsh. On the other hand, the important relation (49), see
also Appendix A, does not seem to have figured in Bose’s thinking. When writing down the crucial expression (12), Bose definitely must have been aware that
he was counting the number of microstates by determining the number of different distributions of quanta over the available quantum cells, regardless of “which
quantum occupies which cell”. He may well have been fully aware that his way
of counting implied the indistinguishability of quanta occupying the same energy
range, but he did not reflect on this curious issue. On the other hand, he didn’t have
to, since his way of counting directly led to one of the most important formulas in
physics, and therefore simply had to be correct.
Many years later, Bose recalled [41]:
I had no idea that what I had done was really novel. . . I was not a statistician to the
extent of really knowing that I was doing something which was really different from
what Boltzmann would have done, from Boltzmann statistics. Instead of thinking of
the light-quantum just as a particle, I talked about these states.
By counting the number of ways to fill a number of photonic states (cells)
Bose obtained Eq. (13) which is exactly the same form as Boltzmann’s Eq. (57)
for zν = 1, but with new meanings attached to the new symbols: ns replaced
by prs and N replaced by As . Bose’s formula leads to an entirely different new
statistics—the quantum statistics for indistinguishable particles, in contrast to
Boltzmann’s distinguishable particles. It took a while for this to sink in even with
Einstein, but that is the nature of research.
Contrary to Bose, Einstein had no experimental motivation when adapting
Bose’s work to particles with nonzero rest mass. He seems to have been guided
by a deep-rooted belief in the essential simplicity of physics, so that he was quite
ready to accept a complete analogy between the gas of light quanta and the ideal
gas of quantum particles, although he may not yet have seen the revolutionary implications of this concept when he submitted his first paper [5] on this matter. But
the arrival at a deep truth on the basis of a well-reflected conviction can hardly
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V.V. Kocharovsky et al.
[2
be called serendipity. His second paper [6] is, by all means, a singular intellectual achievement, combining daring intuition with almost prophetical insight. And
who would blame Einstein for trying to apply, in another section of that second
paper, his quantum theory of the ideal Bose gas to the electron gas in metals?
In view of the outstanding importance which Einstein’s fluctuation formula (70)
has had for the becoming of wave mechanics, it appears remarkable that a puzzling question has long remained unanswered: What happens if one faces, unlike
Einstein in his derivation of this relation (70), a closed system of Bose particles
which does not communicate with some sort of particle reservoir? When the temperature T approaches zero, all N particles are forced into the system’s ground
state, so that the mean square (n0 )2 of the fluctuation of the ground-state occupation number has to vanish for T → 0—but with z0 = 1, the grand canonical
Eq. (69) gives (n0 )2 → N (N +1), clearly indicating that with respect to these
fluctuations the different statistical ensembles are no longer equivalent. What,
then, would be the correct expression for the fluctuation of the ground-state occupation number within the canonical ensemble, which excludes any exchange
of particles with the environment, but still allows for the exchange of energy?
To what extent does the microcanonical ensemble, which applies when even the
energy is kept constant, differ from the canonical one? Various aspects of this
riddle have appeared in the literature over the years [13,15,42], mainly inspired
by academic curiosity, before it resurfaced in 1996 [43–46], this time triggered by
the experimental realization of mesoscopic Bose–Einstein condensates in isolated
microtraps. Since then, much insight into this surprisingly rich problem has been
gained, and some of the answers to the above questions have been given by now.
In the following sections of this article, these new developments will be reviewed
in detail.
2.4. C OMPARISON BETWEEN B OSE ’ S AND E INSTEIN ’ S C OUNTING
OF THE N UMBER OF M ICROSTATES W
In Bose’s original counting (12), he considered the numbers prs of cellsoccupied with r photons, so that the total number of cells is given by As = r prs .
While Einstein still had adopted this way of counting in his first paper [5] on
the ideal Bose gas, as witnessed by his Eq. (29), he used, without comment,
more economical means in his second paper [6], relying on the binomial coefficient (48). Figure 2 shows an example in which N s = 2 particles are distributed
over As = 3 cells, and visualizes that Bose’s formula for counting the number of possible arrangements (or the number of microstates which give the same
macrostate) gives the same result as Einstein’s formula only after one has summed
over all possible configurations, i.e., over those sets p0s , p1s , . . . which simultane-
3]
FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
315
F IG . 2. A simple example showing Bose’s and Einstein’s (textbook) methods of counting the
number of possible ways to put n = 2 particles in a level having A = g = 3 states.
ously obey the two conditions As = r prs and N s = r rprs . In this example,
there are only two such sets; a general proof is given in Appendix A. Thus, it is
actually possible to state the number of microstates without specifying the individual arrangements, by summing over all of them:
s
A
p0s =0
=
s
...
A
s =0
pN
s
Ns
Ns
r=0
r=0
As !
, As =
prs and N s =
rprs
s
s
s
p0 !p1 ! . . . pN s
(N s + As − 1)!
.
N s !(As − 1)!
(71)
3. Grand Canonical versus Canonical Statistics of BEC
Fluctuations
The problem of BEC fluctuations in a Bose gas is well known [13,15]. However,
as with many other problems which are well-posed physically and mathematically, it is highly nontrivial and deep, especially for the interacting Bose gas.
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V.V. Kocharovsky et al.
[3
3.1. R ELATIONS BETWEEN S TATISTICS OF BEC F LUCTUATIONS
IN THE G RAND C ANONICAL , C ANONICAL , AND M ICROCANONICAL
E NSEMBLES
To set the stage, we briefly review here some basic notions and facts from the statistical physics of BEC involving relations between statistics of BEC fluctuations
in the grand canonical, canonical, and microcanonical ensembles.
Specific experimental conditions determine which statistics should be applied
to describe a particular system. In view of the present experimental status the
canonical and microcanonical descriptions of the BEC are of primary importance.
Recent BEC experiments on harmonically trapped atoms of dilute gases deal with
a finite and well-defined number of particles. This number, even if it is not known
exactly, certainly does not fluctuate once the cooling process is over. Magnetic
or optical confinement suggests that the system is also thermally isolated and,
hence, the theory of the trapped condensate based on a microcanonical ensemble is needed. In this microcanonical ensemble the total particle number and the
total energy are both exactly conserved, i.e., the corresponding operators are constrained to be the c-number constants, Nˆ = const and Hˆ = const. On the other
hand, in experiments with two (or many) component BECs, Bose–Fermi mixtures,
and additional gas components, e.g., for sympathetic cooling, there is an energy
exchange between the components. As a result, each of the components can be
described by the canonical ensemble that applies to systems with conserved particle number while exchanging energy with a heat bath of a given temperature.
Such a description is also appropriate for dilute 4 He in a porous medium [22]. In
the canonical ensemble only the total number of particles is constrained to be an
exact, nonfluctuating constant,
N = nˆ 0 +
(72)
nˆ k ,
k=0
but the energy Hˆ has nonzero fluctuations, (Hˆ − H¯ )2 = 0, and only its average
value is a constant, Hˆ = H¯ = E = const, determined by a fixed temperature T
of the system.
However, the microcanonical and canonical descriptions of a many-body system are difficult because of the operator constraints imposed on the total energy
and particle number. As a result, the standard textbook formulation of the BEC
problem assumes either a grand canonical ensemble (describing a system which
is allowed to exchange both energy and particles with a reservoir at a given temperature T and chemical potential μ, which fix only the average total number of
particles, Nˆ = const, and the average total energy Hˆ = const) or some restricted ensemble that selects states so as to ensure a condensate wave function
with an almost fixed phase and amplitude [30,32,47,48]. These standard formulations focus on and provide effective tools for the study of the thermodynamic
3]
FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
317
and hydrodynamic properties of the many-body Bose system at the expense of
an artificial modification of the condensate statistics and dynamics of BEC formation. While the textbook grand canonical prediction of the condensate mean
occupation agrees, in some sense, with the Bose–Einstein condensation of trapped
atomic gases, this is not even approximately true as concerns the grand canonical
counting statistics
nν
1
n¯ ν
GC
ρν (nν ) =
(73)
,
1 + n¯ ν 1 + n¯ ν
which gives the probability to find nν particles in a given single-particle state ν,
where the mean occupation is n¯ ν . Below the Bose–Einstein condensation temperature, where the ground state mean-occupation number is macroscopic, n¯ 0 ∼ N ,
the distribution ρ0GC (n0 ) becomes extremely broad with the squared variance
(n0 − n¯ 0 )2 ≈ n¯ 20 ∝ N 2 even at T → 0 [13,49]. This prediction is surely at odds
with the isolated Bose gas, where for sufficiently low temperature all particles are
expected to occupy the ground state with no fluctuations left. It was argued by
Ziff et al. [13] that this unphysical behavior of the variance is just a mathematical artefact of the standard grand canonical ensemble, which becomes unphysical
below the condensation point. Thus, the grand canonical ensemble is irrelevant to
experiment if not revised properly. Another extremity, namely, a complete fixation
of the amplitude and phase of the condensate wave function, is unable to address
the condensate formation and the fluctuation problems at all.
It was first realized by Fierz [42] that the canonical ensemble with an exactly
fixed total number of particles removes the pathologies of the grand canonical
ensemble. He exploited the fact that the description of BEC in a homogeneous
ideal Bose gas is exactly equivalent to the spherical model of statistical physics,
and that the condensate serves as a particle reservoir for the noncondensed phase.
Recently BEC fluctuations were studied by a number of authors in different statistical ensembles, both in the ideal and interacting Bose gases. The microcanonical
treatment of the ground-state fluctuations in a one-dimensional harmonic trap is
closely related to the combinatorics of partitioning an integer, opening up an interesting link to number theory [43,50,51].
It is worthwhile to compare the counting statistics (73) with the predictions of
other statistical ensembles [46]. For high temperatures, T > Tc , all three ensembles predict the same counting statistics. However, this is not the case for low
temperatures, T < Tc . Here the broad distribution of the grand canonical statistics differs most dramatically from ρ0 (n0 ) in the canonical and microcanonical
statistics, which show a narrow single-peaked distribution around the condensate mean occupation number. In particular, the master equation approach [52,
53] (Section 4) yields a finite negative binomial distribution for the probability
distribution of the ground-state occupation in the ideal Bose gas in the canonical
ensemble (see Fig. 12 and Eqs. (154), (162), and (163)). The width of the peak
318
V.V. Kocharovsky et al.
[3
decreases with decreasing temperature. In fact, it is this sharply-peaked statistical
distribution which one would naively expect for a Bose condensate.
Each statistical ensemble is described by a different partition function. The
microcanonical partition function Ω(E, N ) is equal to the number of N -particle
microstates corresponding to a given total energy E. Interestingly, at low temperatures canonical and microcanonical fluctuations have been found to agree in
the large-N (thermodynamic) limit for one-dimensional harmonic trapping potentials, but to differ in the case of three-dimensional isotropic harmonic traps [54].
More precisely, for the d-dimensional power-law traps characterized by an exponent σ , as considered later in Section 5.3, microcanonical and canonical fluctuations agree in the large-N limit when d/σ < 2, but the microcanonical fluctuations remain smaller than the canonical ones when d/σ > 2 [55]. Thus, in d = 3
dimensions the homogeneous Bose gas falls into the first category, but the harmonically trapped one into the second. The difference between the fluctuations in
these two ensembles is expressed by a formula which is similar in spirit to the one
expressing the familiar difference between the heat capacities at constant pressure
and at constant volume [54,55].
The direct numerical computation of the microcanonical partition function becomes very time consuming or not possible for N > 105 . For N 1, and for
large numbers of occupied excited energy levels, one can invoke, e.g., the approximate technique based on the saddle-point method, which is widely used in
statistical physics [31]. When employing this method, one starts from the known
grand canonical partition function and utilizes the saddle-point approximation
for extracting its required canonical and microcanonical counterparts, which then
yield all desired quantities by taking suitable derivatives. Recently, still another
statistical ensemble, the so called Maxwell’s demon ensemble, has been introduced [54]. Here, the system is divided into the condensate and the particles
occupying excited states, so that only particle transfer (without energy exchange)
between these two subsystems is allowed, an idea that had also been exploited
by Fierz [42] and by Politzer [44]. This ensemble has been used to obtain an approximate analytical expression for the ground-state BEC fluctuations both in the
canonical and in the microcanonical ensemble. The Maxwell’s demon approximation can be understood on the basis of the canonical-ensemble quasiparticle
approach [20,21], which is discussed in Section 5, and which is readily generalizable to the case of the interacting Bose gas (see Section 6). It also directly proves
that the higher statistical moments for a homogeneous Bose gas depend on the
particular boundary conditions imposed, even in the thermodynamic limit [20,21,
56].
The canonical partition function ZN (T ) is defined as
ZN (T ) =
∞
E
e−E/kB T Ω(E, N ),
(74)
3]
FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
319
where the sum runs over all allowed energies, and kB is the Boltzmann constant. Eq. (74) can be used to calculate also the microcanonical partition function
Ω(E, N ) by means of the inversion of this definition. Likewise, the canonical
partition function ZN (T ) can be obtained by the inversion of the definition of the
grand canonical partition function Ξ (μ, T ):
Ξ (μ, T ) =
∞
eμN/kB T ZN (T ).
(75)
N=0
Inserting Eq. (74) into Eq. (75), we obtain the following relation between Ξ (μ, T )
and Ω(E, N ):
Ξ (μ, T ) =
∞
eμN/kB T
∞
e−E/kB T Ω(E, N ).
(76)
E
N=0
As an example, let us consider an isotropic 3-dimensional harmonic trap with
eigenfrequency ω. In this case, a relatively compact expression for the grand
canonical partition function,
(E/h¯ ω+1)(E/h¯ ω+2)/2
∞
1
,
Ξ (μ, T ) =
(77)
1 − e(μ−E)/kB T
E
allows us to find Ω(E, N ) from Eq. (76) by an application of the saddle-point
approximation to the contour integral
'
'
Ξ (z, x)
1
dz
dx N+1 E/hω+1 ,
Ω(E, N) =
(2πi)2
z
x ¯
γz
x=e
−h¯ ω/kB T
, z=e
γx
μ/kB T
,
(78)
where the contours of integration γz and γx include z = 0 and x = 0, respectively. It is convenient to rewrite the function under the integral in Eq. (78) as
exp[ϕ(z, x)], where
ϕ(z, x) = ln Ξ (z, x) − (N + 1) ln z − (E/h¯ ω + 1) ln x.
Taking the contours through the extrema (saddle points) z0 and x0 of ϕ(z, x),
and employing the usual Gaussian approximation, we get for N → ∞ and
E/h¯ ω → ∞ the following asymptotic formula:
−1/2
E/hω+1
Ξ (z0 , x0 )/z0N+1 x0 ¯
,
Ω(E, N) = 2πD(z0 , x0 )
where D(z0 , x0 ) is the determinant of the second derivatives of the function ϕ(z, x), evaluated at the saddle points [45]. For N → ∞ and E/h¯ ω → ∞
the function exp[ϕ(z, x)] is sharply peaked at z0 and x0 , which ensures good
accuracy of the Gaussian approximation. However, the standard result becomes
320
V.V. Kocharovsky et al.
[3
incorrect for E < Nε1 , where ε1 = hω
¯ 1 is the energy of the first excited state
(see [45,57]); in this case, a more refined version of the saddle-point method is
required [58–60]. We discuss this improved version of the saddle-point method in
Appendix F.
An accurate knowledge of the canonical partition function is helpful for the
calculation of the microcanonical condensate fluctuations by the saddle point
method, as has been demonstrated [57] by a numerical comparison with exact
microcanonical simulations. In principle, the knowledge of the canonical partition function allows us to calculate thermodynamic and statistical equilibrium
properties of the system in the standard way (see, e.g., [13,61]). An important
fact is that the usual thermodynamic quantities (average energy, work, pressure,
heat capacities, etc.) and the average number of condensed atoms do not have any
infrared-singular contributions and do not depend on a choice of the statistical
ensemble in the thermodynamic (bulk) limit. However, the variance and higher
moments of the BEC fluctuations do have the infrared anomalies and do depend
on the statistical ensemble, so that their calculation is much more involved and
subtle.
3.2. E XACT R ECURSION R ELATION FOR THE S TATISTICS OF THE N UMBER
OF C ONDENSED ATOMS IN AN I DEAL B OSE G AS
It is worth noting that there is one useful reference result in the theory of BEC
fluctuations, namely, an exact recursion relation for the statistics of the number
of condensed atoms in an ideal Bose gas. Although it does not give any simple
analytical answer or physical insight into the problem, it can be used for “exact”
numerical simulations for traps containing a finite number of atoms. This is very
useful as a reference to be compared against different approximate analytical formulas. This exact recursion relation for the ideal Bose gas had been known for
a long time [61], and rederived independently by several authors [46,62–64]. In
the canonical ensemble, the probability to find n0 particles occupying the singleparticle ground state is given by
ρ0 (n0 ) =
ZN−n0 (T ) − ZN−n0 −1 (T )
;
ZN (T )
Z−1 ≡ 0.
(79)
The recurrence relation for the ideal Bose gas then states [61,62]
ZN (T ) =
N
1 Z1 (T /k)ZN−k (T ),
N
Z1 (T ) =
k=1
∞
e−εν /kB T , Z0 (T ) = 1,
ν=0
(80)
3]
FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
321
which enables one to numerically compute the entire counting statistics (79). Here
ν stands for a set of quantum numbers which label a given single-particle state,
εν is the associated energy, and the ground-state energy is taken as ε0 = 0 by
convention. For an isotropic, three-dimensional harmonic trap one has
Z1 (T ) =
∞
1
n=0
2
(n + 2)(n + 1)e−nh¯ ω/kB T =
1
(1 − e−h¯ ω/kB T )3
,
where 12 (n + 2)(n + 1) is the degeneracy of the level εn = nh¯ ω.
In the microcanonical ensemble the ground-state occupation probability is
given by a similar formula
Ω(E, N − n0 ) − Ω(E, N − n0 − 1)
;
Ω(E, N )
Ω(E, −1) ≡ 0,
ρ0MC (n0 ) =
(81)
where the microcanonical partition function obeys the recurrence relation
Ω(E, N) =
N ∞
1 Ω(E − kεν , N − k),
N
k=1 ν=0
Ω(0, N) = 1, Ω(E > 0, 0) = 0.
(82)
For finite E the sum over ν is finite because of Ω(E < 0, N) = 0.
3.3. G RAND C ANONICAL A PPROACH
Here we discuss the grand canonical ensemble, and show that it loses its validity for the ideal Bose gas in the condensed region. Nevertheless, reasonable
approximate results can be obtained if the canonical-ensemble constraint is properly incorporated in the grand canonical approach, especially if we are not too
close to Tc . In principle, the statistical properties of BECs can be probed with
light [65]. In particular, the variance of the number of scattered photons may distinguish between the Poisson and microcanonical statistics.
3.3.1. Mean Number of Condensed Particles in an Ideal Bose Gas
The Bose–Einstein distribution can be easily derived from the density matrix approach. Consider a collection of particles with the Hamiltonian Hˆ =
†
ˆ k aˆ k (εk − μ), where μ is the chemical potential. The equilibrium state of
ka
the system is described by
ρˆ =
1
exp −β Hˆ ,
Z
(83)
322
V.V. Kocharovsky et al.
[3
&
where Z = Tr{exp(−β Hˆ )} = k (1 − e−β(εk −μ) )−1 . Then the mean number of
particles with energy εk is
† −β aˆ † aˆ (ε −μ) nk
nk aˆ k aˆ k e k k k
nk = Tr ρˆ aˆ k† aˆ k = 1 − e−β(εk −μ)
nk
d(1 − e−β(εk −μ) )−1
1
=
. (84)
= 1 − e−β(εk −μ)
d(−β(εk − μ))
exp[β(εk − μ)] − 1
In the grand canonical ensemble the average condensate particle number n¯ 0 is
determined from the equation for the total number of particles in the trap,
N=
∞
k=0
n¯ k =
∞
k=0
1
,
exp β(εk − μ) − 1
(85)
where εk is the energy spectrum of the trap. In particular, for the three dimensional (3D) isotropic harmonic trap we have εk = h¯ Ω(kx + ky + kz ). For
simplicity, we set ε0 = 0.
For 3D and 1D traps with noninteracting atoms, Eq. (85) was studied by
Ketterle and van Druten [11], and by Grossmann and Holthaus [43,66]. They
calculated the fraction of ground-state atoms versus temperature for various N
and found that BEC also exists in 1D traps, where the condensation phenomenon
looks very similar to the 3D case. Later Herzog and Olshanii [67] used the known
analytical formula for the canonical partition function of bosons trapped in a 1D
harmonic potential [68,69] and showed that the discrepancy between the grand
canonical and the canonical predictions for the 1D condensate fraction becomes
less than a few per cent for N > 104 . The deviation decreases according to a
1/ ln N scaling law for fixed T /Tc . In 3D the discrepancy is even less than in
the 1D system [64]. The ground state occupation number and other thermodynamic properties were studied by Chase, Mekjian and Zamick [64] in the grand
canonical, canonical and microcanonical ensembles by applying combinatorial
techniques developed earlier in statistical nuclear fragmentation models. In such
models there are also constraints, namely the conservation of proton and neutron
number. The specific heat and the occupation of the ground state were found substantially in agreement in all three ensembles. This confirms the essential validity
of the use of the different ensembles even for small groups of particles as long
as the usual thermodynamic quantities, which do not have any infrared singular
contributions, are calculated.
Let us demonstrate how to calculate the mean number of condensed particles
for a particular example of a 3D isotropic harmonic trap. Following Eq. (85), we
can relate the chemical potential μ to the mean number of condensed particles n¯ 0
as 1 + 1/n¯ 0 = exp(−βμ). Thus, we have
N=
∞
∞
nk =
k=0
k=0
1
.
(1 + 1/n¯ 0 ) exp βεk − 1
(86)
3]
FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
323
The standard approach is to consider the case N 1 and separate off the ground
state so that Eq. (86) approximately yields
1
.
N − n¯ 0 =
(87)
exp(βεk ) − 1
k>0
For an isotropic harmonic trap with frequency Ω,
k>0
∞
1
1 (n + 2)(n + 1)
=
exp(βεk ) − 1
2
exp(βnh¯ Ω) − 1
n=1
≈
1
2
∞
1
(x + 2)(x + 1)
dx.
exp(xβ hΩ)
−1
¯
(88)
In the limit kB T h¯ Ω we obtain
k>0
1
1
≈
exp(βεk ) − 1
2
∞
0
x2
dx =
exp(xβ hΩ)
−1
¯
kB T
hΩ
¯
3
ζ (3).
Furthermore, when T = Tc we take n¯ 0 = 0. Then Eqs. (87) and (89) yield
N 1/3
kB Tc = hΩ
¯
ζ (3)
(89)
(90)
and the temperature dependence of the mean condensate occupation with a cusp
at T = Tc in the thermodynamic limit
3 T
.
n¯ 0 (T ) = N 1 −
(91)
Tc
Figure 3 compares the numerical solution of Eq. (86) (solid line) for N = 200
with the numerical calculation of n¯ 0 (T ) from the exact recursion relations in
Eqs. (79) and (80) in the canonical ensemble (large dots). Small dots show the
plot of the solution (91), which is valid only for a large number of particles, N .
Obviously, more accurate solution of the equation for the mean number of condensed particles (86) in a trap with a finite number of particles does not show the
cusp. In Appendix B we derive an analytical solution of Eq. (86) for n¯ 0 (T ) valid
for n¯ 0 (T ) 1. One can see that for the average particle number both ensembles
yield very close answers. However, this is not the case for the BEC fluctuations.
3.3.2. Condensate Fluctuations in an Ideal Bose Gas
s
Condensate
fluctuations are characterized by the central moments (n0 − n0 ) = r rs nr0 n0 s−r . The first few of them are
(n0 − n¯ 0 )2 = n20 − n0 2 ,
(92)
324
V.V. Kocharovsky et al.
[3
F IG . 3. Mean number of condensed particles as a function of temperature for N = 200. The solid
line is the plot of the numerical solution of Eq. (86). The exact numerical result for the canonical
ensemble (Eqs. (79) and (80)) is plotted as large dots.
The dashed line is obtained using Maxwell’s
demon ensemble approach, which yields n¯ 0 = N − ∞
k>0 1/[exp(βεk ) − 1].
(n0 − n¯ 0 )3 = n30 − 3 n20 n0 + 2n0 3 ,
(n0 − n¯ 0 )4 = n40 − 4 n30 n0 + 6 n20 n0 2 − 3n0 4 .
(93)
(94)
At arbitrary temperatures, BEC fluctuations in
the canonical ensemble can be
described via a stochastic variable n0 = N − k=0 nk that depends on and is
complementary to the sum of the independently fluctuating numbers nk , k = 0,
of the excited atoms. In essence, the canonical-ensemble constraint in Eq. (72)
eliminates one degree of freedom (namely, the ground-state one) from the set of
all independent degrees of freedom of the original grand canonical ensemble, so
that only transitions between the ground and excited states remain independently
fluctuating quantities. They describe the canonical-ensemble quasiparticles, or excitations, via the creation and annihilation operators βˆ + and βˆ (see Sections 5
and 6 below).
At temperatures higher than Tc the condensate fraction is small and one can
approximately treat the condensate as being in contact with a reservoir of noncondensate particles. The condensate exchanges particles with the big reservoir.
Hence, the description of particle fluctuations in the grand canonical picture, assuming that the number of atoms in the ground state fluctuates independently from
the numbers of excited atoms, is accurate in this temperature range.
At temperatures close to or below Tc the situation becomes completely different. One can say that at low temperatures, T Tc , the picture is opposite to the
picture of the Bose gas above the BEC phase transition at T > Tc . The canonicalensemble quasiparticle approach, suggested in Refs. [20,21] and valid both for
3]
FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
325
the ideal and interacting Bose gases, states that at low temperatures the noncondensate particles can be treated as being in contact with the big reservoir of the n0
condensate particles. This idea had previously been spelled out by Fierz [42] and
been used by Politzer [44], and was then employed for the construction of the
“Maxwell’s demon ensemble” [54], named so since a permanent selection of the
excited (moving) atoms from the ground state (static) atoms is a problem resembling the famous Maxwell’s demon problem in statistical physics.
Note that although this novel statistical concept can be studied approximately
with the help of the Bose–Einstein expression (84) for the mean number of excited
(only excited, k = 0!) states with some new chemical potential μ, it describes
the canonical-ensemble fluctuations and is essentially different from the standard
grand canonical description of fluctuations in the Bose gas. Moreover, it is approximately valid only if we are not too close to the critical temperature Tc , since
otherwise the “particle reservoir” is emptied. Also, the fluctuations obtained from
the outlined “grand” canonical approach (see Eqs. (95)–(102) and (C10)–(C12)),
an approach complementary to the grand canonical one, provide an accurate description in this temperature range in the thermodynamic limit, but do not take into
account all mesoscopic effects, especially near Tc (for more details, see Sections 4
and 5). Thus, although the mean number of condensed atoms n¯ 0 can be found
from the grand canonical expression used in Eq. (85), we still need to invoke the
conservation of the total particle number N = n0 + n in order to find the higher
moments of condensate fluctuations, nr0 . Namely, we can use the following relation between the central moments of the mth order of the number of condensed
atoms, and that of the noncondensed ones: (n0 − n¯ 0 )m = (−1)m (n − n)
¯ m .
As a result, at low enough temperatures one can approximately write the central
moments in the well-known canonical form via the cumulants in the ideal Bose
gas (see Eqs. (220), (221), (219), (223) and Section 5 below for more details):
(n0 − n¯ 0 )2 = κ2 = κ˜ 2 + κ˜ 1 ≈
(95)
n¯ 2k + n¯ k ,
k>0
(n0 − n¯ 0 ) = −κ3 = −(κ˜ 3 − 3κ˜ 2 − κ˜ 1 ) ≈ −
3
2n¯ 3k + 3n¯ 2k + n¯ k , (96)
k>0
(n0 − n¯ 0 )4 = κ4 + 3κ22 = κ˜ 4 + 6κ˜ 3 + 7κ˜ 2 + κ˜ 1 + 3(κ˜ 2 + κ˜ 1 )2
2
2
6n¯ 4k + 12n¯ 3k + 7n¯ 2k + n¯ k + 3
n¯ k + n¯ k .
≈
k>0
(97)
k>0
These same equations can also be derived by means of the straightforward calculation explained in Appendix C.
Combining the hallmarks of the grand canonical approach, namely, the value
of the chemical potential μ = −β −1 ln(1 + 1/n¯ 0 ) and the mean noncondensate
occupation n¯ k = {exp[β(Ek − μ)] − 1}−1 , with the Eq. (95) describing the fluc-
326
V.V. Kocharovsky et al.
tuations in the canonical ensemble, we find the BEC variance
n20 ≡ (n0 − n¯ 0 )2
"
1
1
=
+
1
2
[exp(βEk )(1 + n¯ 0 ) − 1]
exp(βEk )(1 +
k>0
[3
#
1
n¯ 0 ) − 1
.
(98)
In the case kB T h¯ Ω, this Eq. (98) can be evaluated analytically, as is shown in
Appendix D. The variance up to second order in 1/n¯ 0 from Eq. (D4) is
!
3 2
N
1 1
1 1/2
1
T
π
n0 ≈
.
− (1 + ln n¯ 0 ) + 2
ln n¯ 0 −
ζ (3) Tc
6
n¯ 0
4
n¯ 0 2
(99)
The leading term in this expression yields Politzer’s result [44],
!
!
3
ζ (2)N T 3
T
n0 ≈
≈ 1.17 N
,
ζ (3) Tc
Tc
(100)
2
plotted as a dashed line in Fig. 4, where ζ (2) = π6 ≈ 1.644 9 and ζ (3) ≈ 1.202 1
(compare this with D. ter Haar’s [15] result n0 ≈ N ( TTc )3 , which is missing
the square root). The same formula was obtained later by Navez et al. using the
Maxwell’s demon ensemble [54]. For the microcanonical ensemble the Maxwell’s
demon approach yields [54]
!
!
3
3
ζ (2) 3ζ (3)
T
T
n0 ≈ N
(101)
≈ 0.73 N
;
−
ζ (3) 4ζ (4)
Tc
Tc
higher-order terms have been derived in Ref. [60]. The microcanonical fluctuations are smaller than the canonical ones due to the additional energy constraint.
For 2D and 1D harmonic traps the Maxwell’s demon approach leads to [46,55]
$
√ T
Tc
T
ln
n0 ∼ N
(102)
and n0 ∼ N,
Tc
T
Tc
respectively.
Figure 4 shows the BEC variance n0 (T ) as a function of temperature for a 3D
trap with the total particle number N = 200. The “grand” canonical curve refers
to Eq. (98) and shows good agreement for T < Tc with the numerical result for
n0 (T ) (large dots), obtained within the exact recursion relations (79) and (80)
for the canonical ensemble. The plot of Politzer’s asymptotic formula (100)
(dashed line) does not give good agreement,
since the particle number consid√
ered here is fairly low. We also plot n¯ 0 (n¯ 0 + 1), which is the expression for the
condensate number fluctuations n0 in the grand canonical ensemble; it works
well above Tc . Figure 5 shows the third central moment (n0 − n¯ 0 )3 as a function
3]
FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
327
F IG . 4. Variance (n0 − n0 )2 1/2 of the condensate particle number for N = 200. The solid
line is the “grand” canonical result obtained from Eq. (98) and the numerical solution of n0 from
Eq. (86). Exact numerical data for the canonical ensemble (Eqs. (79) and (80)) are shown as dots. The
asymptotic Politzer approximation [44], given by Eq. (100), is shown by the dashed line, while small
dots result from Eq. (D4). The dash-dotted line is obtained from the master equation approach (see
Eq. (172) below).
of temperature for the total particle number N = 200, plotted using Eqs. (96)
and (86). Dots are the exact numerical result obtained within the canonical ensemble. We also plot the standard grand canonical formula 2n¯ 30 + 3n¯ 20 + n¯ 0 , which
again works well only above Tc .
At high temperatures the main point, of course, is the validity of the standard grand canonical approach where the average occupation of the ground state
alone gives a correct description of the condensate fluctuations, since the excited
particles constitute a valid “particle reservoir”. Condensate fluctuations obtained
from the “grand” canonical approach for the canonical-ensemble quasiparticles
and from the standard grand canonical ensemble provide an accurate description of the canonical-ensemble fluctuations at temperatures not
√ too close to the
narrow crossover region between low (Eq. (98)) and high ( n¯ 0 (n¯ 0 + 1)) temperature regimes; i.e., in the region not too near Tc . In this crossover range both
approximations fail, since the condensate and the noncondensate fractions have
comparable numbers of particles, and there is no any valid particle reservoir. Note,
however, that a better description, that includes mesoscopic effects and works in
the whole temperature range, can be obtained using the condensate master equa-
328
V.V. Kocharovsky et al.
[4
F IG . 5. The third central moment (n0 − n¯ 0 )3 for N = 200, plotted using Eqs. (96) and (86).
Exact numerical data for the canonical ensemble (Eqs. (79) and (80)) are shown as dots. The dashed
line is the result of master equation approach (see Eq. (174) below).
tion, as shown in Section 4; see, e.g., Fig. 13. Another (semi-analytic) technique,
the saddle-point method, is discussed in Appendix F.
4. Dynamical Master Equation Approach and Laser
Phase-Transition Analogy
One approach to the canonical statistics of ideal Bose gases, presented in [52] and
developed further in [53], consists in setting up a master equation for the condensate and finding its equilibrium solution. This approach has the merit of being
technically lucid and physically illuminating. Furthermore, it reveals important
parallels to the quantum theory of the laser. In deriving that master equation, the
approximation of detailed balance in the excited states is used, in addition to the
assumption that given an arbitrary number n0 of atoms in the condensate, the
remaining N − n0 excited atoms are in an equilibrium state at the prescribed temperature T .
In Section 4.2 we summarize the master equation approach against the results
provided by independent techniques. In Section 4.1 we motivate our approach by
sketching the quantum theory of the laser with special emphasis on the points
relevant to BEC.
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329
4.1. Q UANTUM T HEORY OF THE L ASER
In order to set the stage for the derivation of the BEC master equation, let us remind ourselves of the structure of the master equations for a few basic systems
that have some connection with N particles undergoing Bose–Einstein condensation while exchanging energy with a thermal reservoir.
4.1.1. Single Mode Thermal Field
The dissipative dynamics of a small system (s) coupled to a large reservoir (r)
is described by the reduced density matrix equation up to second order in the
interaction Hamiltonian Vˆsr
∂
1
ρˆs (t) = − 2 Trr
∂t
h¯
t
Vˆsr (t), Vˆsr (t ), ρˆs (t) ⊗ ρˆrth dt ,
(103)
0
ρˆrth
is the density matrix of the thermal reservoir, and we take the Markovwhere
ian approximation.
We consider the system as a single radiation mode cavity field (f ) of frequency
ν coupled to a reservoir (r) of two-level thermal atoms, and show how the radiation field thermalizes. The multiatom Hamiltonian in the interaction picture is
gj σˆ j aˆ † ei(ν−ωj )t + adj
Vˆf r = h¯
(104)
j
where σˆ j = |bj aj | is the atomic (spin) operator of the j th particle corresponding to the downward transition, aˆ † is the creation operator for the single mode
field and gj is the coupling constant. The reduced density matrix equation for the
field can be obtained from Eqs. (103) and (104). By using the density matrix for
the thermal ensemble of atoms
|aj aj |e−βEa,j + |bj bj |e−βEb,j /Zj ,
ρˆrth =
(105)
j
where Zj = e−βEa,j + e−βEb,j , we obtain
1 †
∂
κj Paj aˆ aˆ ρˆf − 2aˆ † ρˆf aˆ + ρˆf aˆ aˆ †
ρˆf = −
∂t
2
j
+ Pbj aˆ † aˆ ρˆf − 2aˆ ρˆf aˆ † + ρˆf aˆ † aˆ ,
−βExj
(106)
/Zj with x = a, b and the dissipative constant is 12 κj =
where Pxj = e
t
Re{gj2 0 ei(ν−ωj )(t−t ) dt }. Note that the same structure of the master equation is
obtained for a phonon bath modelled as a collection of harmonic oscillators, as
shown in Appendix E.
330
V.V. Kocharovsky et al.
[4
Taking the diagonal matrix elements ρn,n (t) = n|ρˆf |n of Eq. (106), we have
∂
ρn,n (t) = −κPa (n + 1)ρn,n (t) − nρn−1,n−1 (t)
∂t
− κPb nρn,n (t) − (n + 1)ρn+1,n+1 (t) ,
(107)
where κ is κj times a density of states factor and ρn,n .
The steady state equation gives
ρn,n ≡ pn = e−nβ h¯ ω p0 .
From ∞
n=0 pn = 1 we obtain the thermal photon number distribution
¯
pn = e−nβ h¯ ω 1 − e−β hω
(108)
(109)
which is clearly an exponentially decaying photon number distribution.
4.1.2. Coherent State
Consider the interaction of a single mode field with a classical current J described
by
ˆ
t) d 3 r = h¯ j (t)aˆ + j ∗ (t)aˆ † ,
Vˆcoh (t) = J(r, t) · A(r,
(110)
V
where the complex time dependent coefficient is
A0
J(r, t) · xe
ˆ i(k·r−νt) d 3 r
j (t) =
h¯
V
ˆ
t) of the single mode field, asand A0 is the amplitude vector potential A(r,
sumed to be polarized along x axis. An example of such interaction is a klystron.
Clearly, the unitary time evolution of Vˆcoh is in the form of a displacement operator exp(α ∗ (t)aˆ − α(t)aˆ † ) associated with a coherent state when dissipation is
neglected.
Thus, the density matrix equation for a klystron including coupling with a thermal bath is
∂ ρˆf (t)
1ˆ
=
Vcoh (t), ρˆf (t)
∂t
i h¯
t
1
− 2 Trr
(111)
Vˆf r (t), Vˆf r t , ρˆf (t) ⊗ ρˆrth dt ,
h¯
0
where Vˆf r is given by Eq. (E3). The second term of Eq. (111) describes the damping of the single mode field given by Eq. (E4). By taking the matrix elements of
4]
FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
331
F IG . 6. Photon number distributions for (a) thermal photons plotted from Eq. (108) (dashed line),
(b) coherent state (Poissonian) (thin solid line), and (c) He–Ne laser plotted using Eq. (121) (thick
solid line). Insert shows an atom making a radiation transition.
Eq. (111), after a bit of analysis, we have
√
√
√
∂ρn,n (t)
= −i j (t) n + 1 ρn+1,n + j ∗ (t) n ρn−1,n − j (t) n ρn,n −1
∂t
√
− j ∗ (t) n + 1 ρn,n +1
1 − C n + n ρn,n − 2 (n + 1) n + 1 ρn+1,n +1
2
√
1 − D n + 1 + n + 1 ρn,n − 2 nn ρˆn−1,n −1 .
(112)
2
Clearly, the first line of Eq. (112) shows that the change in the photon number
is effected by the off-diagonal field density matrix or the coherence between two
states of different photon number. On the other hand, the damping mechanism
only causes a change in the photon number through the diagonal matrix element
or the population of the number state. This is depicted in Fig. 7.
It can be shown that the solution of Eq. (112) for D = 0 is the matrix element
n
2
of a coherent state |β = e|β| /2 n √β |n, i.e.,
n!
β(t)n β ∗ (t)n −|β(t)|2
e
,
ρn,n (t) = n |ββ| n = √ √
n! n !
(113)
332
V.V. Kocharovsky et al.
[4
F IG . 7. Diagonal (star) and off-diagonal (circle) density matrix elements that govern temporal
dynamics in (a) klystron and (b) thermal field.
t
where β(t) = α(t) − 12 C 0 α(t ) dt . This can be verified if we differentiate
Eq. (113),
"
#
√
dρn,n
dα(t) 1
=
− Cα(t) n ρn−1,n
dt
dt
2
#
" ∗
√
dα (t) 1 ∗
+ n
− Cα (t) ρn,n −1
dt
2
"
#
√
dα(t) 1
−
− Cα(t) n + 1 ρn,n +1
dt
2
" ∗
#
√
dα (t) 1 ∗
−
(114)
− Cα (t) n + 1 ρn+1,n
dt
2
and using
√
√
n ρn,n = ρn−1,n ,
n ρn,n = ρn,n −1 ,
√
√
n + 1 ρn+1,n +1 = ρn,n +1 ,
n + 1 ρn+1,n +1 = ρn+1,n ,
where
dα(t)
dt
(115)
= −ij ∗ (t) is found by comparing Eq. (114) with Eq. (112).
4.1.3. Laser Master Equation
The photon number equation (107) for thermal field is linear in photon number, n,
and it describes only the thermal damping and pumping due to the presence of a
thermal reservoir. Now, we introduce a laser pumping scheme to drive the single
4]
FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
333
F IG . 8. Typical setup of a laser showing an ensemble of atoms driving a single mode field. A competition between lasing and dissipation through cavity walls leads to a detailed balance.
mode field and show how the atom–field nonlinearity comes into the laser master
equation.
We consider a simple three level system where the cavity field couples level a
and level b of lasing atoms in a molecular beam injected into a cavity in the excited
state at rate r (see Fig. 8). The atoms undergo decay from level b to level c. The
pumping mechanism from level c up to level a can thus produce gain in the single
mode field. As shown in [19], we find
∂ρn,n (t)
= − A(n + 1) − B(n + 1)2 ρn,n
∂t
gain
+ An − Bn2 ρn−1,n−1 ,
(116)
2
2
where A = 2rg
is the linear gain coefficient and B = 4g
A is the self-saturation
γ2
γ2
coefficient. Here g is the atom–field coupling constant and γ is the b → c decay
rate. We take the damping of the field to be
∂ρn,n (t)
= −Cnρn,n + C(n + 1)ρn+1,n+1 .
(117)
∂t
loss
Thus, the overall master equation for the laser is
∂ρn,n (t)
= − A(n + 1) − B(n + 1)2 ρn,n + An − Bn2 ρn−1,n−1
∂t
− Cnρn,n + C(n + 1)ρn+1,n+1 ,
(118)
334
V.V. Kocharovsky et al.
[4
which is valid for small B/A 1.
We emphasize that the nonlinear process associated with B is a key physical
process in the laser (but not in a thermal field) because the laser field is so large.
We proceed with detailed balance equation between level n − 1 and n,
− A(n + 1) − B(n + 1)2 pn + C(n + 1)pn+1 = 0,
(119)
An − Bn2 pn−1 − Cnpn = 0.
(120)
By iteration of pk = A−CBk pk−1 , we have
n
pn = p0
k=1
A − Bk
,
C
(121)
∞
&n A−Bk
where p0 = 1/(1 + ∞
n=1
k=1
n=0 pn = 1. Eq. (121) is
C ) follows from
plotted in Fig. 6. There we clearly see that the photon statistics of, e.g., a He–Ne,
laser is not Poissonian pn = nn e−n /n!, as would be expected for a coherent
state.
4.2. L ASER P HASE -T RANSITION A NALOGY
Bose–Einstein condensation of atoms in a trap has intriguing similarities with
the threshold behavior of a laser which also can be viewed as a kind of a phase
transition [70,71]. Spontaneous formation of a long range coherent-order parameter, i.e., macroscopic wave function, in the course of BEC second-order phase
transition is similar to spontaneous generation of a macroscopic coherent field
in the laser cavity in the course of lasing. In both systems stimulated processes
are responsible for the appearance of the macroscopic-order parameter. The main
difference is that for the Bose gas in a trap there is also interaction between the
atoms which is responsible for some processes, including stimulated effects in
BEC. Whereas for the laser there are two subsystems, namely the laser field and
the active atomic medium. The crucial point for lasing is the interaction between
the field and the atomic medium which is relatively small and can be treated perturbatively. Thus, the effects of different interactions in the laser system are easy
to trace and relate to the observable characteristics of the system. This is not the
case in BEC and it is more difficult to separate different effects.
As is outlined in the previous subsection, in the quantum theory of laser, the
dynamics of laser light is conveniently described by a master equation obtained
by treating the atomic (gain) media and cavity dissipation (loss) as reservoirs
which when “traced over” yield the coarse grained equation of motion for the
reduced density matrix for laser radiation. In this way we arrive at the equation of
motion for the probability of having n photons in the cavity given by Eq. (118).
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FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
335
From Eq. (121) we have the important result that partially coherent laser light has
a sharp photon distribution (with width several times Poissonian for a typical He–
Ne laser) due to the presence of the saturation nonlinearity, B, in the laser master
equation. Thus, we see that the saturation nonlinearity in the radiation–matter
interaction is essential for laser coherence.
One naturally asks: is the corresponding nonlinearity in BEC due to atom–
atom scattering? Or is there a nonlinearity present even in an ideal Bose gas? The
master equation presented in this Section proves that the latter is the case.
More generally we pose the question: Is there a similar nonequilibrium approach for BEC in a dilute atomic gas that helps us in understanding the underlying physical mechanisms for the condensation, the critical behavior, and the
associated nonlinearities? The answer to this question is “yes” [52,53].
4.3. D ERIVATION OF THE C ONDENSATE M ASTER E QUATION
We consider the usual model of a dilute gas of Bose atoms wherein interatomic
scattering is neglected. This ideal Bose gas is confined inside a trap, so that the
number of atoms, N , is fixed but the total energy, E, of the gas is not fixed. Instead,
the Bose atoms exchange energy with a reservoir which has a fixed temperature T .
As we shall see, this canonical-ensemble approach is a useful tool in studying the
current laser cooled dilute gas BEC experiments [24–27,88]. It is also directly
relevant to the He-in-vycor BEC experiments [22].
This “ideal gas + reservoir” model allows us to demonstrate most clearly the
master equation approach to the analysis of dynamics and statistics of BEC, and
in particular, the advantages and mathematical tools of the method. Its extension
for the case of an interacting gas which includes usual many-body effects due to
interatomic scattering will be discussed elsewhere.
Thus, we are following the so-called canonical-ensemble approach. It describes, of course, an intermediate situation as compared with the microcanonical
ensemble and the grand canonical ensemble. In the microcanonical ensemble, the
gas is completely isolated, E = const, N = const, so that there is no exchange
of energy or atoms with a reservoir. In the grand canonical ensemble, only the
average energy per atom, i.e., the temperature T and the average number of atoms
N are fixed. In such a case there is an exchange of both energy and atoms with
the reservoir.
The “ideal gas + thermal reservoir” model provides the simplest description
of many qualitative and, in some cases, quantitative characteristics of the experimental BEC. In particular, it explains many features of the condensate dynamics
and fluctuations and allows us to obtain, for the first time, the atomic statistics of
the BEC as discussed in the introduction and in the following.
336
V.V. Kocharovsky et al.
[4
F IG . 9. Simple harmonic oscillators as a thermal reservoir for the ideal Bose gas in a trap.
4.3.1. The “Ideal Gas + Thermal Reservoir” Model
For many problems a concrete realization of the reservoir system is not very important if its energy spectrum is dense and flat enough. For example, one expects
(and we find) that the equilibrium (steady state) properties of the BEC are largely
independent of the details of the reservoir. For the sake of simplicity, we assume
that the reservoir is an ensemble of simple harmonic oscillators whose spectrum
is dense and smooth, see Fig. 9. The interaction between the gas and the reservoir
is described by the interaction picture Hamiltonian
V =
(122)
gj,kl bj† ak al† e−i(ωj −νk +νl )t + h.c.,
j
k>l
where bj† is the creation operator for the reservoir j oscillator (“phonon”), and ak†
and ak (k = 0) are the creation and annihilation operators for the Bose gas atoms
in the kth level. Here hν
¯ k is the energy of the kth level of the trap, and gj,kl is the
coupling strength.
4.3.2. Bose Gas Master Equation
The motion of the total “gas + reservoir” system is governed by the equation for the total density matrix in the interaction representation, ρ˙total (t) =
−i[V (t), ρtotal (t)]/h¯ . Integrating the above equation for ρtotal , inserting it back
into the commutator in Eq. (123), and tracing over the reservoir, we obtain the
exact equation of motion for the density matrix of the Bose-gas subsystem
ρ(t)
˙ =−
1
t
h2
¯
dt Trres V (t), V t , ρtotal t ,
(123)
0
where Trres stands for the trace over the reservoir degrees of freedom.
We assume that the reservoir is large and remains unchanged during the interaction with the dynamical subsystem (Bose gas). As discussed in [53], the
4]
FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
337
density operator for the total system “gas + reservoir” can then be factored,
i.e., ρtotal (t ) ≈ ρ(t ) ⊗ ρres , where ρres is the equilibrium density matrix of the
reservoir. If the spectrum is smooth, we are justified in making the Markov approximation, viz. ρ(t ) → ρ(t). We then obtain the following equation for the
reduced density operator of the Bose-gas subsystem,
κ
ρ˙ = −
(ηkl + 1) ak† al al† ak ρ − 2al† ak ρak† al + ρak† al al† ak
2
k>l
κ † †
−
(124)
ηkl ak al al ak ρ − 2al ak† ρak al† + ρak al† al ak† .
2
k>l
In deriving Eq. (124), we replaced the summation over reservoir modes by an integration (with the density of reservoir modes D(ωkl )) and neglected the frequency
dependence of the coefficient κ = 2πDg 2 /h¯ 2 . Here
−1
ηkl = η(ωkl ) = Trres b† (ωkl )b(ωkl ) = exp(h¯ ωkl /T ) − 1
(125)
is the average occupation number of the heat bath oscillators at frequency ωkl ≡
νk − νl . Equation (124) is then the equation of motion for an N atom Bose gas
driven by a heat bath at temperature T .
4.3.3. Condensate Master Equation
What we are most interested in is the probability distribution
pn0 =
pn0 ,{nk }n0
{nk }n0
for the number of condensed atoms n0 , i.e., the number of atoms in the ground
level of the trap. Let us introduce pn0 ,{nk }n0 = n0 , {nk }n0 |ρ|n0 , {nk }n0 as a
diagonal element of the density matrix in the canonical ensemble where n0 +
k>0 nk = N and |n0 , {nk }n0 is an arbitrary state of N atoms with occupation
numbers of the trap’s energy levels, nk , subject to the condition that there are n0
atoms in the ground state of the trap.
In order to get an equation of motion for the condensate probability distribution
pn0 , we need to perform the summation over all possible occupations {nk }n0 of the
excited levels in the trap. The resulting equation of motion for pn0 , from Eq. (124),
is
dpn0
(ηkl + 1) (nl + 1)nk pn0 ,{nk }n0
= −κ
dt
{nk }n0 k>l>0
− nl (nk + 1)pn0 ,{...,nl −1,...,nk +1,...}n0
+ ηkl nl (nk + 1)pn0 ,{nk }n0
338
V.V. Kocharovsky et al.
[4
− (nl + 1)nk pn0 ,{...,nl +1,...,nk −1,...}n0
(ηk + 1)(n0 + 1)nk pn0 ,{nk }n0
−κ
{nk }n0 k >0
− (ηk + 1)n0 (nk + 1)pn0 −1,{nk +δk,k }n0 −1
+ ηk n0 (nk + 1)pn0 ,{nk }n0
− ηk (n0 + 1)nk pn0 +1,{nk −δk,k }n0 +1 ,
(126)
k
where
ηk = η(νk ) is the mean number of thermal phonons of mode and the
sum k runs over all excited levels.
To simplify Eq. (126) we assume that the atoms in the excited levels with a
given number of condensed atoms n0 are in an equilibrium state at the temperature T , i.e.,
exp(− Th¯ k>0 νk nk )
,
pn0 ,{nk }n0 = pn0 (127)
h¯ {nk }n0 exp(− T
k>0 νk nk )
where k>0 nk = N −n0 , and we assume that the sum k>0 runs over all energy
are treated
states of the trap, including degenerate states whose occupations nk as different stochastic variables. Equation (127) implies that the sum k>l>0 in
Eq. (126) is equal to zero, since as depicted in Fig. 10,
(ηkl + 1)pn0 ,{nk }n0 = ηkl pn0 ,{...,nl +1,...,nk −1,...}n0 ,
(ηkl + 1)pn0 ,{...,nl −1,...,nk +1,...}n0 = ηkl pn0 ,{nk }n0 .
(128)
Equation (128) is precisely the detailed balance condition. The average number
of atoms in an excited level, subject to the condition that there are n0 atoms in the
ground state, from Eq. (127), is
pn0 ,{nk }n0
nk .
nk n0 =
(129)
pn0
{nk }n0
Therefore, the equation of motion for pn0 can be rewritten in the symmetrical and
transparent form
d
pn0 = −κ Kn0 (n0 + 1)pn0 − Kn0 −1 n0 pn0 −1
dt
+ Hn0 n0 pn0 − Hn0 +1 (n0 + 1)pn0 +1 ,
where
Kn0 =
k >0
(ηk + 1)nk n0 ,
Hn0 =
ηk nk n0 + 1 .
(130)
(131)
k >0
We can obtain the steady state distribution of the number of atoms condensed
in the ground level of the trap from Eq. (130). The mean value and the variance of
4]
FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
339
F IG . 10. Detailed balance and the corresponding probability flow diagram. We call Kn0 cooling
(kooling) rate since the laser loss rate is denote by “C”.
the number of condensed atoms can then be determined. It is clear from Eq. (130)
that there are two processes, cooling and heating. The cooling process is represented by the first two terms with the cooling coefficient Kn0 while the heating
by the third and fourth terms with the heating coefficient Hn0 . The detailed balance condition yields the following expression for the number distribution of the
condensed atoms
n0
pn0 = p0
i=1
Ki−1
,
Hi
(132)
where the partition function
ZN =
N
1
=
pN
N
n0 =0 i=n0 +1
Hi
Ki−1
(133)
is determined from the normalization condition N
n0 =0 pn0 = 1. The functions Hi
and Ki as given by Eq. (131), involve, along with ηk (Eq. (125)), the function nk n0 (Eq. (129)). In the following sections, we shall derive closed form
expressions for these quantities under various approximations. The master equation (130) for the distribution function for the condensed atoms is one of our main
results. It yields explicit expressions for the statistics of the condensed atoms and
the canonical partition function. Physical interpretation of various coefficients in
the master equations is summarized in Fig. 11.
Under the above assumption of a thermal equilibrium for noncondensed atoms,
we have
h¯ {nk }n0 nk exp(− T
k>0 νk nk )
nk n0 = (134)
.
h¯
{n }n exp(− T
k>0 νk nk )
k
0
In the next two sections we present different approximations that clarify the general result (132).
340
System coefficients
−1
Laser Gm = A[1 + B
A (m + 1)]
Lm = C loss
Physics
gain
A: Linear stimulated emission gain
B: Nonlinear saturation
C: Loss “through” mirrors
BEC: With cross excitations (CNB II)
Gm ⇒ Kn0 = [N + 1 − (n0 + 1)][1 + η]
Lm ⇒ Hn0 = H + (N − n0 )η
η = “cross-excitation” parameter
N atom cooling coefficient due to spontaneous emission of phonons,
adds atoms to condensate.
N atom heating coefficient due to phonon absorption from bath at temperature T removes atoms from condensate.
N atom cooling coefficient due to stimulated emission of phonon as
enhanced by atoms.
V.V. Kocharovsky et al.
BEC: Low temp limit (CNB I)
Gm ⇒ Kn0 = N + 1 − (n0 + 1)
Lm ⇒ Hn0 = k [eβεk − 1]−1 ≡ H
≈ N (T /Tc )3 weak trap
N atom heating coefficient due to absorption of phonon. Absorption
rate is enhanced by (N − n0 )η due to interaction with multiple phonons
(stimulated absorption) and increased absorption due to presence of
atoms.
F IG . 11. Physical interpretation of various coefficients in the master equations
[4
4]
FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
341
Short summary of this subsection is as follows. We introduce the probability of
having n0 atoms in the ground level and nk atoms in the kth level Pn0 ,n1 ,...,nk ,... .
We assume that the atoms in the excited levels with a given number of condensed
atoms n0 are in equilibrium state at the temperature T , then
Pn0 ,n1 ,...,nk ,... =
1
exp −β(E0 n0 + E1 n1 + · · · + Ek nk +) .
ZN
(135)
This equation yields
PN ≡ Pn0 =N,n1 =0,...,nk =0,... =
1
exp[−βE0 N ].
ZN
(136)
Assuming E0 = 0 we obtain the following expression for the partition function
ZN =
1
.
PN
(137)
We assume that Bose gas can exchange heat (but not particles) with a harmonicoscillator thermal reservoir. The reservoir has a dense and smooth spectrum. The
average occupation number of the heat bath oscillator at a frequency ωq = cq is
ηq =
1
.
exp(h¯ ωq /kB T ) − 1
(138)
The master equation for the distribution function of the condensed bosons pn0 ≡
ρn0 ,n0 takes the form
p˙ n0 = −
κkq nk n0 (ηq + 1) (n0 + 1)pn0 − n0 pn0 −1 Kool
k,q
−
κkq nk + 1n0 ηq n0 pn0 − (n0 + 1)pn0 +1 Heat .
(139)
k,q
The factors κkq embody the spectral density of the bath and the coupling strength
of the bath oscillators to the gas particles, and determine the rate of the condensate
evolution since there is no direct interaction between
the particles
of an ideal Bose
gas. Since κkq = κ · δ(h¯ Ωk − h¯ cq) the sum k,q reduces to k ,
1
p˙ n0 = −
nk n0 (ηk + 1) (n0 + 1)pn0 − n0 pn0 −1 Kool
κ
k
−
nk + 1n0 ηk n0 pn0 − (n0 + 1)pn0 +1 Heat .
k
Particle number constraint comes in a simple way:
k nk n0
= N − n¯ 0 .
(140)
342
V.V. Kocharovsky et al.
[4
4.4. L OW T EMPERATURE A PPROXIMATION
At low enough temperatures, the average occupations in the reservoir are small
and ηk + 1 1 in Eq. (131). This suggests the simplest approximation for the
cooling coefficient
Kn0 (141)
nk n0 = N − n¯ 0 .
k
In addition, at very low temperatures the number of noncondensed atoms is also
very small, we can therefore approximate nk n0 + 1 by 1 in Eq. (131). Then
the heating coefficient is a constant equal to the total average number of thermal
excitations in the reservoir at all energies corresponding to the energy levels of
the trap,
−1
eh¯ νk /T − 1 .
ηk =
Hn0 H, H ≡
(142)
k>0
k>0
Under these approximations, the condensate master equation (130) simplifies
considerably and contains only one nontrivial parameter H. We obtain
d
pn0 = −κ (N − n0 )(n0 + 1)pn0 − (N − n0 + 1)n0 pn0 −1
dt
+ H n0 pn0 − (n0 + 1)pn0 +1 .
(143)
It may be noted that Eq. (143) has the same form as Eq. (107) of motion for the
photon distribution function in a laser operating not too far above threshold. The
identification is complete if we define the gain, saturation, and loss parameters in
laser master equation by κ(N + 1), κ, and κH, respectively. The mechanism for
gain, saturation, and loss are however different in the present case.
A laser phase transition analogy exists via the P -representation [70,71]. The
steady-state solution of the Fokker–Planck equation for laser near threshold is [19]
1
B
A−C
∗
2
4
exp
|α| −
|α|
P (α, α ) =
(144)
N
A
2A
which clearly indicates a formal similarity between
ln P (α, α ∗ ) = − ln N + 1 − H/(N + 1) n0 − 1/2(N + 1) n20
(145)
for the laser equation and the Ginzburg–Landau type free energy [19,70,71]
G(n0 ) = ln pn0 ≈ const + a(T )n0 + b(T )n20 ,
(146)
where |α|2 = n0 , a(T ) = −(N − H)/N and b(T ) = 1/(2N ) for large N near Tc .
The resulting steady state distribution for the number of condensed atoms is
given by
pn0 =
1 HN−n0
,
ZN (N − n0 )!
(147)
4]
FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
343
where ZN= 1/pN is the partition function. It follows from the normalization
condition n0 pn0 = 1 that
ZN = eH #(N + 1, H)/N!,
(148)
∞ α−1 −t
where #(α, x) = x t
e dt is an incomplete gamma-function.
The distribution (147) can be presented as a probability distribution for the total
number of noncondensed atoms, n = N − n0 ,
Pn ≡ pN−n =
e−H N ! Hn
.
#(N + 1, H) n!
(149)
It looks somewhat like a Poisson distribution, however, due to the additional normalization factor, N !/ #(N + 1, H) = 1, and a finite number of admissible values
of n = 0, 1, . . . , N , it is not Poissonian. The mean value and the variance can be
calculated from the distribution (147) for an arbitrary finite number of atoms in
the Bose gas,
n0 = N − H + HN+1 /ZN N!,
n20 ≡ n20 − n0 2 = H 1 − n0 + 1 HN /ZN N ! .
(150)
(151)
As we shall see from the extended treatment in the next section, the approximations (141), (142) and, therefore, the results (150), (151) are clearly valid at
low temperatures, i.e., in the weak trap limit, T ε1 , where ε1 is an energy gap
between the first excited and the ground levels of a single-particle spectrum in
the trap. However, in the case of a harmonic trap the results (150), (151) show
qualitatively correct behavior for all temperatures, including T ε1 and T ∼ Tc
[52].
In particular, for a harmonic trap we have from Eq. (142) that the heating rate
is
1
ηk =
H=
exp[β hΩ(l
+
m + n)] − 1
¯
k
l,m,n
3
T
kB T 3
ζ (3) = N
.
≈
(152)
T
hΩ
¯
c
Thus, in the low temperature region the master equation (143) for the condensate
in the harmonic trap becomes
1
p˙ n0 = − (N + 1)(n0 + 1) − (n0 + 1)2 pn0 + (N + 1)n0 − n20 pn0 −1
κ
3
T
−N
(153)
n0 pn0 − (n0 + 1)pn0 +1 .
Tc
344
V.V. Kocharovsky et al.
[4
4.5. Q UASITHERMAL A PPROXIMATION FOR N ONCONDENSATE
O CCUPATIONS
At arbitrary temperatures, a very reasonable approximation for the average
noncondensate occupation numbers in the cooling and heating coefficients in
Eq. (131) is suggested by Eq. (134) in a quasithermal form,
(
(N − n¯ 0 )
nk n0 = ηk
(154)
nk n0
ηk = ε /T
,
(e k − 1)H
k>0
k
where εk = hν
Eq. (142). Equation (154)
¯ k , ηk is given by Eq. (125) and H by satisfies the canonical-ensemble constraint, N = n0 + k>0 nk , independently of
the resulting distribution pn0 . This important property is based on the fact that a
quasithermal distribution (154) provides the same relative average occupations in
excited levels of the trap as in the thermal reservoir, Eq. (125).
To arrive at the quasithermal approximation in Eq. (154) one can go along the
following logic. In the low temperature limit we assumed ηk 1 and took
nk n0 ηk + 1 ≈
nk n0 = N − n¯ 0 .
k
k
To go further, still in the low temperature limit, we can write
exp(−βEk )
nk n0 ≈ (N − n¯ 0 ) .
k exp(−βEk )
This is physically motivated since the thermal factor in [. . .] is the fraction of the
excited atoms in the state k, and N − n¯ 0 is the total number of excited atoms. Note
that
ηk =
1
exp(βEk ) − 1
⇒
exp(−βEk ) =
ηk
.
1 + ηk
Since we are at low temperature we take exp(−βEk ) ≈ ηk and therefore
ηk
(N − n¯ 0 )
=
,
nk n0 ≈ (N − n¯ 0 ) [exp(βEk ) − 1]H
k ηk
(155)
where
H=
ηk .
k
Now this ansatz is good for arbitrary temperatures. As a result,
nk n0 ηk + 1 ≈ (N − n0 )(1 + η),
k
(156)
4]
FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
345
where
η=
1
1 2
1 1
nk n0 ηk =
ηk =
.
(N − n0 )
H
H
[exp(βEk ) − 1]2
k
k>0
k>0
Another line of thought is the following:
n¯ k
.
nk n0 ≈ (N − n0 ) ¯k
kn
But by detailed balance in the steady state
κ(n¯ 0 + 1)n¯ k (η¯ k + 1) ≈ κ n¯ 0 (n¯ k + 1)η¯ k
and if the ground level is macroscopically
occupied then n¯ 0 ≈ n¯ 0 ± 1. Since
√
even at T = Tc one finds n¯ 0 ∼ N, one “always” has n¯ 0 1. Therefore,
n¯ k (η¯ k + 1) ≈ (n¯ k + 1)η¯ k and, hence, n¯ k ≈ η¯ k . As a result we again obtain
Eq. (155).
Calculation of the heating and cooling rates in this approximation is very simple. For example, for the heating rate we have
nk + 1n0 ηk =
ηk +
nk n0 ηk ≈ H + η(N − n0 ). (157)
k
k
k
In summary, the cooling and heating coefficients (131) in the quasithermal approximation of Eq. (154) are
Kn0 = (N − n0 )(1 + η),
Hn0 = H + (N − n0 )η.
(158)
Compared with the low temperature approximation (141) and (142), these coefficients acquire an additional contribution (N − n0 )η due to the cross-excitation
parameter
η=
1
1
1 ηk nk n0 =
.
ε
/T
k
N − n0
H
(e
− 1)2
k>0
(159)
k>0
4.6. S OLUTION OF THE C ONDENSATE M ASTER E QUATION
Now, at arbitrary temperatures, the condensate master equation (130) contains
two nontrivial parameters, H and η,
dpn0
= −κ (1 + η) (N − n0 )(n0 + 1)pn0 − (N − n0 + 1)n0 pn0 −1
dt
+ H + (N − n0 )η n0 pn0
− H + (N − n0 − 1)η (n0 + 1)pn0 +1 .
(160)
346
V.V. Kocharovsky et al.
[4
It can be rewritten also in the equivalent form
1 dpn0
= − (N + 1)(n0 + 1) − (n0 + 1)2 pn0
κ dt
+ (N + 1)n0 − n0 2 pn0 −1
− (T /Tc )3 N n0 pn0 − (n0 + 1)pn0 +1 .
(161)
The steady-state solution of Eq. (160) is given by
N−n0
1 (N − n0 + H/η − 1)!
η
ZN (H/η − 1)!(N − n0 )! 1 + η
N−n0
1
η
N − n0 + H
−1
η
=
,
N − n0
ZN
1+η
pn0 =
(162)
where the canonical partition function ZN = 1/pN is
ZN =
N−n0
N η
N − n0 + H/η − 1
.
N − n0
1+η
(163)
n0 =0
It is worth noting that the explicit formula (162) satisfies exactly the general relation between the probability distribution of the number of atoms in the ground
state, pn0 , and the canonical partition function [46], Eq. (79).
The master equation (160) for pn0 , and the analytic approximate expressions (162) and (163) for the condensate distribution function pn0 and the partition
function ZN , respectively, are among the main results of the condensate master
equation approach. As we shall see later, they provide a very accurate description
of the Bose gas for a large range of parameters and for different trap potentials.
Now we are able to present the key picture of the theory of BEC fluctuations,
that is the probability distribution pn0 , Fig. 12. Analogy with the evolution of the
photon number distribution in a laser mode (from thermal to coherent, lasing) is
obvious from a comparison of Fig. 12 and Fig. 6. With an increase of the number
of atoms in the trap, N , the picture of the ground-state occupation distribution
remains qualitatively the same, just a relative width of all peaks becomes more
narrow.
The canonical partition function (163) allows us to calculate also the microcanonical partition function Ω(E, N) by means of the inversion of the definition
in Eq. (74). Moreover, in principle, the knowledge of the canonical partition
function allows us to calculate all thermodynamic and statistical equilibrium properties of the system in the standard way (see, e.g., [13,61] and discussion in the
Introduction).
4]
FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
347
F IG . 12. Probability distribution of the ground-state occupation, pn0 , at the temperature T =
0.2Tc in an isotropic harmonic trap with N = 200 atoms as calculated from the solution of the
condensate master equation (130) in the quasithermal approximation, Eq. (162), (solid line) and from
the exact recursion relations in Eqs. (79) and (80) (dots).
Previously, a closed-form expression for the canonical partition function (74)
was known only for one-dimensional harmonic traps [68,69]
N
ZN (T ) =
k=1
1
.
1 − e−k h¯ ω/T
(164)
In the general case, there exists only the recursion relation (80) that is quite complicated, and difficult for analysis [46,61–63].
The distribution (162) can also be presented as a probability distribution for the
total number of noncondensed atoms, n = N − n0 ,
n
1
η
n + H/η − 1
Pn = pN−n =
(165)
.
n
ZN
1+η
The distribution (165) can be named as a finite negative binomial distribution,
since it has the form of the well-known negative binomial distribution [72],
n+M −1
q n (1 − q)M , n = 0, 1, 2, . . . , ∞,
Pn =
(166)
n
that was so named due to a coincidence of the probabilities Pn with the terms in
the negative-power binomial formula
∞ 1
n+M −1
=
(167)
q n.
n
(1 − q)M
n=0
348
V.V. Kocharovsky et al.
[4
It has a similar semantic origin as the well-known binomial distribution,
M
(1 − q)n q M−n ,
Pn =
n
which was named after a Newton’s binomial formula
M M M
q + (1 − q) =
(1 − q)n q M−n .
n
n=0
The finite negative binomial distribution (165) tends to the well-known distribution (166) only in the limit N (1 + η)H.
The average number of atoms condensed in the ground state of the trap is
n0 ≡
N
n0 pn0 .
(168)
n0 =0
It follows, on substituting for pn0 from Eq. (162), that
n0 = N − H + p0 η(N + H/η).
(169)
The central moments of the mth order, m > 1, of the number-of-condensedatom and number-of-noncondensed-atom fluctuations are equal to each other for
even orders and have opposite signs for odd orders,
¯ m.
(n0 − n¯ 0 )m = (−1)m (n − n)
(170)
The squared variance can be represented as
N
n(n − 1)Pn + n − n2
n20 = n2 − n2 =
(171)
n=0
and calculated analytically. We obtain
n20 = (1 + η)H − p0 (ηN + H)(N − H + 1 + η)
− p02 (ηN + H)2 ,
(172)
where
p0 =
N
1 (N + H/η − 1)!
η
ZN N !(H/η − 1)! 1 + η
(173)
is the probability that there are no atoms in the condensate.
All higher central moments
of the distribution Eq. (162) can be calculated ana
s
lytically using n0 s = N
n0 =0 n0 pn0 and Eqs. (162), (163). In particular, the third
central moment is
4]
FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
349
F IG . 13. The first four central moments for the ideal Bose gas in an isotropic harmonic
trap with N = 200 atoms as calculated via the solution of the condensate master equation
(solid lines—quasithermal approximation, Eq. (162); dashed lines—low temperature approximation,
Eq. (147)) and via the exact recursion relations in Eqs. (79) and (80) (dots).
3 n0 − n0 = −(1 + η)(1 + 2η)H
+ p0 (H + ηN ) 1 + (H − N )2 + 2 η2 + N (1 + η)
+ 3 η − H(1 + η)
+ 3p02 (H + ηN )2 (1 + η − H + N )
+ 2p03 (H + ηN )3 .
(174)
The first four central moments for the Bose gas in a harmonic trap with N =
200 atoms are presented in Fig. 13 as the functions of temperature in different
approximations.
For the “condensed phase” in the thermodynamic limit, the probability p0 vanishes exponentially if the temperature is not very close to the critical temperature.
In this case only the first term in Eq. (172) remains, resulting in
n20 = (1 + η)H ≡
(175)
nk 2 + nk .
k>0
350
V.V. Kocharovsky et al.
[4
This result was obtained earlier by standard statistical methods (see [13] and references therein).
It is easy to see that the result (165) reduces to the simple approximation (149)
in the formal limit η → 0, H/η → ∞, when
−n0
#(N − n0 + H/η)
H
(176)
.
→
#(N + H/η)
η
The limit applies to only very low temperatures, T ε1 . However, due to
Eqs. (142) and (159), the parameter H/η tends to 1 as T → 0, but never to
infinity. Nevertheless, the results (169) and (172) agree with the low temperature
approximation results (150) and (151) for T ε1 . In this case the variance n20
is determined mainly by a square root of the mean value n which is correctly
approximated by Eq. (150) as n ≡ N − n0 ≈ H.
4.7. R ESULTS FOR BEC S TATISTICS IN D IFFERENT T RAPS
As we have seen, the condensate fluctuations are governed mainly by two parameters, the number of thermal excitations H and the cross-excitation parameter η.
They are determined by a single-particle energy spectrum of the trap. We explicitly present them below for arbitrary power-law trap. We discuss mainly the
three-dimensional case. A generalization to other dimensions is straightforward
and is given in the end of this subsection. First, we discuss briefly the case of the
ideal Bose gas in a harmonic trap. It is the simplest case since the quadratic energy spectrum implies an absence of the infrared singularity in the variance of the
BEC fluctuations. However, because of the same reason it is not robust relative to
an introduction of a realistic weak interaction in the Bose gas as is discussed in
Section 5.
4.7.1. Harmonic Trap
The potential in the harmonic trap has, in general, an asymmetrical profile
in space, Vext (x, y, z) = m2 (x 2 ωx2 + y 2 ωy2 + z2 ωz2 ), with eigenfrequencies
{ωx , ωy , ωz } = ω, ωx ωy ωx > 0. Here m is the mass of the atom. The
single-particle energy spectrum of the trap,
εk = hkω
≡ h(k
¯
¯ x ωx + ky ωy + kz ωz ),
(177)
can be enumerated by three nonnegative integers {kx , ky , kz } = k, kx,y,z 0. We
have
1
1
H=
(178)
,
ηH =
.
h
kω/T
h
kω/T
¯
¯
e
−1
(e
− 1)2
k>0
k>0
4]
FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
351
The energy gap between the ground state and the first excited state in the trap is
equal to ε1 = h¯ ωx .
If the sums can be replaced by the integrals (continuum approximation), i.e., if
hω
¯ x T , the parameters H and ηH are equal to
3
T
T3
ζ (3) =
N,
H= 3
(179)
Tc
h¯ ωx ωy ωz
3
T3
T
ζ (2) − ζ (3)
ηH = 3
(180)
ζ (2) − ζ (3) =
N
,
Tc
ζ (3)
h¯ ωx ωy ωz
where a standard critical temperature is introduced as
ωx ωy ωz N 1/3
Tc = h¯
;
ζ (3) = 1.202 . . . ,
ζ (3)
ζ (2) =
π2
.
6
(181)
Therefore, the cross-excitation parameter η is a constant, independent of the temperature and the number of atoms, η = [ζ (2) − ζ (3)]/ζ (3) ≈ 0.37. The ratio
H/η = N(T /Tc )3 [ζ (3)/(ζ (2) − ζ (3))] goes to infinity in the thermodynamic
limit proportionally to the number of atoms N .
In the opposite case of very low temperatures, T h¯ ωx , we have
hω
h¯ ωz
h¯ ωx
¯ y
+ exp −
+ exp −
,
H ≈ exp −
(182)
T
T
T
2h¯ ωy
2h¯ ωz
2h¯ ωx
+ exp −
+ exp −
ηH ≈ exp −
(183)
T
T
T
with an exponentially good accuracy. Now the cross-excitation parameter η depends exponentially on the temperature and, instead of the number 0.37, is exponentially small. The ratio
¯ z 2
¯ x ) + exp(− ¯ y ) + exp(− hω
[exp(− hω
H
T
T
T )]
=
∼1
2h¯ ωy
2h¯ ωx
η
exp(− T ) + exp(− T ) + exp(− 2h¯Tωz )
hω
(184)
becomes approximately a constant. The particular case of an isotropic harmonic
trap is described by the same equations if we substitute ωx = ωy = ωz = ω.
4.7.2. Arbitrary Power-Law Trap
We now consider the general case of a d-dimensional trap with an arbitrary powerlaw single-particle energy spectrum [46,55,73]
εk = h¯
d
j =1
ωj kjσ ,
k = {kj ; j = 1, 2, . . . , d},
(185)
352
V.V. Kocharovsky et al.
[4
where kj 0 is a nonnegative integer and σ > 0 is an index of the energy
spectrum. We assume 0 < ω1 ω2 · · · ωd , so that the energy gap between
the ground state and the first excited state in the trap is ε1 = h¯ ω1 . We then have
H=
k>0
1
eεk /T
−1
ηH =
,
k>0
1
.
− 1)2
(eεk /T
(186)
In the case ε1 T , the sum can be replaced by the integral only for the parameter H (Eq. (186)) if d > σ ,
d/σ
d
T
d/σ
H = Aζ
(187)
=
N, d > σ,
T
σ
Tc
where the critical temperature is
σ/d
[#( 1 + 1)]d
N
Tc =
, A = &d σ
.
Aζ (d/σ )
( j =1 h¯ ωj )1/σ
(188)
The second parameter can be calculated by means of this continuum approximation only if 0 < σ < d/2,
d
d
d/σ
ηH = AT
−1 −ζ
ζ
σ
σ
d/σ
d
ζ ( − 1) − ζ ( σd )
T
=
(189)
, 0 < σ < d/2.
N σ
Tc
ζ ( σd )
If σ > d/2, it has a formal infrared divergence and should be calculated via a
discrete sum,
2
T
aσ,d
ηH =
(190)
N 2σ/d
, σ > d/2,
1
Tc
[#( σ + 1)]2σ [ζ ( σd )]2σ/d
where
aσ,d =
&
( dj =1 hω
¯ j )2/d
εk2
k>0
.
The traps with the dimension lower than the critical value, d σ , can be analyzed
on the basis of Eqs. (186) as well. We omit this analysis here since there is no
phase transition in this case.
The cross-excitation parameter η has different dependence on the number of
atoms for high, d > 2σ , or low, d < 2σ , dimensions,
η=
ζ ( σd − 1) − ζ ( σd )
ζ ( σd )
,
d > 2σ > 0,
(191)
4]
FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
η=
T
Tc
2−d/σ
N 2σ/d−1
aσ,d
[#( σ1
+ 1)]2σ [ζ ( σd )]2σ/d
,
d < 2σ.
353
(192)
Therefore, the traps with small index of the energy spectrum, 0 < σ < d/2, are
similar to the harmonic trap. The traps with larger index of the energy spectrum,
σ > d/2, are similar to the box with “homogeneous” Bose gas. For the latter
traps, the cross-excitation parameter η goes to infinity in the thermodynamic limit,
proportionally to N 2σ/d−1 . The ratio H/η goes to infinity in the thermodynamic
limit only for 0 < σ < d. In the opposite case, σ > d, it goes to zero. We obtain
d/σ
ζ ( σd )
H
T
, d > 2σ > 0,
N d
(193)
=
η
Tc
ζ ( σ − 1) − ζ ( σd )
2(d/σ −1)
2σ 2σ/d
T
1
d
H
−1
2(1−σ/d)
N
aσ,d
,
=
+1
#
ζ
η
Tc
σ
σ
d < 2σ.
(194)
It is remarkable that BEC occurs only for those spatial dimensions, d > σ , for
which H/η → ∞ at N → ∞. (We do not consider here the case of the critical dimension d = σ , e.g., one-dimensional harmonic trap, where a quasi-condensation
occurs at a temperature Tc ∼ h¯ ω1 N/ log N .) For spatial dimensions lower than
the critical value, d < σ , BEC does not occur (see, e.g., [46]). Interestingly, even
for the latter case there still exists a well-defined single peak in the probability distribution pn0 at low enough temperatures. With the help of the explicit formulas
in Section 3 we can describe this effect as well.
In the opposite case of very low temperatures, T ε1 , the parameters
H≈
d
j =1
e−
h¯ ωj
T
,
ηH ≈
d
j =1
e−
2h¯ ωj
T
,
ε1
η ∼ e− T
(195)
d
−(h¯ ωj −ε1 )/T ∼ d becomes a conare exponentially small. The ratio H
j =1 e
η ∼
stant.
Formulas (185)–(195) for the arbitrary power-law trap contain all particular
formulas for the three-dimensional harmonic trap (d = 3, σ = 1) and the box,
i.e., the “homogeneous gas” with d = 3 and σ = 2, as the particular cases.
In Fig. 13, numerical comparison of the results obtained from the exact recursion relations in Eqs. (79)–(80) and from our approximate explicit formulas from
Section 4 in the particular case of the ideal Bose gas in the three-dimensional
isotropic harmonic trap for various temperatures is demonstrated. The results indicate an excellent agreement between the exact results and the results based on
quasithermal approximation, including the mean value n0 , the squared variance
n20 as well as the third and fourth central moments. The low temperature approximation, Eq. (147), is good only at low temperatures. That is expected since
it neglects by the cross-excitation parameter η.
354
V.V. Kocharovsky et al.
[4
4.8. C ONDENSATE S TATISTICS IN THE T HERMODYNAMIC L IMIT
The thermodynamic, or bulk [13] limit implies an infinitely large number of
atoms, N → ∞, in an infinitely large trap under the condition of a fixed critical temperature, i.e., N ωx ωy ωz = const in the harmonic trap, L3 N = const in
&
the box, and N σ dj =1 ωj = const in an arbitrary d-dimensional power-law trap
with an energy spectrum index σ . Then, BEC takes place at the critical temperature Tc (for d > σ ) as a phase transition, and for some lower temperatures the
factor p0 is negligible. As a result, we have the following mean value and the
variance for the number of condensed atoms
1
n0 = N − H ≡ N −
(196)
,
eεk /T − 1
k>0
n20
= (1 + η)H ≡
1
k>0
eεk /T − 1
+
k>0
1
,
(eεk /T − 1)2
(197)
which agree with the results obtained for the ideal Bose gas for different traps in
the canonical ensemble by other authors [13,14,46,73–77]. In particular, we find
the following scaling of the fluctuations of the number of condensed atoms:
T d/σ
( Tc ) N, d > 2σ > 0
n20 ∼ C ×
(198)
, ε 1 T < Tc ,
( TTc )2 N 2σ/d , d < 2σ
"
σ/d
d
ωi
d
2
exp − &d
ζ
n0 ≈ n ≈
1/d
σ
[ j =1 ωj ]
i=1
σ
#
Tc
1
× #
(199)
+1
, T ε1 ,
σ
T N σ/d
where C is a constant. From Eq. (198), we see that in the high dimensional traps,
d > 2σ , e.g., in the three-dimensional harmonic trap, fluctuations display the
proper thermodynamic behavior, n20 ∝ N . However, fluctuations become anomalously large [46,55,74,78], n20 ∝ N 2σ/d N , in the low dimensional traps,
σ < d < 2σ . In the quantum regime, when the temperature is less than the
energy gap between the ground and the first excited level in the trap, it follows
from Eq. (199) that condensate fluctuations become exponentially small. For all
temperatures, when BEC exists (d > σ ), the root-mean-square fluctuations normalized
to the mean number of condensed atoms vanishes in the thermodynamic
limit: n20 /n0 → 0 as N → ∞.
Another remarkable property of the distribution function obtained in Section 4
is that it yields the proper mean value and variance of the number of atoms in the
ground level of the trap even for temperatures higher than the critical temperature.
In particular, it can be shown that its asymptotic for high temperatures, T Tc ,
4]
FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
355
yields the standard thermodynamic relation n0 ≈ n0 known from the analysis
of the grand canonical ensemble [13]. This nice fact indicates that the present
master equation approach to the statistics of the cool Bose gas is valuable in the
study of mesoscopic effects as well, both at T < Tc and T > Tc . Note that,
in contrast, the validity of the Maxwell’s demon ensemble approach [54] to the
statistics of BEC remains restricted to temperatures well below the onset of BEC,
T < Tc .
4.9. M ESOSCOPIC AND DYNAMICAL E FFECTS IN BEC
In recent experiments on BEC in ultracold gases [24–29], the number of condensed atoms in the trap is finite, i.e., mesoscopic rather than macroscopic,
N ∼ 103 –106 . Therefore, it is interesting to analyze mesoscopic effects associated with the BEC statistics.
The mean number of atoms in the ground state of the trap with a finite number
of atoms is always finite, even at high temperatures. However, it becomes macroscopically large only at temperatures lower than some critical temperature, Tc ,
that can be defined via the standard relation
ηk (Tc ) ≡ H(Tc ) = N.
(200)
This equation has an elementary physical meaning, namely it determines the temperature at which the total average number of thermal excitations at all energy
levels of the trap becomes equal to the total number of atoms in the trap. The results (162), (169), (172) shown in Fig. 13 explicitly describe a smooth transition
from a mesoscopic regime (finite number of atoms in the trap, N < ∞) to the
thermodynamic limit (N = ∞) when the threshold of the BEC becomes very
sharp so that we have a phase transition to the Bose–Einstein condensed state
at the critical temperature given by Eq. (200). This can be viewed as a specific
demonstration of the commonly accepted resolution to the Uhlenbeck dilemma in
his famous criticism of Einstein’s pioneering papers on BEC [6,9,10,12].
Although for systems containing a finite number of atoms there is no sharp
critical point, as is obvious from Figs. 3, 12, and 14, it is useful to define a critical
characteristic value of a temperature in such a case as well. It should coincide with
the standard definition (200) in the thermodynamic limit. Different definitions for
Tc were proposed and discussed in [11,66,79–85]. We follow a hint from laser
physics. There we know that fluctuations dominate near threshold. However, we
define a threshold inversion as that for which gain (in photon number for the lasing
mode) equals loss. Let us use a similar dynamical approach for BEC on the basis
of the master equation, see also [86].
We note that, for a laser operating near the threshold where B/A 1, the
equation (118) of motion for the probability pn of having n photons in the cavity
356
V.V. Kocharovsky et al.
[4
implies the following rate of the change for the average photon number:
d
(201)
n = (A − C)n − B (n + 1)2 + A.
dt
Here A, B, and C are the linear gain, nonlinear saturation, and linear loss coefficients, respectively. On neglecting the spontaneous emission term A and noting
that the saturation term B(n + 1)2 is small compared to (A − C)n near threshold, we define the threshold (critical) inversion to occur when the linear gain rate
equals the linear loss rate, i.e., A = C.
Similar to laser physics, the condensate master equation (130) implies a coupled hierarchy of moment equations which are useful in the analysis of time
evolution. In the quasithermal approximation (160), we find
l−1 i+1 i+2 dnl0 l − n0 − n0
=κ
(1 + η) N ni0 + ni+1
0
dt
i
i=0
+ (−1)l−i (H + ηN ) ni+1
− (−1)l−i η ni+2
.
0
0
(202)
Similar moment equations in the low-temperature approximation (143) follow
from Eq. (202) with η = 0,
l−1 i+2 dnl0 l i
− n0 + (−1)l−i H ni+1
.
=κ
N n0 + (N − 1) ni+1
0
0
dt
i
i=0
(203)
The dynamical equation for the first moment, as follows from Eq. (202), has the
following form:
dn0 (204)
= κ (1 + η)N + (N − 1 − η − H)n0 − n20 .
dt
Near the critical temperature, T ≈ Tc , the mean number of the condensed atoms
is small, n0 N , and it is reasonable to neglect the second moment n20 compared to N n0 and the spontaneous cooling (spontaneous emission in lasers) term
κ(1 + η)N compared to κN n0 . In this way, neglecting fluctuations, we arrive at
a simple equation for the competition between cooling and heating processes,
dn0 ≈ κ(N − H − η)n0 .
(205)
dt
In analogy with the laser threshold we can define the critical temperature, T = Tc ,
as a point where cooling equals heating, i.e., dn0 /dt = 0. This definition of the
critical temperature
H(Tc ) + η(Tc ) = N,
(206)
5]
FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
357
is valid even for mesoscopic systems and states that at T = Tc the rate of the
removal of atoms from the ground state equals to the rate of the addition, in the
approximation neglecting fluctuations. In the thermodynamic limit it corresponds
to the standard definition, Eqs. (181) and (188). For a mesoscopic system, e.g., of
N = 103 atoms in a trap, the critical temperature as given by Eq. (206) is about
few per cent shifted from the thermodynamic-limit value given by Eqs. (181)
and (188). Other definitions for Tc also describe the effect of an effective-Tc
shift [11,66,79–85], which is clearly seen in Fig. 14, and agree qualitatively with
our definition.
Note that precisely the same definition of the critical temperature follows from
a statistical mechanics point of view, which in some sense is alternative to the dynamical one. We may define the critical temperature as the temperature at which
the mean number of condensed atoms in the steady-state solution to the master equation vanishes when neglecting fluctuations and spontaneous cooling. We
make the replacement n20 ≈ n0 2 in Eq. (204) and obtain the steady-state solution to this nonlinear equation, n0 = N − H − η. Now we see that n0 vanishes at the same critical temperature (206). Finally, we remind again that a
precise definition of the critical temperature is not so important and meaningful
for the mesoscopic systems as it is for the macroscopic systems in the thermodynamic limit since for the mesoscopic systems, of course, there is not any sharp
phase transition and an onset of BEC is dispersed over a whole finite range of
temperatures around whatever Tc , as is clearly seen in Figs. 3, 12, and 14.
5. Quasiparticle Approach and Maxwell’s Demon Ensemble
In order to understand relations between various approximate schemes, we formulate a systematic analysis of the equilibrium canonical-ensemble fluctuations of
the Bose–Einstein condensate based on the particle number conserving operator
formalism of Girardeau and Arnowitt [87], and the concept of the canonicalensemble quasiparticles [20,21]. The Girardeau–Arnowitt operators can be interpreted as the creation and annihilation operators of the canonical-ensemble
quasiparticles which are essentially different from the standard quasiparticles
in the grand canonical ensemble. This is so because these operators create and
annihilate particles in the properly reduced many-body Fock subspace. In this
way, we satisfy the N -particle constraint of the canonical-ensemble problem in
Eq. (72) from the very beginning. Furthermore, we do this while taking into account all possible correlations in the N -boson system in addition to what one has
in the grand canonical ensemble. These canonical-ensemble quasiparticles fluctuate independently in the ideal Bose gas and form dressed canonical-ensemble
quasiparticles in the dilute weakly interacting Bose gas due to Bogoliubov coupling (see Section 6 below).
358
V.V. Kocharovsky et al.
[5
Such an analysis was elaborated in [20,21] and resulted in the explicit expressions for the characteristic function and all cumulants of the ground-state occupation statistics both for the dilute weakly interacting and ideal Bose gases. We
present it here, including the analytical formulas for the moments of the groundstate occupation fluctuations in the ideal Bose gases in an arbitrary power-law
trap, and, in particular, in a box (“homogeneous gas”) and in an arbitrary harmonic trap. In Section 6 we extend this analysis to the interacting Bose gas. In
particular, we calculate the effect of Bogoliubov coupling between quasiparticles
on suppression of the ground-state occupation fluctuations at moderate temperatures and their enhancement at very low temperatures and clarify a crossover
between ideal-gas and weakly-interacting-gas statistics which is governed by a
pair-correlation, squeezing mechanism. The important conclusion is that in most
cases the ground-state occupation fluctuations are anomalously large and are not
Gaussian even in the thermodynamic limit.
Previous studies focused mainly on the mean value, n¯ 0 , and squared variance,
(n0 − n¯ 0 )2 , of the number of condensed atoms.1 Higher statistical moments are
more difficult to calculate, and it was often assumed that the condensate fluctuations have vanishing higher cumulants (semi-invariants). That is, it was assumed
that the condensate fluctuations are essentially Gaussian with all central moments
determined by the mean value and the variance. We here show that this is not
true even in the thermodynamic limit. In, particular, we prove that in the general
case the third and higher cumulants normalized by the corresponding power of
the variance do not vanish even in the thermodynamic limit.
The results of the canonical-ensemble quasiparticle approach are valid for temperatures a little lower than a critical temperature, namely, when the probability
of having zero atoms in the ground state of the trap is negligibly small and the
higher-order effects of the interaction between quasiparticles are not important.
We outline also the Maxwell’s demon ensemble approximation introduced and
studied for the ideal Bose gas in [42,44,46,54,55] and show that it can be justified
on the basis of the method of the canonical-ensemble quasiparticles, and for the
case of the ideal Bose gas gives the same results.
This section is organized as follows: We start with the reduction of the Hilbert
space and the introduction of the canonical-ensemble quasiparticles appropriate to the canonical-ensemble problem in Section 5.1. Then, in Section 5.2, we
analytically calculate the characteristic function and all cumulants of the groundstate occupation distribution for the ideal Bose gas in a trap with an arbitrary
1 The only exception known to the authors is the paper [14] where the third moment of the groundstate occupation for the ideal gas in a harmonic trap was discussed in the Maxwell’s demon ensemble
approximation. Higher moments were also discussed on the basis of the master equation approach in
Refs. [52,53].
5]
FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
359
single-particle energy spectrum. We also discuss the Maxwell’s demon ensemble approach and compare it with the canonical-ensemble quasiparticle approach.
In Section 5.3 we apply these results to the case of an arbitrary d-dimensional
power-law trap which includes a three-dimensional box with periodic boundary
conditions (“homogeneous gas”) and a three-dimensional asymmetric harmonic
trap as the particular cases.
5.1. C ANONICAL -E NSEMBLE Q UASIPARTICLES IN THE R EDUCED
H ILBERT S PACE
In principle, to study the condensate fluctuations, we have to fix only the external
macroscopical and global, topological parameters of the system, like the number
of particles, temperature, superfluid flow pattern (“domain” or vortex structure),
boundary conditions, etc. We then proceed to find the condensate density matrix
via a solution of the von Neumann equation with general initial conditions admitting all possible quantum states of the condensate. In particular, this is a natural
way to approach the linewidth problem for the atom laser [88,89]. Obviously, this
is a complicated problem, especially for the interacting finite-temperature Bose
gas; because of the need for an efficient technique to account for the additional
correlations introduced by the constraint in such realistic ensembles. The latter
is the origin of the difficulties in the theory of the canonical or microcanonical ensembles (see discussion in Section 3.1). According to [32], a calculation
of equilibrium statistical properties using the grand canonical ensemble and a
perturbation series will be impossible since the series will have zero radius of
convergence.
One way out of this problem is to develop a technique which would allow us to
make calculations in the constrained many-body Hilbert space, e.g., on the basis
of the master equation approach as discussed in Section 4. Another possibility is
to solve for the constraint from the very beginning by a proper reduction of the
many-body Hilbert space so that we can work with the new, already unconstrained
quasiparticles. This approach is demonstrated in the present Section. Working in
the canonical ensemble, we solve for the fluctuations of the number of atoms
in the ground state in the ideal Bose gas in a trap (and similarly in the weakly
interacting Bose gas with the Bogoliubov coupling between excited atoms, see
Section 6). More difficult problems involving phase fluctuations of the condensate
with an accurate account of the quasiparticle renormalization due to interaction at
finite temperatures and the dynamics of BEC will be discussed elsewhere.
We begin by defining an occupation number operator in the many-body Fock
space as usual,
(n) (n) √
(n) (n+1) . (207)
nˆ k = aˆ k+ aˆ k , nˆ k ψk = nψk , aˆ k+ ψk = n + 1ψk
360
V.V. Kocharovsky et al.
[5
The particle number constraint (72) determines a canonical-ensemble (CE) subspace of the Fock space. Again we would like to work with the particle-number
conserving creation and annihilation operators. The latter are given in the Girardeau and Arnowitt paper [87],
βˆk+ = aˆ k+ βˆ0 ,
βˆk = βˆ0+ aˆ k ,
βˆ0 = (1 + nˆ 0 )−1/2 aˆ 0 .
(208)
These operators for k = 0 can be interpreted as describing new canonicalensemble quasiparticles which obey the Bose canonical commutation relations
on the subspace n0 = 0,
βˆk , βˆk+ = δk,k .
(209)
We are interested in the properties of the fraction of atoms condensed in the
ground level of the trap, k = 0. We focus on the important situation when the
ground-state occupation distribution is relatively well peaked, i.e., its variance is
much less than the mean occupation of the ground level of a trap,
1/2
n¯ 0 .
(n0 − n¯ 0 )2
(210)
In such a case, the relative role of the states with zero ground-state occupation,
n0 = 0, is insignificant, so that we can approximate the canonical-ensemble sub. Obviously, this approximation is valid only
space HCE by the subspace HnCE
0 =0
for temperatures T < Tc .
The physical meaning of the canonical-ensemble quasiparticles, βˆk = βˆ0+ aˆ k ,
is that they describe transitions between ground (k = 0) and excited (k = 0)
states. All quantum properties of the condensed atoms have to be expressed via
the canonical-ensemble quasiparticle operators in Eq. (208). In particular, we have
the identity
nˆ 0 = N −
(211)
nˆ k ,
k=0
where the occupation operators of the excited states are
nˆ k = aˆ k+ aˆ k = βˆk+ βˆk .
(212)
Note that in Refs. [90,91] quasiparticle operators similar in spirit to those of
Ref. [87] were introduced which, unlike βˆk , did not obey the Bose commutation
relations (209) exactly, if noncommutation of the ground-state occupation operators aˆ 0 and aˆ 0+ is important. As was shown by Girardeau [92], this is important
because the commutation corrections can accumulate in a perturbation series for
quantities like an S-matrix. Warning concerning a similar subtlety was stressed
some time ago [93].
We are interested in fluctuations in the number of atoms condensed in the
ground state of a trap, n0 . This is equal to the difference between the total number
5]
FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
361
of atoms in a trap and the number of excited atoms, n0 = N − n. In principle, a
condensed state can be defined via the bare trap states as their many-body mixture fixed by the interaction and external conditions. Hence, occupation statistics
of the ground as well as excited states of a trap is a very informative feature of
the BEC fluctuations. Of course, there are other quantities that characterize BEC
fluctuations, e.g., occupations of collective, dressed or coherent excitations and
different phases.
5.2. C UMULANTS OF BEC F LUCTUATIONS IN AN I DEAL B OSE G AS
Now we can use the reduced Hilbert space and the equilibrium canonicalensemble density matrix ρˆ to conclude that the occupation numbers of the
canonical-ensemble quasiparticles, nk , k = 0, are independent stochastic variables with the equilibrium distribution
ρk (nk ) = exp(−nk εk /T ) 1 − exp(−εk /T ) .
(213)
The statistical distribution of the number of excited atoms, n = k=0 nk , which
is equal, according to Eq. (211), to the number of noncondensed atoms, is a simple
“mirror” image of the distribution of the number of condensed atoms,
ρ(n) = ρ0 (n0 = N − n).
(214)
A useful way to find and to describe it is via the characteristic function
Θn (u) = Tr eiunˆ ρˆ .
(215)
Thus upon taking the Fourier transform of Θn (u) we obtain the probability distribution
π
1
ρ(n) =
(216)
e−iun Θn (u) du.
2π
−π
Taylor expansions of Θn (u) and log Θn (u) give explicitly initial (noncentral)
moments and cumulants, or semi-invariants [72,94]:
∞
um
dm
,
αm , αm ≡ nm =
Θ
(u)
Θn (u) =
(217)
n
m
m!
du
u=0
m=0
log Θn (u) =
∞
κm
m=1
Θn (u = 0) = 1.
(iu)m
m!
,
κm =
dm
d(iu)m
log Θn (u)
,
u=0
(218)
¯ m
The cumulants κr , initial moments αm , and central moments μm ≡ (n − n)
are related to each other by the simple binomial formulas [72,94] via the mean
362
V.V. Kocharovsky et al.
[5
number of the noncondensed atoms n¯ = N − n¯ 0 ,
r
r r
r
μr =
(−1)k
αr−k n¯ k , αr =
μr−k n¯ k ,
k
k
k=0
k=0
¯ 2 ≡ μ2 = κ2 , (n − n)
¯ 3 ≡ μ3 = κ3 ,
n¯ = κ1 , (n − n)
(n − n)
¯ 4 ≡ μ4 = κ4 + 3κ22 , (n − n)
¯ 5 ≡ μ5 = κ5 + 10κ2 κ3 ,
(n − n)
¯ 6 ≡ μ6 = κ6 + 15κ2 κ4 + κ22 + 10κ32 , . . . .
(219)
Instead of calculation of the central moments, μm = (n − n)
¯ m , it is more
convenient, in particular so for the analysis of the non-Gaussian properties, to
solve for the cumulants κm , which are related to the moments by simple binomial
expressions. The first six are
¯
κ1 = n,
κ2 = μ2 ,
κ5 = μ5 − 10μ2 μ3 ,
κ3 = μ3 ,
κ4 = μ4 − 3μ22 ,
κ6 = μ6 − 15μ2 μ4 − 2μ22 .
(220)
As discussed in detail below, the essence of the BEC fluctuations and the most
simple formulas are given in terms of the “generating cumulants” κ˜ m which are
related to the cumulants κm by the combinatorial formulas in Eq. (223),
κ1 = κ˜ 1 ,
κ2 = κ˜ 2 + κ˜ 1 ,
κ3 = κ˜ 3 + 3κ˜ 2 + κ˜ 1 ,
κ4 = κ˜ 4 + 6κ˜ 3 + 7κ˜ 2 + κ˜ 1 , . . . .
(221)
The main advantage of the cumulant analysis of the probability distribution
ρ(n) is the simple fact that the cumulant of a sum of independent stochastic
(k)
variables is equal to a sum of the partial cumulants,
r . This is a con& κr = k=0 κ
sequence of the equalities log Θn (u) = log k=0 Θnk (u) = k=0 log Θnk (u).
For each canonical-ensemble quasiparticle, the characteristic function can be easily calculated from the equilibrium density matrix as follows:
zk − 1
.
Θnk (u) = Tr eiunˆ k ρˆk = Tr eiunˆ k e−εk nˆ k /T 1 − e−εk /T =
zk − z
(222)
Here we introduced the exponential function of the single-particle energy spectrum εk , namely zk = exp(εk /T ), and a variable z = exp(iu) which has the
character of a “fugacity”. As a result, we obtain an explicit formula for the characteristic function and all cumulants of the number of excited (and, according to
the equation n0 = N − n, condensed) atoms in the ideal Bose gas in an arbitrary
trap as follows:
∞
∞
zk − 1 (eiu − 1)m
(iu)r
log
κ˜ m
κr
=
=
,
log Θn (u) =
zk − z
m!
r!
k=0
m=1
r=1
5]
FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
κ˜ m = (m − 1)!
e
k=0
εk /T
−1
−m
;
κr =
r
σr(m) κ˜ m .
363
(223)
m=1
Here we use the Stirling numbers of the 2nd kind [72],
m
∞
x
k
xn
1 m
σr(m) =
(−1)m−k
e − 1 = k!
σn(k) , (224)
kr ,
k
m!
n!
k=0
n=k
that yield a simple expression for the cumulants κr via the generating cumulants κ˜ m . In particular, the first four cumulants are given in Eq. (221).
Thus, due to the standard relations (219), the result (223) yields all moments
of the condensate fluctuations. Except for the average value, all cumulants are
independent of the total number of atoms in the trap; they depend only on the
temperature and energy spectrum of the trap. This universal temperature dependence of the condensate fluctuations was observed and used in [42,44,46,54,57]
to study the condensate fluctuations in the ideal Bose gas on the basis of the socalled Maxwell’s demon ensemble approximation. The method of the canonicalensemble quasiparticles also agrees with and provides further justification for the
“demon” approximation. The main point is that the statistics is determined by
numbers and fluctuations of the excited, noncondensed atoms which behave independently of the total atom number N for temperatures well below the critical
temperature since all “excess” atoms stay in the ground state of the trap. Therefore, one can calculate statistics in a formal limit as if we have an infinite number
of atoms in the condensate. That is we can say that the condensate plays the part of
an infinite reservoir for the excited atoms, in agreement with previous works [14,
42,44,46,54,57].
Obviously, our approximation in Eq. (210) as well as the Maxwell’s demon
ensemble approximation does not describe all mesoscopic effects that can be important very close to the critical temperature or for a very small number of atoms
in the trap. However, it takes into account the effect of a finite size of a trap via
the discreteness of the energy levels εk , i.e., in this sense the approximation (210)
describes not only the thermodynamic limit but also systems with a finite number of atoms N . In addition, the mesoscopic effects can be partially taken into
account by a “grand” canonical approximation for the probability distribution of
the canonical-ensemble quasiparticle occupation numbers
ρ˜k (nk ) = exp(−nk ε˜ k /T ) 1 − exp(−˜εk /T ) , ε˜ k = εk − μ,
(225)
where the chemical potential is related to the mean number of the condensed
atoms n¯ 0 = 1/(1 − exp(−βμ)) and should be found self-consistently from the
grand-canonical equation
−1
N − n¯ 0 =
(226)
eε˜ k /T − 1 .
k=0
364
V.V. Kocharovsky et al.
[5
The canonical-ensemble quasiparticle result for all cumulants remains the same
as is given by Eq. (223) above, with the only difference that now all quasiparticle
energies are shifted by a negative chemical potential (˜εk = εk − μ),
−m
eε˜ k /T − 1
, m = 1, 2, . . . ;
κ˜ m = (m − 1)!
k=0
κr =
r
σr(m) κ˜ m .
(227)
m=1
The first, m = 1, equation in Eq. (227) is a self-consistency equation (226).
The way it takes into account the mesoscopic effects (within this mean-number
“grand” canonical approximation) is similar to the way in which the selfconsistency equation (264) of the mean-field Popov approximation takes into
account the effects of weak atomic interaction. The results of this canonicalensemble quasiparticle approach within the “grand” canonical approximation for
the quasiparticle occupations (225) were discussed in Section 3 for the case of the
isotropic harmonic trap. Basically, the “grand” canonical approximation improves
only the result for the mean number of condensed atoms n¯ 0 (T ), but not for the
fluctuations.
5.3. I DEAL G AS BEC S TATISTICS IN A RBITRARY P OWER -L AW T RAPS
The explicit formulas for the cumulants demonstrate that the BEC fluctuations
depend universally and only on the single-particle energy spectrum of the trap, εk .
There are three main parameters that enter this dependence, namely,
(a) the ratio of the energy gap between the ground level and the first excited level
in the trap to the temperature, ε1 /T ,
(b) the exponent of the energy spectrum in the infrared limit, εk ∝ k σ at k → 0,
and
(c) the dimension of the trap, d.
The result (223) allows to easily analyze the condensate fluctuations in a general
case of a trap with an arbitrary dimension d 1 of the space and with an arbitrary
power-law single-particle energy spectrum [46,55,73]
εl = h¯
d
ωj ljσ ,
l = {lj ; j = 1, 2, . . . , d},
(228)
j =1
where lj 0 is a nonnegative integer and σ > 0 is an index of the energy
spectrum. The results for the particular traps with a trapping potential in the form
of a box or harmonic potential well can be immediately deduced from the general
case by setting the energy spectrum exponent to be equal to σ = 2 for a box and
5]
FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
365
σ = 1 for a harmonic trap. We assume that the eigenfrequencies of the trap are
ordered, 0 < ω1 ω2 · · · ωd , so that the energy gap between the ground
state and the first excited state in the trap is ε1 = h¯ ω1 . All cumulants (223) of the
condensate occupation fluctuations are given by the following formula:
κ˜ m = (m − 1)!
)
l≡(l1 ,...,ld )>0
κr =
r
d
h¯ ωj ljσ
exp
T
*−m
−1
,
j =1
σr(m) κ˜ m .
(229)
m=1
Let us consider first again the case of moderate temperatures larger than the
energy gap, ε1 T < Tc . The first cumulant, i.e., the mean number of noncondensed atoms, can be calculated by means of a continuum approximation of
the discrete sum by an integral if the space dimension of a trap is higher than a
critical value, d > σ . Namely, one has an usual BEC phase transition with the
mean value
d/σ
d
T
d/σ
=
N,
T
κ1 ≡ n¯ ≡ N − n¯ 0 = Aζ
σ
Tc
d > σ, ε1 T < Tc ,
(230)
where the standard critical temperature is
N
Tc =
Aζ (d/σ )
σ/d
,
[#( 1 + 1)]d
A = &d σ
.
( j =1 h¯ ωj )1/σ
(231)
The second-order generating cumulant can be calculated by means of this continuum approximation only if d > 2σ ,
d/σ
ζ ( d − 1) − ζ ( σd )
d
T
d
,
N σ
−1 −ζ
=
κ˜ 2 = AT d/σ ζ
σ
σ
Tc
ζ ( σd )
d > 2σ.
(232)
In the opposite case it has to be calculated via a discrete sum because of a formal
infrared divergence of the integral. Keeping only the main term in the expansion
of the exponent in Eq. (229), exp( Th¯ dj =1 ωj ljσ ) − 1 ≈ Th¯ dj =1 ωj ljσ , we find
"
σ σ/d #2
+ T σ/d
1
d
(2)
N
aσ,d ,
+1
#
ζ
κ˜ 2 =
Tc
σ
σ
σ < d < 2σ,
(233)
366
V.V. Kocharovsky et al.
(2)
where aσ,d =
[5
&d
¯ ωj )2/d /εl2 .
j =1 h
l>0 (
The ratio of the variance to the mean
√
number of noncondensed atoms is equal to κ2 /κ1 = κ˜ 1−1 + κ˜ 2 /κ˜ 12 , i.e.,
+ d/(2σ ) ! (n − n)
¯ 2
d
d
−1/2 Tc
ζ
=N
−1
ζ
,
n¯
T
σ
σ
d > 2σ,
(234)
(n − n)
¯ 2
=
n¯
σ < d < 2σ.
−2σ −2σ/d
2( d −1)
σ
1
d
1 Tc d/σ
2( σ −1) Tc
(2)
+N d
aσ,d #
,
+1
ζ
N T
T
σ
σ
(235)
We see that the traps with a relatively high dimension of the space, d > 2σ ,
produce normal thermodynamic fluctuations (234) ∝ N −1/2 and behave similar to the harmonic trap. However, the traps with a relatively low dimension of
the space, σ < d < 2σ , produce anomalously large fluctuations (235) in the
thermodynamic limit, ∝N σ/d−1 N −1/2 and behave similar to the box with
a homogeneous Bose gas, where there is a formal infrared divergence in the
continuum-approximation integral for the variance.
The third and higher-order central moments (n − n)
¯ m , or the third and higherorder cumulants κm , provide further parameters to distinguish different traps with
respect to their fluctuation behavior. The mth-order generating cumulant can be
calculated by means of the continuous approximation only if d > mσ ,
κ˜ m =
AT d/σ
#( σd )
d > mσ.
∞
0
d
t σ −1
dt =
t
(e − 1)m
T
Tc
d/σ
N
#( σd )ζ ( σd )
∞
0
d
t σ −1
dt,
t
(e − 1)m
(236)
In the opposite case we have to use a discrete sum because of a formal infrared
divergence of the integral. Again, keeping only the main
term in the expansion of
the exponent in Eq. (229), exp( Th¯ dj =1 ωj ljσ ) − 1 ≈ Th¯ dj =1 ωj ljσ , we find
"
σ σ/d #m
+ T σ/d
1
d
(m)
N
aσ,d ,
+1
#
ζ
κ˜ m =
Tc
σ
σ
σ < d < mσ,
(237)
&d
(m)
where aσ,d =
¯ ωj )m/d /εlm . (For the sake of simplicity, as in
l>0 ( j =1 h
Eq. (248), we again omit here a discussion of an obvious
logarithmic factor that
suppresses the ultraviolet divergence in the latter sum l>0 for the marginal case
d = mσ ; see, e.g., Eq. (250).)
We conclude that all cumulants up to the order m < d/σ have normal behavior,
κm ∝ N , but for the higher orders, m > d/σ , they acquire an anomalous growth
in the thermodynamic limit, κm κ˜ m ∝ N mσ/d . This result provides a simple
5]
FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
367
classification of the relative strengths of the higher-order fluctuation properties
of the condensate in different traps. In particular, it makes it obvious that for all
power–law traps with 1 < d/σ < 2 the condensate fluctuations are not Gaussian,
since
κm
m/2
κ2
∝ N 0 → const = 0 at N → ∞, m 3,
(238)
so that the asymmetry coefficient, γ1 ≡ (n0 − n¯ 0 )3 /(n0 − n¯ 0 )2 3/2 = 0, and
the excess coefficient, γ2 ≡ (n0 − n¯ 0 )4 /(n0 − n¯ 0 )2 2 − 3 = 0, are not zero.
Traps with d/σ > 2 show Gaussian condensate fluctuations, since all higherm/2
order cumulant coefficients κm /κ2 vanish, namely,
κm
m/2
κ2
κm
m/2
κ2
∝ N 1−m/2 → 0 at N → ∞ if 3 m <
σ
∝ N m( d − 2 ) → 0
1
at N → ∞ if m >
d
.
σ
d
,
σ
(239)
(240)
(For the sake of simplicity, we omit here an analysis of the special cases when
d/σ is an integer. It also can be done straightforwardly on the basis of the result (229).) Very likely, a weak interaction also violates this nonrobust property
and makes properties of the condensate fluctuations in the traps with a relatively
high dimension of the space, d/σ > 2, similar to that of the box with the homogeneous Bose gas (see Section 6 below), as it is stated below for the particular
case of the harmonic traps.
For traps with a space dimension lower than the critical value, d < σ , it is
known that a BEC phase transition does not exist (see, e.g., [46]). Nevertheless,
even in this case there still exists a well-peaked probability distribution ρ0 (n0 ) at
low enough temperatures, so that the condition (210) is satisfied and our general
result (223) describes this effect as well. In this case there is a formal infrared
divergence in the corresponding integrals for all cumulants (223), starting with
the mean value. Hence, all of them should be calculated as discrete sums. For
moderate temperatures we find approximately
−m
m d
T
ωj ljσ
κ˜ m (m − 1)!
h¯
l>0 j =1
d T m
∼ (m − 1)!
,
h¯ ωj
j =1
(241)
368
V.V. Kocharovsky et al.
[5
so that higher cumulants have larger values. In particular, the mean number of
noncondensed atoms is of the same order as the variance,
d
−1
d
T
T κ1 ≡ n¯ ≡ N − n¯ 0 ωj ljσ
∼
h¯
h¯ ωj
j =1
l>0 j =1
d
T 2
2
∼ n0 ∼ , d < σ.
(242)
h¯ ωj
j =1
Therefore, until
n¯ ∼ n20 N,
i.e.,
d
−1
d
, ,
1
σ
T Tc = N
∼N
,
hω
¯ j lj
h¯ ωj
j =1
l>0 j =1
(243)
there is a well-peaked condensate distribution, n20 n¯ 0 .
The marginal case of a trap with the critical space dimension, d = σ , is also
described by our result (229), but we omit its discussion in the present paper.
We mention only that there is also a formal infrared divergence, in this case a
logarithmic divergence, and, at the same time, it is necessary to keep the whole
exponent in Eq. (229); because otherwise in an approximation like (241) there
appears an ultraviolet divergence. The physical result is that in such traps, e.g., in
a one-dimensional harmonic trap, a quasicondensation of the ideal Bose gas takes
place at the critical temperature [43,46] Tc ∼ hω
¯ 1 N/ ln N .
For very low temperatures, T ε1 , the second and higher energy levels in
the trap are not thermally excited and atoms in the ideal Bose gas in any trap
go to the ground level with an exponential accuracy. This situation is also described by Eq. (229) that yields n¯ 0 N and proves that all cumulants of the
number-of-noncondensed-atom distribution become the same, since all higherorder generating cumulants exponentially vanish faster than κ1 ,
κm κ1 ≡ κ˜ 1 ,
κ˜ m (m − 1)!
d
e−mh¯ ωj /T ,
T ε1 .
(244)
j =1
The conclusion is that for the ideal Bose gas in any trap the distribution of the
number of noncondensed atoms becomes Poissonian at very low temperatures. It
means that the distribution of the number of condensed atoms is not Poissonian,
but a “mirror” image of the Poisson’s distribution. We see, again, that the complementary number of noncondensed atoms, n = N − n0 , is more convenient for
5]
FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
369
the characterization of the condensate statistics. A physical reason for this is that
the noncondensed atoms in different excited levels fluctuate independently.
The formulas (229)–(244) for the power-law traps contain all corresponding
formulas for a box (d = 3, σ = 2) and for a harmonic trap (d = 3, σ = 1) in a
3-dimensional space as particular cases. It is worth to stress that BEC fluctuations
in the ideal gas for the latter two cases are very different. In the box, if the temperature is larger than the trap energy gap, ε1 T < Tc , all cumulants, starting with
the variance, m 2, are anomalously large and dominated by the lowest energy
modes, i.e., formally infrared divergent,
κm ≈ κ˜ m ∝ (T /Tc )m N 2m/3 ,
m 2.
(245)
Only the mean number of condensed atoms,
n¯ 0 = N − n¯ = N − κ1 = N 1 − (T /Tc )3/2 ,
2/3
2π h¯ 2
N
,
Tc =
m
L3 ζ (3/2)
(246)
can be calculated
correctly via replacement of the discrete sum in Eq. (223) by an
∞
integral 0 . . . d 3 k. The correct value of the squared variance,
2
s4
2
4/3 T
,
κ2 ≡ (n0 − n¯ 0 ) = N
Tc π 2 (ζ (3/2))4/3
1
= 16.53,
s4 =
(247)
l4
l=0
can be calculated from Eq. (223) only as a discrete sum. Thus, for the box the condensate fluctuations are anomalous and non-Gaussian even in the thermodynamic
limit. To the contrary, for the harmonic trap with temperature much larger than
the energy gap, ε1 T < Tc , the condensate fluctuations are Gaussian in the
thermodynamic limit. This is because, contrary to the case of the homogeneous
gas, in the harmonic trap only the third and higher-order cumulants, m 3, are
lowest-energy-mode dominated, i.e., formally infrared divergent,
−3
κ3 ≈ κ˜ 3 = 2
eh¯ ωl/T − 1 ,
l>0
m T
(ωl)−m ∝ N m/3 ,
κm ≈ κ˜ m ≈ (m − 1)!
h¯
l>0
m > 3,
(248)
and they are small compared with an appropriate power of the variance squared
ζ (2) T 3
κ2 ≡ (n0 − n¯ 0 )2 =
(249)
N.
ζ (3) Tc
370
V.V. Kocharovsky et al.
[5
The asymmetry coefficient behaves as
γ1 ≡
κ3
3/2
κ2
≡
(n0 − n¯ 0 )3 log N
∝ 1/2 → 0,
(n0 − n¯ 0 )2 3/2
N
(250)
and all higher normalized cumulants (m 3) vanish in the thermodynamic limit,
N → ∞, as follows:
κm
m/2
κ2
∝
1
→ 0.
N m/6
(251)
It is important to realize that a weak interaction violates this nonrobust property
and makes properties of the condensate fluctuations in a harmonic trap similar to
that of the homogeneous gas in the box (see Section 6 below). For the variance,
the last fact was first pointed out in [78].
5.4. E QUIVALENT F ORMULATION IN T ERMS OF THE P OLES
OF THE G ENERALIZED Z ETA F UNCTION
Cumulants of the BEC fluctuations in the ideal Bose gas, Eq. (223), can be written in an equivalent form which is quite interesting mathematically (see [95]
and references therein). Namely, starting with the cumulant generating function
ln Ξex (β, z), where β = 1/kB T and z = eβμ ,
ln Ξex (β, z) = −
=
∞
ln 1 − z exp −β(εν − ε0 )
ν=1
∞ n
∞
z exp[−nβ(εν
n
ν=1 n=1
− ε0 )]
,
(252)
we use the Mellin–Barnes transform
e
−a
1
=
2πi
τ
+i∞
dt a −t #(t)
τ −i∞
to write
τ
+i∞
∞ n
∞ z 1
1
dt #(t)
ln Ξex (β, z) =
n 2πi
[nβ(εν − ε0 )]t
ν=1 n=1
=
1
2πi
τ −i∞
τ
+i∞
dt #(t)
τ −i∞
∞
ν=1
∞
1
zn
.
[β(εν − ε0 )]t
nt+1
n=1
(253)
5]
FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
371
Recalling the series representation of the Bose functions
gα (z) =
∞
zn /nα
n=1
and introducing the generalized, “spectral” Zeta function
−t
Z(β, t) = β(εν − ε0 ) ,
we arrive at the convenient (and exact) integral representation
ln Ξex (β, z) =
τ
+i∞
1
2πi
dt #(t)Z(β, t)gt+1 (z).
(254)
τ −i∞
d
gα (z) = gα−1 (z) and gα (1) = ζ (α),
Utilizing the well-known relations z dz
where ζ (α) denotes the original Riemann Zeta function, Eq. (254) may be written
in the appealing compact formula
∂ k
κk (β) = z
ln Ξex (β, z)
∂z
z=1
1
=
2πi
τ
+i∞
dt #(t)Z(β, t)ζ (t + 1 − k).
(255)
τ −i∞
Thus, by means of the residue theorem, Eq. (255) links all cumulants of the
canonical distribution in the condensate regime to the poles of the generalized
Zeta function Z(β, t), which embodies all the system’s properties, and to the pole
of a system-independent Riemann Zeta function, the location of which depends on
the order k of the respective cumulant. The formula (255) provides a systematic
asymptotic expansion of the cumulants κk (β) through the residues of the analytically continued integrands, taken from right to left. The large-system behavior is
extracted from the leading pole, finite-size corrections are encoded in the nextto-leading poles, and the non-Gaussian nature of the condensate fluctuations is
definitely seen. The details and examples of such analysis can be found in [95].
Concluding the Section 5, it is worthwhile to mention that previously only first
two moments, κ1 and κ2 , were analyzed for the ideal gas [13,46,53,57,73–77], and
the known results coincide with ours. Our explicit formulas provide a complete
answer to the problem of all higher moments of the condensate fluctuations in the
ideal gas. The canonical-ensemble quasiparticle approach, taken together with the
master equation approach gives a fairly complete picture of the central moments.
For the interacting Bose gas this problem becomes much more involved. We
address it in the next, last part of this review within a simple approximation that
takes into account one of the main effects of the interaction, namely, the Bogoliubov coupling.
372
V.V. Kocharovsky et al.
[6
6. Why Condensate Fluctuations in the Interacting Bose Gas
are Anomalously Large, Non-Gaussian, and Governed by
Universal Infrared Singularities?
In this section, following the Refs. [20,21], the analytical formulas for the statistics, in particular, for the characteristic function and all cumulants, of the
Bose–Einstein condensate in the dilute, weakly interacting gases in the canonical ensemble are derived using the canonical-ensemble quasiparticle method. We
prove that the ground-state occupation statistics is not Gaussian even in the thermodynamic limit. We calculate the effect of Bogoliubov coupling on suppression
of ground-state occupation fluctuations and show that they are governed by a paircorrelation, squeezing mechanism.
It is shown that the result of Giorgini, Pitaevskii and Stringari (GPS) [78] for
the variance of condensate fluctuations is correct, and the criticism of Idziaszek
and others [96,97] is incorrect. A crossover between the interacting and ideal
Bose gases is described. In particular, it is demonstrated that the squared variance
of the condensate fluctuations for the interacting Bose gas, Eq. (271), tends to a
half of that for the ideal Bose gas, Eq. (247), because the atoms are coupled in
strongly correlated pairs such that the number of independent degrees of freedom
contributing to the fluctuations of the total number of excited atoms is only 1/2 the
atom number N . This pair correlation mechanism is a consequence of two-mode
squeezing due to Bogoliubov coupling between k and −k modes. Hence, the fact
that the fluctuation in the interacting Bose gas is 1/2 of that in the ideal Bose gas
is not an accident, contrary to the conclusion of GPS. Thus, there is a deep (not
accidental) parallel between the fluctuations of ideal and interacting bosons.
Finally, physics and universality of the anomalies and infrared singularities of
the order parameter fluctuations for different systems with a long range order below a critical temperature of a second-order phase transition, including strongly
interacting superfluids and ferromagnets, is discussed. In particular, an effective
nonlinear σ model for the systems with a broken continuous symmetry is outlined and the crucial role of the Goldstone modes fluctuations combined with an
inevitable geometrical coupling between longitudinal and transverse order parameter fluctuations and susceptibilities in the constrained systems is demonstrated.
6.1. C ANONICAL -E NSEMBLE Q UASIPARTICLES
IN THE ATOM -N UMBER -C ONSERVING B OGOLIUBOV
A PPROXIMATION
We consider a dilute homogeneous Bose gas with a weak interatomic scattering
described by the well-known Hamiltonian [30]
6]
FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
H =
h¯ 2 k2
2M
k
aˆ k+ aˆ k +
1 k3 k4 |U |k1 k2 aˆ k+4 aˆ k+3 aˆ k2 aˆ k1 ,
2V
373
(256)
{ki }
where V = L3 is a volume of a box confining the gas with periodic boundary
conditions. The main effect of the weak interaction is the Bogoliubov coupling
between bare canonical-ensemble quasiparticles, βˆk = βˆ0+ aˆ k , via the condensate.
It may be described, to a first approximation, by a quadratic part of the Hamiltonian (256), i.e., by the atom-number-conserving Bogoliubov Hamiltonian [92]
HB =
N (N − 1)U0 h¯ 2 k2
(nˆ 0 + 1/2)Uk +
+
+
βˆk βˆk
2V
2M
V
k=0
+
1 +
Uk (1 + nˆ 0 )(2 + nˆ 0 ) βˆk+ βˆ−k
+ h.c. ,
2V
(257)
k=0
where we will make an approximation nˆ 0 n¯ 0 1, which is consistent with our
main assumption (210) of the existence of a well-peaked condensate distribution
function. Then, the Bogoliubov canonical transformation,
+
βˆk = uk bˆk + vk bˆ−k
;
βˆk+ = uk bˆk+ + vk bˆ−k ;
1
Ak
uk = , vk = ,
1 − A2k
1 − A2k
V
h¯ 2 k2
n¯ 0 Uk
Ak =
−
,
εk −
n¯ 0 Uk
2M
V
(258)
describes the condensate canonical-ensemble quasiparticles which have a “gapless” Bogoliubov energy spectrum and fluctuate independently in the approximation (257), since
εk bˆk+ bˆk ,
HB = E0 +
!
εk =
k=0
n¯ 0 Uk
h¯ 2 k2
+
2M
V
2
−
n¯ 0 Uk
V
2
.
(259)
In other words, we again have an ideal Bose gas although now it consists of
the dressed quasiparticles which are different both from the atoms and bare
(canonical-ensemble) quasiparticles introduced in Section 5. Hence, the analysis
of fluctuations can be carried out in a similar fashion to the case of the noninteracting, ideal Bose gas of atoms. This results in a physically transparent and
analytical theory of BEC fluctuations that was suggested and developed in [20,
21].
374
V.V. Kocharovsky et al.
[6
The only difference with the ideal gas is that now the equilibrium density matrix,
ˆ+ ˆ
ρˆk , ρˆk = e−εk bk bk /T 1 − e−εk /T ,
ρˆ =
(260)
k=0
is not diagonal in the bare atomic occupation numbers, the statistics of which we
are going to calculate. This feature results in the well-known quantum optics effect of squeezing of the fluctuations. The number of atoms with coupled momenta
k and −k is determined by the Bogoliubov coupling coefficients according to the
following equation:
+
+ ˆ
aˆ −k = βˆk+ βˆk + βˆ−k
aˆ k+ aˆ k + aˆ −k
β−k
2
+
+ ˆ b−k
= uk + vk2 bˆk bˆk + bˆ−k
+ +
ˆ
ˆ
ˆ
ˆ
+ 2uk vk bk b−k + bk b−k + 2vk2 .
(261)
6.2. C HARACTERISTIC F UNCTION AND ALL C UMULANTS OF BEC
F LUCTUATIONS
The characteristic function for the total number of atoms in the two, k and −k,
modes squeezed by Bogoliubov coupling is calculated in [20,21] as
2 ˆ+ ˆ ˆ+ ˆ
ˆ+ ˆ
ˆ+ ˆ
Θ±k (u) ≡ Tr eiu(βk βk +β−k β−k ) e−εk (bk bk +b−k b−k )/T 1 − e−εk /T
(z(Ak ) − 1)(z(−Ak ) − 1)
=
,
(z(Ak ) − eiu )(z(−Ak ) − eiu )
Ak − eεk /T
z(Ak ) =
(262)
.
Ak eεk /T − 1
The term “squeezing” originates from the studies of a noise reduction in quantum
optics (see the discussion after Eq. (278)).
The characteristic function for the distribution of the total number of the excited atoms
& is equal to the product of the coupled-mode characteristic functions,
(k, −k)-modes are indeΘn (u) = k=0,mod{±k} Θ±k (u), since different pairs of&
pendent to the first approximation (257). The product
runs over all different
pairs of (k, −k)-modes.
It is worth noting that by doing all calculations via the canonical-ensemble
quasiparticles (Section 5) we automatically take into account all correlations introduced by the canonical-ensemble constraint. As a result, similar to the ideal gas
(Eq. (223)), we obtain the explicit formula for all cumulants in the dilute weakly
interacting Bose gas,
1
1
1
+
,
κ˜ m = (m − 1)!
2
(z(Ak ) − 1)m
(z(−Ak ) − 1)m
k=0
6]
FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
κr =
r
σr(m) κ˜ m .
375
(263)
m=1
In comparison with the ideal Bose gas, Eq. (223), we have effectively a mixture
of two species of atom pairs with z(±Ak ) instead of exp(εk /T ).
It is important to emphasize that the first equation in (263), m = 1, is a nonlinear self-consistency equation,
N − n¯ 0 = κ1 (n¯ 0 ) ≡
1 + A2k eεk /T
k=0
(1 − A2k )(eεk /T − 1)
,
(264)
to be solved for the mean number of ground-state atoms n¯ 0 (T ), since the Bogoliubov coupling coefficient (258), Ak , and the energy spectrum (259), εk , are
themselves functions of the mean value n¯ 0 . Then, all the other equations in (263),
m 2, are nothing else but explicit expressions for all cumulants, m 2, if
one substitutes the solution of the self-consistency equation (264) for the mean
value n¯ 0 . The Eq. (264), obtained here for the interacting Bose gas (257) in
the canonical-ensemble quasiparticle approach, coincides precisely with the selfconsistency equation for the grand-canonical dilute gas in the so-called first-order
Popov approximation (see a review in [36]). The latter is well established as a reasonable first approximation for the analysis of the finite-temperature properties of
the dilute Bose gas and is not valid only in a very small interval near Tc , given
by Tc − T < a(N/V )1/3 Tc Tc , where a = MU0 /4π h¯ 2 is a usual s-wave
scattering length. The analysis of the Eq. (264) shows that in the dilute gas the
self-consistent value n¯ 0 (T ) is close to that given by the ideal gas model, Eq. (246),
and for very low temperatures goes smoothly to the value given by the standard
Bogoliubov theory [30,33,93] for a small condensate depletion, N − n¯ 0 N .
This is illustrated by Fig. 14a. (Of course, near the critical temperature Tc the
number of excited quasiparticles is relatively large, so that along with the Bogoliubov coupling (257) other, higher-order effects of interaction should be taken into
account to get a complete theory.) Note that the effect of a weak interaction on
the condensate fluctuations is very significant (see Fig. 14b–d) even if the mean
number of condensed atoms changes by relatively small amount.
6.3. S URPRISES : BEC F LUCTUATIONS ARE A NOMALOUSLY L ARGE AND
N ON -G AUSSIAN E VEN IN THE T HERMODYNAMIC L IMIT
According to the standard textbooks on statistical physics, e.g., [30,61,77,98],
any extensive variable Cˆ of a thermodynamic system has vanishing relative rootmean-square fluctuations. Namely, in the thermodynamic limit, a relative squared
¯ 2 /C¯ 2 = Cˆ 2 /C
ˆ 2 − 1 ∝ V −1 goes inversely proportional
variance (Cˆ − C)
376
V.V. Kocharovsky et al.
[6
F IG . 14. Temperature scaling of the first four cumulants, the mean value n¯ 0 /N = N − κ1 /N ,
√
1/3
the variance κ2 /N = (n0 − n¯ 0 )2 1/2 /N 1/2 , the third central moment −κ3 /N 1/2 =
2
3
1/3
1/2
1/4
1/2
4
(n0 − n¯ 0 ) /N , the fourth cumulant |κ4 | /N
= |(n0 − n¯ 0 ) − 3κ2 |/N 2 , of the
ground-state occupation fluctuations for the dilute weakly interacting Bose gas (Eq. (263)), with
U0 N 1/3 /ε1 V = 0.05 (thick solid lines), as compared with Eq. (223) (thin solid lines) and with the
exact recursion relation (80) (dot-dashed lines) for the ideal gas in the box; N = 1000. For the ideal
gas our results (thin solid lines) are almost indistinguishable from the exact recursion calculations
(dot-dashed lines) in the condensed region, T < Tc (N ). Temperature is normalized by the standard
thermodynamic-limit critical value Tc (N = ∞) that differs from the finite-size value Tc (N ), as is
clearly seen in graphs.
to the system volume V , or total number of particles N . This fundamental property originates from the presence of a finite correlation length ξ that allows us
to partition a large system into an extensive number V /ξ 3 of statistically independent subvolumes, with a finite variance in each subvolume. As a result, the
central limit theorem of probability theory yieldsa Gaussian distribution for the
¯ 2 ∼ V 1/2 . Possible
variable Cˆ with a standard scaling for variance, (Cˆ − C)
deviations from this general rule are especially interesting. It turns out that BEC
6]
FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
377
in a trap is one of the examples of such peculiar systems. A physical reason for the
anomalously large and non-Gaussian BEC fluctuations is the existence of the long
range order below the critical temperature of the second-order phase transition.
Let us show in detail that the result (263) implies, similar to the case of the
ideal homogeneous gas (Section 5), that the ground-state occupation fluctuations
in the weakly interacting gas are not Gaussian in the thermodynamic limit, and
anomalously large. The main fact is that the anomalous contribution to the BEC
fluctuation cumulants comes from the modes which have the most negative Bogoliubov coupling coefficient Ak ≈ −1 since in this case one has z(Ak ) → 1, so
that this function produces a singularity in the first term in Eq. (263). (The second
term in Eq. (263) never makes a singular contribution since always z(−Ak ) < −1
or z(−Ak ) > exp(εk /T ).) These modes, with Ak ≈ −1, exist only if the interaction energy g = n¯ 0 Uk /V in the Bogoliubov Hamiltonian (257) is larger than
the energy gap between the ground state and the first excited states in the trap,
1
, i.e.,
ε1 = ( 2πLh¯ )2 2M
g
2a n¯ 0
(265)
≡
1,
ε1
πL
and are the infrared modes with the longest wavelength of the trap energy spectrum. In terms of the scattering length for atom–atom collisions, a = MUk /4π h¯ 2 ,
the latter condition (265) coincides with a familiar condition for the Thomas–
Fermi regime, in which the interaction energy is much larger then the atom’s
kinetic energy.
Let us use a representation which is obvious from Eqs. (258) and (259),
2εk
Ak (1 − Ak )
εk
εk
1+
tanh
,
≈
z(Ak ) − 1 ≡
(266)
g(1 − Ak )
1 + eεk /T
g
2T
where in the last, approximate equality we set Ak ≈ −1, and neglect the contribution from the second term in Eq. (263), assuming that the singular contribution
from the modes with Ak ≈ −1 via the first term in Eq. (263) is dominant. Then,
for all infrared-dominated cumulants of higher orders m 2, the result (263)
reduces to a very transparent form
1
1
κ˜ m ≈ (m − 1)!
(267)
εk
εk m .
2
[ g tanh( 2T
)]
k=0
Finally, using the Bogoliubov energy spectrum in Eq. (259),
εk = ε1 l4 + (2g/ε1 )l2
with a set of integers l = (lx , ly , lz ), we arrive to the following simple formulas
for the higher-order generating cumulants in the Thomas–Fermi regime (265):
m 1
1
T
κ˜ m ≈ (m − 1)!
, g 2T 2 /ε1 ; ε1 g T , (268)
2m
2
ε1
l
l=0
378
V.V. Kocharovsky et al.
[6
1
1
g m/2 ,
κ˜ m ≈ (m − 1)!
√
2
2ε1
[|l| tanh(|l| 2gε1 /T )]m
l=0
2T /ε1 g T ; g ε1 ,
1
g m/2 1
,
κ˜ m ≈ (m − 1)!
2
2ε1
|l|m
2
(269)
g 2T 2 /ε1 ;
g ε1 ,
(270)
l=0
for high, moderate, and very low temperatures T , respectively, as √
compared to the
geometrical mean of the interaction and gap energies in the trap, gε1 /2.
In particular, for the Thomas–Fermi regime (265) and relatively high temperatures, the squared variance, as given by Eq. (268),
1
1
1
1
(n0 − n¯ 0 )2 =
+
+
2
z(Ak ) − 1
(z(Ak ) − 1)2
(z(−Ak ) − 1)2
k=0
1
+
z(−Ak ) − 1
→
N 4/3 (T /Tc )2 s4
2π 2 (ζ (3/2))4/3
(271)
scales as (n0 − n¯ 0 )2 ∝ N 4/3 . Here the arrow indicates the limit of sufficiently
strong interaction, g ε1 and T g ε1 .
The behavior (271), (245), and (275) is essentially different from that of the normal fluctuations of most extensive physical observables, which are Gaussian with
the squared variance proportional to N. The only exception for the Eqs. (267)–
(270) is the low temperature limit of the variance that is not infrared-dominated
and should be calculated not via the Eq. (270), but directly from Eq. (263) using the fact that all modes are very poorly occupied at low temperatures, i.e.,
exp(−εk /T ) 1, if g 2T 2 /ε1 , ε1 . Thus, we immediately find
κ2 = κ˜ 2 + κ˜ 1 ≈
2A2k
(1 − A2k )2
k=0
3/2
=
1 g2
g2
≈
2π
2
εk2
ε12
k=0
3/2
∞
0
dr
r 2 + 2g/ε1
√ a n¯ 0
g
=2 π
.
(272)
L
2 ε1
The above results extend and confirm the result of the pioneering paper [78]
where only the second moment, (n0 − n¯ 0 )2 , was calculated. (The result of [78]
was rederived by a different way in [99], and generalized in [100].) The higherorder cumulants κm , m > 2, given by Eqs. (263) and (267)–(270), are not zero,
do not go to zero in the thermodynamic limit and, moreover, are relatively large
compared with the corresponding exponent of the variance (κm )m/2 that proves
and measures the non-Gaussian character of the condensate fluctuations. For the
π2
=√
6]
FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
379
Thomas–Fermi regime (265) and relatively high temperatures, the relative values
of the higher-order cumulants are given by Eq. (268) as
1
κm
s2m
m/2−1
(273)
≈
2
(m
−
1)!
,
s
=
,
2m
(κ2 )m/2
(s4 )m/2
l2m
l=0
where l = (lx , ly , lz ) are integers, s4 ≈ 16.53, s6 ≈ 8.40, and s2m ≈ 6 for m 1.
In particular, the asymmetry coefficient of the ground-state occupation probability
distribution
,
3/2
γ1 = (n0 − n¯ 0 )3 (n0 − n¯ 0 )2
√
3/2
= −κ3 /κ2 ≈ 2 2 s6 /(s4 )3/2 ≈ 0.35
(274)
is not very small at all.
This non-Gaussian statistics stems from an infrared singularity that exists for
the fluctuation cumulants κm , m 2, for the weakly interacting gas in a box
despite of the acoustic (i.e., linear, like for the ideal gas in the harmonic trap) behavior of the Bogoliubov–Popov energy spectrum (259) in the infrared limit. The
reason is that the excited mode squeezing (i.e., linear mixing of atomic modes)
via Bogoliubov coupling affects the BEC statistics (263) for the interacting gas
also directly (not only via a modification of the quasiparticle energy spectrum),
namely, via the function (262), z(Ak ), which is different from a simple exponent
of the bare energy, exp(εk /T ), that enters into the corresponding formula for the
noninteracting gas (223).
6.4. C ROSSOVER BETWEEN I DEAL AND I NTERACTION -D OMINATED BEC:
Q UASIPARTICLES S QUEEZING AND PAIR C ORRELATION
Now we can use the analytical formula (263) to describe explicitly a crossover
between the ideal-gas and interaction-dominated regimes of the BEC fluctuations.
Obviously, if the interaction energy g = n¯ 0 Uk /V is less than the energy gap
between the ground state and the first excited state in the trap, g < ε1 , the Bogoliubov coupling (258) becomes small for all modes, |Ak | 1, so that both terms
in Eq. (263) give similar contributions and all fluctuation cumulants κ˜ m tend to
their ideal gas values in the limit of vanishing interaction, g ε1 . Namely, in the
near-ideal gas regime n¯ 0 a/L 1 the squared variance linearly decreases from
its ideal-gas value with an increase of the weak interaction as follows:
N 4/3 (T /Tc )2 s4
2
2 s6 g
(n0 − n¯ 0 ) ≈ 2
1−π
s4 ε1
π (ζ (3/2))4/3
4/3
2
N (T /Tc ) s4
n¯ 0 a
= 2
(275)
.
1
−
3.19
L
π (ζ (3/2))4/3
380
V.V. Kocharovsky et al.
[6
With a further increase of the interaction energy over the energy gap in the trap,
g > ε1 , the essential differences between the weakly interacting and ideal gases
appear. First, the energy gap is increased by the interaction, that is
2π h¯ 2 1
2
ε˜ 1 = ε1 + 2ε1 n¯ 0 (T )U0 /V > ε1 =
(276)
,
L
2M
so that the border T ∼ ε˜ 1 between the moderate temperature√and very low temperature regimes is shifted to a higher temperature, T ∼√ gε1 /2. Thus, the
interaction strength g determines also the temperature T ∼ gε1 /2 above which
another important effect of the weak interaction comes into play. Namely, according to Eqs. (267)–(270), the suppression of all condensate-fluctuation cumulants
by a factor of 1/2, compared with the ideal gas values (see Eq. (247)), takes place
for moderate temperatures, ε˜ 1 T < Tc , when a strong coupling (Ak ≈ −1)
contribution dominates in Eq. (263). The factor 1/2 comes from the fact that,
according to Eq. (262),
z(Ak = −1) = 1,
z(Ak = 1) = −1,
so that the first term in Eq. (263) is resonantly large but the second term is relatively small. In this case, the effective energy spectrum, which can be introduced
for the purpose of comparison with the ideal gas formula (223), is
1
1
k2
(277)
= T ln z(Ak ) (1 + Ak )εk εk2 V /U0 n¯ 0 .
2
2
2M
That is, the occupation of a pair of strongly coupled modes in the weakly interacting gas can be characterized by the same effective energy spectrum as that of a free
eff
atom. It is necessary to emphasize that the effective energy εk = T ln(z(Ak )),
introduced in Eq. (277), describes only the occupation of a pair of bare atom
excitations k and −k (see Eqs. (260)–(262)) and, according to Eq. (263), the
ground-state occupation. That is, it would be wrong to reduce the analysis of
the thermodynamics and, in particular, the entropy of the interacting gas to this
effective energy. Thermodynamics is determined by the original energy spectrum
of the dressed canonical-ensemble quasiparticles, Eq. (259).
This remarkable property explains why the ground-state occupation fluctuations in the interacting gas in this case are anomalously large to the same extent
as in the noninteracting gas except factor of 1/2 suppression in the cumulants of
all orders. These facts were considered in [78] to be an accidental coincidence.
We see now that, roughly speaking, this is so because the atoms are coupled in
strongly correlated pairs such that the number of independent stochastic occupation variables (“degrees of freedom”) contributing to the fluctuations of the total
number of excited atoms is only 1/2 the atom number N . This strong pair correlation effect is clearly seen in the probability distribution of the total number of
eff
εk
6]
FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
381
F IG . 15. The probability distribution (278) P2 as a function of the number of atoms in k and −k
modes, nk + n−k , for the interaction energy U0 n¯ 0 /V = 103 εk and temperature T = εk . The pair
correlation effect due to Bogoliubov coupling in the weakly interacting Bose gas is clearly seen for
low occupation numbers, i.e., even occupation numbers nk + n−k are more probable than odd ones.
atoms in the two coupled k and −k modes,
[z(Ak ) − 1][z(−Ak ) − 1]
P2 (nk + n−k ) =
z(−Ak ) − z(Ak )
nk +n−k +1 1
−
z(−Ak )
1
z(Ak )
nk +n−k +1
(278)
(see Fig. 15). The latter formula follows from Eq. (262). Obviously, a higher probability for even occupation numbers nk + n−k as compared to odd numbers at
low occupations means that the atoms in the k and −k modes have a tendency
to appear or disappear simultaneously, i.e., in pairs. This is a particular feature
of the well-studied in quantum optics phenomenon of two-mode squeezing (see,
e.g., [18,37]). This squeezing means a reduction in the fluctuations of the population difference nk − n−k , and of the relative phases or so-called quadrature-phase
amplitudes of an interacting state of two bare modes aˆ k and aˆ −k compared with
their appropriate uncoupled state, e.g., coherent or vacuum state. The squeezing
is due to the quantum correlations which build up in the bare excited modes via
Bogoliubov coupling (258) and is very similar to the noise squeezing in a nondegenerate parametric amplifier studied in great details by many authors in quantum
optics in 80s [18,37]. We note that fluctuations of individual bare excited modes
are not squeezed, but there is a high degree of correlation between occupation
numbers in each mode.
It is very likely that in the general case of an arbitrary power-law trap the interaction also results in anomalously large fluctuations of the number of ground-state
382
V.V. Kocharovsky et al.
[6
atoms, and a formal infrared divergence due to excited mode squeezing via Bogoliubov coupling and renormalization of the energy spectrum. In the particular
case of the isotropic harmonic trap this was demonstrated in [78] for the variance
of the condensate fluctuations. Therefore, the ideal gas model for traps with a
low spectral index σ < d/2 (such as a three-dimensional harmonic trap where
σ = 1 < d/2 = 3/2), showing Gaussian, normal thermodynamic condensate
fluctuations with the squared variance proportional to N instead of anomalously
large fluctuations (see Eqs. (232), (234) and (239), (240)), is not robust with respect to the introduction of a weak interatomic interaction.
At the same time, the ideal gas model for traps with a high spectral index σ >
d/2 (e.g., for a three-dimensional box with σ = 2 > d/2 = 3/2) exhibits nonGaussian, anomalously large ground-state occupation fluctuations with a squared
variance proportional to N 2σ/d N (see Eqs. (233), (234)) similar to those found
for the interacting gas. Fluctuations in the ideal Bose gas and in the Bogoliubov
Bose gas differ by a factor of the order of 1, which, of course, depends on the
trap potential and is equal to 1/2 in the particular case of the box, where n20 ∝
N 4/3 . We conclude that, contrary to the interpretation formulated in [78], similar
behavior of the condensate fluctuations in the ideal and interacting Bose gases in
a box is not accidental, but is a general rule for all traps with a high spectral index
σ > d/2, or a relatively low dimension of space, d < 2σ .
As follows from Eq. (263), the interaction essentially modifies the condensate
fluctuations also at very low temperatures, T ε˜ 1 (see Fig. 14). Namely, in the
interacting Bose gas a temperature-independent quantum noise,
κ˜ m → κ˜ m (T = 0) = 0,
m 2,
(279)
additional to vanishing (at T → 0) in the ideal Bose gas noise, appears due to
quantum fluctuations of the excited atoms, which are forced by the interaction to
occupy the excited levels even at T = 0, so that n¯ k (T = 0) = 0. Thus, in the
limit of very low temperatures the results of the ideal gas model (Section 5) are
essentially modified by weak interaction and do not describe condensate statistics
in the realistic weakly interacting Bose gases.
The temperature scaling of the condensate fluctuations described above is depicted in Fig. 14 both for the weakly interacting and ideal gases. A comparison
with the corresponding quantities calculated numerically from the exact recursion relation in Eqs. (79) and (80) for the ideal gas in a box is also indicated. It
is in good agreement with our approximate analytical formula (263) for all temperatures in the condensed phase, T < Tc , except of a region near to the critical
temperature, T ≈ Tc . It is worth stressing that the large deviations of the asymmetry coefficient, γ1 = (n0 − n¯ 0 )3 /(n0 − n¯ 0 )2 3/2 , and of the excess coefficient,
γ2 = (n0 − n¯ 0 )4 /(n0 − n¯ 0 )2 2 − 3 from zero, which are of the order of 1 at
T ∼ Tc /2 or even more at T ≈ 0 and T ≈ Tc , indicate how far the ground-state
occupation fluctuations are from being Gaussian. (In the theory of turbulence,
6]
FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
383
the coefficients γ1 and γ2 are named as skewness and flatness, respectively.) This
essentially non-Gaussian behavior of the ground-state occupation fluctuations remains even in the thermodynamic limit.
Mesoscopic effects near the critical temperature are also clearly seen in Fig. 14,
and for the ideal gas are taken into account exactly by the recursion relations (79),
(80). The analytical formulas (263) take them into account only via a finite-size
effect, L ∝ N −1/3 , of the discreteness of the single-particle spectrum εk . This
finite-size (discreteness) effect produces, in particular, some shift of the characteristic BEC critical temperature compared to its thermodynamic-limit value, Tc . For
the case shown in Fig. 14, it is increased by a few per cent. Similarly to the meannumber “grand” canonical approximation described at the end of Section 3.2 for
the case of the ideal gas, the canonical-ensemble quasiparticle approach can partially accommodates for the mesoscopic effects by means of the grand-canonical
shift of all quasiparticle energies by a chemical potential, ε˜ k = εk −μ. In this case
the self-consistency equation (264) acquires an additional nonlinear contribution
due to the relation exp(−βμ) = 1 + 1/n¯ 0 . Obviously, this “grand” canonical approximation works only for T < Tc , whereas at T > Tc we can use the standard
grand canonical approach, since the ground-state occupation is not macroscopic
above the critical temperature. However, the “grand” canonical approach takes
care only of the mean number of condensed atoms n¯ 0 , and does not improve the
results of the canonical-ensemble quasiparticle approach for BEC fluctuations.
6.5. U NIVERSAL A NOMALIES AND I NFRARED S INGULARITIES
OF THE O RDER PARAMETER F LUCTUATIONS IN THE S YSTEMS
WITH A B ROKEN C ONTINUOUS S YMMETRY
The result that there are singularities in the central moments of the condensate
fluctuations, emphasized in [20,21] and discussed above in detail for the BEC in
a trap, can be generalized for other long-range ordered systems below the critical temperature of a second-order phase transition, including strongly interacting
systems. That universality of the infrared singularities was discussed in [99,100]
and can be traced back to a well-known property of an infrared singularity in
a longitudinal susceptibility χ (k) of such systems [30,79,101]. The physics of
these phenomena is essentially determined by long wavelength phase fluctuations, which describe the noncondensate statistics, and is intimately related to the
fluctuation–dissipation theorem, Bogoliubov’s 1/k 2 theorem for the static susceptibility χϕϕ (k) of superfluids, and the presence of Goldstone modes, as will be
detailed in the following.
384
V.V. Kocharovsky et al.
[6
6.5.1. Long Wavelength Phase Fluctuations, Fluctuation–Dissipation Theorem,
and Bogoliubov’s 1/k 2 Theorem
First, let us refer to a microscopic derivation given in [102] that demonstrates the
fact that the phase fluctuations alone dominate the low-energy physics. Also, let
us assume that, as was shown by Feynman, the spectrum of excitations in the
infrared limit k → 0 is exhausted by phonon-like modes with a linear dispersion
ωk = ck, where c is an actual velocity of sound. Then, following a textbook [30],
we can approximate an atomic field operator via a phase fluctuation operator as
follows:
ˆ
,
Ψˆ (x) = n˜ 0 ei ϕ(x)
(h¯ k)−1/2 cˆk eikx + cˆk+ e−ikx ,
ϕ(x)
ˆ
= (mc/2V ntot )1/2
(280)
k=0
where n˜ 0 is the bare condensate density, ntot the mean particle density, m the particle mass, cˆk and cˆk+ are phonon annihilation and creation operators, respectively.
An omission of the k = 0 term in the sum in Eq. (280) can be rigorously justified on the basis of the canonical-ensemble quasiparticle approach (see Section 5)
and is related to the fact that a global phase factor is irrelevant to any observable
gauge-invariant quantity. In a homogeneous superfluid, the renormalized condensate density is determined by the long range order parameter,
n¯ 0 = lim Ψˆ + (x)Ψˆ (0) ≈ n˜ 0 exp − ϕˆ 2 (0) ,
(281)
x→∞
if we use the approximation (280) and neglect phonon interaction. Thus, the mean
number of condensate particles is depleted with an increase of temperature in
accordance with an increase of the variance of the phase fluctuations, n¯ 0 (T ) ∝
exp(−ϕˆ 2 (0)T ), which yields a standard formula [30] for the thermal depletion,
n¯ 0 (T ) − n¯ 0 (T = 0) = −n¯ 0 (T = 0)m(kB T )2 /12ntot ch¯ 3 .
Similar to the mean value, higher moments of the condensate fluctuations of the
homogeneous superfluid in the canonical ensemble can be represented in terms of
the phase fluctuations of the noncondensate via the operator
nˆ = N/V − nˆ 0 = Ψˆ + (x)Ψˆ (x) d 3 x,
(282)
where
Ψˆ (x) = Ψˆ (x) −
2
n¯ 0 (T ) = n˜ 0 ei ϕˆ (x) − e−ϕˆ (0)/2 .
(283)
In particular, the variance of the condensate occupation is determined by the correlation function of the phase fluctuations as follows [99]:
2 3 3
2
ˆ ϕ(y)
ˆ
d x d y.
(nˆ 0 − n¯ 0 )2 ≈ 2n˜ 20 e−2ϕˆ (0) ϕ(x)
(284)
6]
FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
385
The last formula can be evaluated with the help of a classical form of the
fluctuation–dissipation theorem,
2
2 3 3
kB T χϕϕ (k) .
d xd y =
ϕ(x)
ˆ ϕ(y)
ˆ
(285)
k=0
For a homogeneous superfluid, the static susceptibility in the infrared limit k → 0
does not depend on the interaction strength and, in accordance with Bogoliubov’s
1/k 2 theorem for the static susceptibility of superfluids, is equal to [99]
χϕϕ (k) = m/ntot h¯ 2 k 2 ,
k → 0.
(286)
Then Eq. (284) immediately yields the squared variance of the condensate fluctuations in the superfluid with arbitrary strong interaction in exactly the same
form as is indicated by the arrow in Eq. (271), only with the additional factor
(n¯ 0 (0)/ntot )2 . The latter factor is almost 1 for the BEC in dilute Bose gases but
can be much less in liquids, for example in 4 He superfluid it is about 0.1. Thus,
indeed, the condensate fluctuations at low temperatures can be calculated via the
long wavelength phase fluctuations of the noncondensate.
6.5.2. Effective Nonlinear σ Model, Goldstone Modes, and Universality
of the Infrared Anomalies
Following [100], let us use an effective nonlinear σ model [103] to demonstrate that the infrared singularities and anomalies of the order parameter fluctuations exist in all systems with a broken continuous symmetry, independently
on the interaction strength, and are similar to that of the BEC fluctuations in the
Bose gas. That model describes directional fluctuations of an order parameter
(0)
(0)
Ψ (x) = ms Ω(x) with a fixed magnitude ms in terms of an NΩ -component unit
vector Ω(x). It is inspired by the classical theory of spontaneous magnetization
in ferromagnets. The constraint |Ω(x)| = 1 suggests a standard decomposition of
the order parameter
Ω(x) = Ω0 (x), Ωi (x); i = 1, . . . , NΩ − 1 ,
N
Ω −1
Ω 2 (x),
Ω (x) = 1 −
(287)
0
i
i=1
into a longitudinal component Ω0 (x) and NΩ − 1 transverse
Goldstone fields
Ω02 (x) = 1 − i Ωi2 (x), resembles the
Ωi (x). The above σ model constraint,
particle-number constraint nˆ 0 = N − k=0 nˆ k in Eq. (211) for the many-body
atomic Bose gas in a trap. Although the former is a local, more stringent, constraint and the latter is only a global, integral constraint, in both cases it results
in the infrared singularities, anomalies, and non-Gaussian properties of the order parameter fluctuations. Obviously, in the particular case of a homogeneous
386
V.V. Kocharovsky et al.
[6
system the difference between the local and integral constraints disappears entirely. Within the σ model, a superfluid can be described as a particular case of
an NΩ = 2 system, with the superfluid (condensate) and normal (noncondensate)
component.
At zero external field, the
effective action for the fluctuations of the order parameter is S[Ω] = (ρs /2T ) [∇ Ω(x)]2 d 3 x, where the spin stiffness ρs is the only
parameter. Below the critical temperature Tc , a continuous symmetry becomes
broken and there appears an intensive nonzero-order parameter with a mean value
Ψ (x) · Ψ (0) d 3 x = V m2L → V m2s ,
(288)
V
which gives the spontaneous magnetization m2s of the infinite system in the thermodynamic limit V → ∞. The leading long distance behavior of the two-point
correlation function G(x) = Ψ (x) · Ψ (0) may be obtained from a simple
Gaussian spin wave calculation. Assuming low enough temperatures, we can
neglect the spin wave interactions and consider the transverse Goldstone fields
Ωi (k) as the Gaussian random functions of the momentum k with the correlation
function Ωi (k)Ωi (k ) = δi,i δk,−k T /ρs k 2 . As a result, the zero external field
correlation function below the critical temperature is split into longitudinal and
transverse parts G(x) = m2s [1 + G (x) + (NΩ − 1)G⊥ (x)], where m2s = G(∞)
now is the renormalized value of the spontaneous magnetization. To the lowest
nontrivial order in the small fluctuations of the Goldstone fields, the transverse
correlation function decays very slowly with a distance r, G⊥ ∝ T /ρs r, in accordance with Bogoliubov’s 1/k 2 theorem of the divergence of the transverse
susceptibility in the infrared limit,
χ⊥ (k) = m2s G⊥ (k)/T = m2s /ρs k 2 .
(289)
The longitudinal correlation function is simply related to the transverse one [103],
G (x) ≈
.
1 - 2
1
Ωi2 (0) = (NΩ − 1)G2⊥ (x),
Ωi (x)
c
4
2
(290)
and, hence, decays slowly with a 1/r 2 power law. That means that contrary to the
naive mean field picture, where the longitudinal susceptibility χ (k → 0) below
the critical temperature is finite, the Eq. (290) leads to an infrared singularity in
the longitudinal susceptibility and correlation function [101],
χ (k → 0) ∼ T /ρs2 k;
G ∼ 1/r 2 .
(291)
Although the Eq. (290) is obtained by means of perturbation theory, the result
for the slow power-law decay of the longitudinal correlation function, Eq. (291),
holds for arbitrary temperatures T < Tc [103].
6]
FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
387
Knowing susceptibilities, it is immediately possible to find the variance of the
operator
Mˆ s = V −1
(292)
Ψ (x) · Ψ (y) d 3 x d 3 y
V
V
that describes the fluctuations of the spontaneous magnetization in a finite system
at zero external field. Its mean value is given by Eq. (288) as Mˆ s = M¯ s = V m2L .
Its fluctuations are determined by the connected four-point correlation function
G4 = Ψ (x1 ) · Ψ (x2 )Ψ (x3 ) · Ψ (x4 )c . In the 4th-order of the perturbation theory
for an infinite system, it may be expressed via the squared transverse susceptibilities as follows:
1
G4 = m4s (NΩ − 1) G⊥ (x1 − x3 ) − G⊥ (x1 − x4 ) − G⊥ (x2 − x3 )
2
2
+ G⊥ (x2 − x4 ) .
(293)
For a finite system, it is more convenient to do similar calculations in the momentum representation and replace all integrals by the corresponding discrete sums,
which yields the squared variance
2 Mˆ s − M¯ s = 2m4s (NΩ − 1)T 2 ρs−2
(294)
k −4 ∝ T 2 V 4/3 .
k=0
This expression has the same anomalously large scaling ∝V 4/3 in the thermodynamic limit V → ∞ (due to the same infrared singularity) as the variance of
the BEC fluctuations in the ideal or interacting Bose gas in Eqs. (247) or (271),
respectively. Again, although the Eq. (294) is obtained by means of perturbation
theory, the latter scaling is universal below the critical temperature, just like the
1/k infrared singularity of the longitudinal susceptibility. Of course, very close
to Tc there is a crossover to the critical singularities, as discussed, e.g., in [104].
Thus, the averagefluctuation of the order parameter is still vanishing in the thermodynamic limit (Mˆ s − M¯ s )2 /M¯ s ∝ V −1/3 → 0; that is, the order parameter
(e.g., spontaneous magnetization or macroscopic wave function) is still a welldefined self-averaging quantity. However, this self-averaging in systems with a
broken continuous symmetry is much weaker than expected naively from the standard Einstein theory of the Gaussian fluctuations in macroscopic thermodynamics. Note that, in accordance with the well-known Hohenberg–Mermin–Wagner
theorem, in systems with lower dimensions, d 2, the strong fluctuations of the
direction of the magnetization completely destroy the long range order, and the
self-averaging order parameter does not exist anymore. An important point also is
that the scaling result in Eq. (294) holds for any dynamics and temperature dependence of the average order parameter M¯ s (T ) although the function M¯ s (T ) is, of
course, different, say, for ferromagnets, anti-ferromagnets, or a BEC in different
traps. It is the constraint, either |Ω| = 1 in the σ model or N = nˆ 0 + k=0 nˆ k
388
V.V. Kocharovsky et al.
[6
in the BEC, that predetermines the anomalous scaling in Eq. (294). The temperature dependence of the variance in Eq. (294) at low temperatures is also universal,
since ρs → const at T → 0. The reason for this fact is that the dominant finite
size dependence is determined by the leading low energy constant in the effective
field theory for fluctuations of the order parameter, which is precisely ρs in the
effective action S[Ω].
6.5.3. Universal Scaling of Condensate Fluctuations in Superfluids
In homogeneous superfluids the translational invariance requires the superfluid density to be equal to the full density ntot , so that the associated stiffness ρs (T → 0) = h¯ 2 ntot /m is independent of the interaction strength. Thus,
Eq. (294) in accord with the Eq. (271) yields the remarkable conclusion that the
relative variance of the ground-state occupation at low temperatures
is a universal
√
function of the density and the thermal wavelength λT = h/ 2πmT , as well as
the system size L = V 1/3 and the boundary conditions,
2
(295)
(n0 − n¯ 0 )2 /n¯ 0 = B/ ntot λ2T L .
The boundary conditions determine the low-energy spectrum of quasiparticles in the trap in the infrared limit and, hence, the numerical prefactor B in
the
singularity of the variance, namely, the coefficient B in the sum
infrared
−4 = BV 4/3 /8π 2 . In particular, in accord with Eqs. (247) and (271),
k=0 k
one has B = 0.8375 for the box with periodic boundary conditions, and
2
B = 8E3 (2)/π
Dirichlet boundary conditions. Here
∞ = 0.5012for the box2with
Ed (t) = n1 =1,...,nd =1 (n1 + · · · + nd )−t is the generalized Epstein zeta function [95], convergent for d < 2t. Of course, in a finite trap at temperatures of the
order of or less than the energy of the first excited quasiparticle, T < ε1 , the scaling law (294) is no longer valid and the condensate fluctuations acquire a different
temperature scaling, due to the temperature-independent quantum noise produced
by the excited atoms, which are forced by the interaction to occupy the excited
energy levels even at zero temperature, as it was discussed for the Eqs. (279)
and (244).
6.5.4. Constraint Mechanism of Anomalous Order Parameter Fluctuations and
Susceptibilities versus Instability in the Systems with a Broken Continuous
Symmetry
It is important to realize that the anomalously large order parameter fluctuations
and susceptibilities have a simple geometrical nature, related to the fact that the
direction of the order parameter is only in a neutral, rather than in a stable equilibrium, and does not violate an overall stability of the system with a broken
continuous symmetry at any given temperature below phase transition, T < Tc .
6]
FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
389
On one hand, on the basis of a well-known relation between the longitudinal susceptibility and the variance M of the order parameter fluctuations,
χαα ≡
2M
1 ∂Mα
=
,
N ∂Bα
N kB T
2 2M = Mˆ α − M¯ α ,
(296)
it is obvious that if the fluctuations are anomalous, i.e., go as 2M ∼ N γ with
γ > 1 instead of the standard macroscopic thermodynamic scaling 2M ∼ N ,
then the longitudinal susceptibility diverges in the thermodynamic limit N → ∞
as 2M /N ∼ N γ −1 → ∞. On the other hand, the susceptibilities in stable systems should be finite, since otherwise any spontaneous perturbation will result in
an infinite response and a transition to another phase. One could argue (as it was
done in a recent series of papers [97]) that the anomalous fluctuations cannot exist
since they break the stability condition and make the system unstable. However,
such an argument is not correct.
First of all, the anomalously large transverse susceptibility in the infrared limit
χ⊥ ∼ k −2 ∼ L2 in Bogoliubov’s 1/k 2 theorem (289) originates from the obvious
property of a system with a broken continuous symmetry that it is infinitesimally
easy to change the direction of the order parameter and, of course, does not violate
stability of the system. However, it implies anomalously large fluctuations in the
direction of the order parameter. Therefore, an anomalously large variance of the
ˆ − M)
¯ ⊥ M,
¯ which are perpendicular
fluctuations comes from the fluctuations (M
to the mean value of the order parameter. This is the key issue. It means that a
¯ that easily rotates these transverse fluctuations
longitudinal external field B M,
¯
towards M when χ⊥ B ∼ M , will
change the longitudinal order parameter by a
¯
¯ 2 − 2 ≈ 2 /2M.
¯ It is obvious that this
large increment, χ B ∼ |M| − M
M
M
pure geometrical rotation of the order parameter has nothing to do with an instability of the system, but immediately reveals the anomalously large longitudinal
susceptibility and relates its value to the anomalous transverse susceptibility and
variance of the fluctuations,
¯
χ ∼ 2M /2B M¯ ∼ M χ⊥ /2M.
(297)
This basically geometrical mechanism of an anomalous behavior of constrained
systems constitutes an essence of the constraint mechanism of the infrared anomalies in fluctuations and susceptibilities of the order parameter for all systems
with a broken continuous symmetry. The latter qualitative estimate yields the
anomalous scaling of the longitudinal susceptibility discussed above, χ ∼ L
with increase of the system size L, Eq. (291), since χ⊥ ∼ L2 , M¯ ∼ L3 , and
M ∼ L2 .
A different question is whether the Bose gas in a trap is unstable in the grand
canonical ensemble when an actual exchange of atoms with a reservoir is allowed,
and only the mean number of atoms in the trap is fixed, N¯ = const. In this case,
390
V.V. Kocharovsky et al.
[7
for example, the isothermal compressibility is determined by the variance of the
number-of-atoms fluctuations κT ≡ −V −1 (∂V /∂P )T = V (Nˆ − N¯ )2 /N¯ 2 kB T ,
and diverges in the thermodynamic limit if fluctuations are anomalous. In particular, the ideal Bose gas in the grand canonical ensemble does not have a
well-defined condensate order parameter, since the variance is of the order of the
mean value, (Nˆ − N¯ )2 ∼ N¯ , and it is unstable against a collapse [13,15,97].
In summary, one could naively expect that the order parameter fluctuations
below Tc are just like that of a standard thermodynamic variable, because there
is a finite restoring force for deviations from the equilibrium value. However,
in all systems with a broken continuous symmetry the universal existence of infrared singularities in the variance and higher moments ensures anomalously large
and non-Gaussian fluctuations of the order parameter. This effect is related to the
long-wavelength phase fluctuations and the infrared singularity of the longitudinal susceptibility originating from the inevitable geometrical coupling between
longitudinal and transverse order parameter fluctuations in constrained systems.
7. Conclusions
It is interesting to note that the first results for the average and variance of occupation numbers in the ideal Bose gas in the canonical ensemble were obtained about
fifty years ago by standard statistical methods [42,75,76] (see also [74,77] and
the review [13]). Only later, in the 60s, laser physics and its byproduct, the master equation approach, was developed (see, e.g., [16,17]). In this paper we have
shown that the latter approach provides very simple and effective tools to calculate statistical properties of an ideal Bose gas in contact with a thermal reservoir.
In particular, the results (169) and (172) reduce to the mentioned old results in the
“condensed region” in the thermodynamic limit.
However, the master equation approach gives even more. It yields simple analytical expressions for the distribution function of the number of condensed
atoms (162) and for the canonical partition function (163). In terms of cumulants, or semi-invariants [72,94], for the stochastic variables n0 or n = N − n0 , it
was shown [21] that the quasithermal approximation (154), with the results (169)
and (172), gives correctly both the first and the second cumulants. The analysis of
the higher-order cumulants is more complicated and includes, in principle, a comparison with more accurate calculations of the conditioned average number of
noncondensed atoms (129) as well as higher-order corrections to the second-order
master equation (124). It is clear that the master equation approach is capable of
giving the correct answer for higher-order cumulants and, therefore, moments of
the condensate fluctuations. Even without these complications, the approximate
result (162) reproduces the higher moments, calculated numerically via the exact
7]
FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
391
recursion relation (80), remarkably well for all temperatures T < Tc and T ∼ Tc
(see Fig. 12).
As we demonstrated in Section 4, the simple formulas yielded by the master
equation approach allow us to study mesoscopic effects in BECs for a relatively
small number of atoms that is typical for recent experiments [24–29]. Moreover,
it is interesting in the study of the dynamics of BEC. This technique for studying statistics and dynamics of BEC shows surprisingly good results even within
the simplest approximations. Thus, the analogy with phase transitions and quantum fluctuations in lasers (see, e.g., [19,52,70,71]) clarifies some problems in
BEC. The equilibrium properties of the number-of-condensed-atom statistics in
the ideal Bose gas are relatively insensitive to the details of the model. The origin
of dynamical and coherent properties of the evaporatively cooling gas with an interatomic interaction is conceptually different from that in the present “ideal gas
+ thermal reservoir” model. The present model is rather close to the dilute 4 He
gas in porous gel experiments [22] in which phonons in the gel play the role of
the external thermal reservoir. Nevertheless, the noncondensed atoms always play
a part of some internal reservoir, and the condensate master equation probably
contains terms similar to those in Eq. (130) for any cooling mechanism.
For the ideal Bose gas in the canonical ensemble the statistics of the condensate fluctuations below Tc in the thermodynamic limit is essentially the statistics
of the sum ofthe noncondensed modes of a trap, nˆ k , that fluctuate independently,
nˆ 0 = N − k=0 nˆ k . This is well understood, especially, due to the Maxwell’s
demon ensemble approximation elaborated in a series of papers [14,42,44,46,54,
55], and is completed and justified to a certain extent in [20,21] by the explicit
calculation of the moments (cumulants) of all orders, and by the reformulation of the canonical-ensemble problem in the properly reduced subspace of the
original many-particle Fock space. The main result (263) of [20,21] explicitly describes the non-Gaussian properties and the crossover between the ideal-gas and
interaction-dominated regimes of the BEC fluctuations.
The problem of dynamics and fluctuations of BEC for the interacting gas is
much more involved. The master equation approach provides a powerful tool
for the solution of this problem as well. Of course, to take into account higherorder effects of interaction between atoms we have to go beyond the second-order
master equation, i.e., to iterate Eq. (123) more times and to proceed with the
higher-order master equation similarly to what we discussed above. It would be
interesting to show that the master equation approach could take into account
all higher-order effects in a way generalizing the well-known nonequilibrium
Keldysh diagram technique [30,105,106]. As a result, the second-order master
equation analysis presented above can be justified rigorously, and higher-order
effects in condensate fluctuations at equilibrium, as well as nonequilibrium stages
of cooling of both ideal and interacting Bose gases can be calculated.
392
V.V. Kocharovsky et al.
[7
The canonical-ensemble quasiparticle method, i.e., the reformulation of the
problem in terms of the proper canonical-ensemble quasiparticles, gives even
more. Namely, it opens a way to an effective solution of the canonical-ensemble
problems for the statistics and nonequilibrium dynamics of the BEC in the interacting gas as well. The first step in this direction is done in Section 6, where
the effect of the Bogoliubov coupling between excited atoms due to a weak interaction on the statistics of the fluctuations of the number of ground-state atoms
in the canonical ensemble was analytically calculated for the moments (cumulants) of all orders. In this case, the BEC statistics is essentially the statistics
of the sum of the dressed quasiparticles that fluctuate independently. In particular, a suppression of the condensate fluctuations at the moderate temperatures and
their enhancement at very low temperatures immediately follow from this picture.
There is also the problem of the BEC statistics in the microcanonical ensemble, which is closely related to the canonical-ensemble problem. In particular,
the equilibrium microcanonical statistics can be calculated from the canonical
statistics by means of an inversion of a kind of Laplace transformation from
the temperature to the energy as independent variable. Some results concerning
the BEC statistics in the microcanonical ensemble for the ideal Bose gas were
presented in [14,45,46,54,55,60]. We can calculate all moments of the microcanonical fluctuations of the condensate from the canonical moments found in the
present paper. Calculation of the microcanonical statistics starting from the grand
canonical ensemble and applying a saddle-point method twice, first, to obtain the
canonical statistics and, then, to get the microcanonical statistics [45], meets certain difficulties since the standard saddle-point approximation is not always good
and explicit to restore the canonical statistics from the grand canonical one with
sufficient accuracy [57]. The variant of the saddle-point method discussed in Appendix F is not subject to these restrictions.
Another important problem is the study of mesoscopic effects due to a relatively small number of trapped atoms (N ∼ 103 –106 ). The canonical-ensemble
, takes
quasiparticle approach under the approximation (210), i.e., HCE ≈ HnCE
0 =0
into account only a finite-size effect of the discreteness of the single particle spectrum, but does not include all mesoscopic effects. Hence, other methods should
be used (see, e.g., [14,44,46,49,61–64,67]). In particular, the master equation approach provides amazingly good results in the study of mesoscopic effects, as was
demonstrated recently in [52,53].
The canonical-ensemble quasiparticle approach also makes it clear how to extend the Bogoliubov and more advanced diagram methods for the solution of
the canonical-ensemble BEC problems and ensure conservation of the number
of particles. The latter fact cancels the main arguments of Refs. [96,97] against
the Giorgini–Pitaevskii–Stringari result [78] and shows that our result (263) and,
in particular, the result of [78] for the variance of the number of ground-state
atoms in the dilute weakly interacting Bose gas, correctly take into account one
7]
FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
393
of the main effects of the interaction, namely, dressing of the excited atoms by
the macroscopic condensate via the Bogoliubov coupling. If one ignores this and
other correlation effects, as it was done in [96,97], the result cannot be correct.
This explains a sharp disagreement of the ground-state occupation variance suggested in [96] with the predictions of [78] and our results as well. Note also
that the statement from [96] that “the phonon spectrum plays a crucial role in
the approach of [78]” should not be taken literally since the relative weights of
bare modes in the eigenmodes (quasiparticles) is, at least, no less important than
eigenenergies themselves. In other words, our derivation of Eq. (263) shows that
squeezing of the excited states due to Bogoliubov coupling in the field of the
macroscopic condensate is crucial for the correct calculation of the BEC fluctuations. Besides, the general conclusion that very long wavelength excitations have
an acoustic, “gapless” spectrum (in the thermodynamic limit) is a cornerstone fact
of the many-body theory of superfluidity and BEC [93]. Contrary to a pessimistic
picture of a mess in the study of the condensate fluctuations in the interacting gas
presented in [96], we are convinced that the problem can be clearly formulated
and solved by a comparative analysis of the contributions of the main effects of
the interaction in the tradition of many-body theory. In particular, the result (263)
corresponds to a well-established first-order Popov approximation in the diagram
technique for the condensed phase [36].
We emphasize here an important result of an analytical calculation of all higher
cumulants (moments) [21]. In most cases (except, e.g., for the ideal gas in the
harmonic trap and similar high dimensional traps where d > 2σ ), both for the
ideal Bose gas and for the interacting Bose gas, the third and higher cumulants
of the number-of-condensed-atom fluctuations normalized to the corresponding
power of the variance do not tend to the Gaussian zero value in the thermodynamic
limit, e.g., (n0 − n0 )3 /(n0 − n0 )2 3/2 does not vanish in the thermodynamic
limit.
Thus, fluctuations in BEC are not Gaussian, contrary to what is usually assumed
following the Einstein theory of fluctuations in the macroscopic thermodynamics.
Moreover, BEC fluctuations are, in fact, anomalously large, i.e., they are not normal at all. Both these remarkable features originate from the universal infrared
anomalies in the order parameter fluctuations and susceptibilities in constrained
systems with a broken continuous symmetry. The infrared anomalies come from
a long range order in the phases below the critical temperature of a second-order
phase transition and have a clear geometrical nature, related to the fact that the
direction of the order parameter is only in a neutral, rather than in a stable equilibrium. Hence, the transverse susceptibility and fluctuations are anomalously
large and, through an inevitable geometrical coupling between longitudinal and
transverse order parameter fluctuations in constrained systems, produce the anomalous order parameter fluctuations. In other words, the long wavelength phase
fluctuations of the Goldstone modes, in accordance with the Bogoliubov 1/k 2
394
V.V. Kocharovsky et al.
[8
theorem for the transverse susceptibility, generate anomalous longitudinal fluctuations in the order parameter of the systems below the critical temperature of
the second-order phase transition. Obviously, this constraint mechanism of the
infrared anomalies in fluctuations and susceptibilities of the order parameter is
universal for all systems with a broken continuous symmetry, including BEC in
ideal or weakly interacting gases as well as superfluids, ferromagnets and other
systems with strong interaction. It would be interesting to extend the analysis of
the order parameter fluctuations presented in this review from the BEC in gases
to other systems.
The next step should be an inclusion of the effects of a finite renormalization
of the energy spectrum as well as the interaction of the canonical-ensemble quasiparticles at finite temperatures on the statistics and dynamics of BEC. It can
be done on the level of the second-order Beliaev–Popov approximation, which is
considered to be enough for the detailed account of most many-body effects (for a
review, see [36]). A particularly interesting problem is the analysis of phase fluctuations of the condensate in the trap, or of the matter beam in the atom laser [88],
because the interaction is crucial for the existence of the coherence in the condensate [21,30,32,33,48,93,107–109]. As far as the equilibrium or quasiequilibrium
properties are concerned, the problem can be solved effectively by applying either
the traditional methods of statistical physics to the canonical-ensemble quasiparticles, or the master equation approach, that works surprisingly well even without
any explicit reduction of the many-particle Hilbert space [52,53]. For the dynamical, nonequilibrium properties, the analysis can be based on an appropriate
modification of the well-known nonequilibrium Keldysh diagram technique [105,
106,110,111] which incorporates both the standard statistical and master equation
methods.
Work in the directions mentioned above is in progress and will be presented
elsewhere. Clearly, the condensate and noncondensate fluctuations are crucially
important for the process of the second-order phase transition, and for the overall
physics of the Bose–Einstein-condensed interacting gas as a many-particle system.
8. Acknowledgements
We would like to acknowledge the support of the Office of Naval Research
(Award No. N00014-03-1-0385) and the Robert A. Welch Foundation (Grant No.
A-1261). One of us (MOS) wishes to thank Micheal Fisher, Joel Lebowitz, Elliott
Lieb, Robert Seiringer for stimulating discussions and Leon Cohen for suggesting
this review article.
9]
FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES
395
9. Appendices
Appendix A. Bose’s and Einstein’s Way
of Counting Microstates
When discussing Einstein’s 1925 paper [6], we referred to the identity (49),
Z!
N +Z−1
,
(A1)
=
N
p0 ! . . . pN !
{p0 ,p1 ,...,pN }
which expresses the number of ways to distribute N Bose particles over Z quantum cells in two different manners: On the left-hand side, which corresponds to
Bose’s way of counting microstates, numbers pr specify how many cells contain r
quanta; of course, with only N quanta being available, one has pr = 0 for r > N.
As symbolically indicated by the prime on the summation sign, the sum thus extends only over those sets {p0 , p1 , . . . , pN } with comply with the conditions
N
pr = Z,
(A2)
r=0
stating that there are Z cells to accommodate the quanta, and
N
rpr = N,
(A3)
r=0
stating that the number of quanta be N . The right-hand side of Eq. (A1) gives the
total number of microstates, taking into account all possible sets of occupation
numbers in the single expression already used by Einstein in his final derivation [6] of the Bose–Einstein distribution which can still be found in today’s
textbooks.
The validity of Eq. (A1) is clear for combinatorial reasons. Nonetheless, since
this identity (A1) constitutes one of the less known relations in the theory of Bose–
Einstein statistics, we give its explicit proof in this appendix.
As for most mathematical proofs, one needs tools, an idea, and a conjurer’s
trick. In the present case, the tools are two generalizations of the binomial theorem
n n
(a + b)n =
(A4)
a k bn−k ,
k
k=0
where
n!
n
=
k
k!(n − k)!
(A5)
396
V.V. Kocharovsky et al.
[Appendix A
denotes the familiar binomial coefficients. The first such generalization is the
multinomial theorem
n!
p p
p
(a1 + a2 + · · · + aN )n =
(A6)
a 1 a 2 . . . aNN ,
p1 !p2 ! . . . pN ! 1 2
pr =n
which is easily understood: When multiplying out the left-hand side, every product obtained contains one factor ai from each bracket (a1 +a2 +· · ·+aN ). Hence,
p p
p
in every product a1 1 a2 2 . . . aNN the exponents add up to the number of brackets,
which is n. Therefore, for such a product there are n! permutations of the individual factors ai . However, if identical factors ai are permuted among themselves,
for which there are pi ! possibilities, one obtains the same value. Hence, the coefficient of each product on the right-hand side in Eq. (A6) corresponds to the
number of possible arrangements of its factors, divided by the number of equivalent arrangements. Note that the reasoning here is essentially the same as for the
justification of Bose’s expression (12), which is, of course, not accidental.
The second generalization of the binomial theorem (A4) required for the proof
of the identity (A1) emerges when we replace the exponent n by a nonnatural
number γ : One then has
∞ γ
γ
(a + b) =
(A7)
a k bγ −k ,
k
k=0
with the definition
γ (γ − 1)(γ − 2) · · · · · (γ − k + 1)
γ
=
.
k
k!
(A8)
If γ is not a natural number, this series (A7) converges for any complex numbers a, b, provided |a/b| < 1. This generalized binomial theorem (A7), which
is treated in introductory analysis courses, is useful, e.g., for writing down the
Taylor expansion of (1 + x)γ .
Given these tools, the idea for proving the identity (A1) now consists in considering the expression
Z
PN Z (x) = 1 + x + x 2 + x 3 + · · · + x N ,
(A9)
where x is some variable which obeys |x| < 1, but need not be specified further.
According to the multinomial theorem (A6), one has
PN Z (x) =
N
r=0 pr =Z
p
0 p0 1 p1
Z!
x
. . . xN N .
x
p0 !p1 ! . . . pN !
(A10)
Directing the attention then to a systematic ordering of this series with respect to
powers of x, the coefficient of x N equals the sum of all coefficients encountered
Appendix B] FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES 397
here which accompany terms with r rpr = N , which is precisely the left-hand
side of the desired equation (A1), keeping in mind the restrictions (A2) and (A3).
Now comes the conjurer’s trick: The coefficient of x N in this expression (A10),
equaling the left-hand side of Eq. (A1), also equals the coefficient of x N in the
expression
Z
P∞ (x) = 1 + x + x 2 + x 3 + · · · + x N + x N+1 + · · · ,
(A11)
since this differs from PN Z (x) only by powers of x higher than N . But this involves a geometric series, which is immediately summed:
∞ Z
xr
= (1 − x)−Z .
P∞ (x) =
(A12)
r=0
According to the generalized binomial theorem (A7), one has
(1 − x)−Z =
∞ −Z
k=0
k
(−x)k ,
so that, in view of the definition (A8), the coefficient of x N equals
(−Z)(−Z − 1) · · · · · (−Z − N + 1)
N −Z
= (−1)N
(−1)
N
N!
(Z + N − 1) · · · · · (Z + 1)Z
=
N!
(Z + N − 1)!
Z+N −1
=
.
=
N
N !(Z − 1)!
(A13)
(A14)
This is the right-hand side of the identity (A1), which completes the proof.
Appendix B. Analytical Expression for the Mean Number
of Condensed Atoms
We obtain an analytical expression for n¯ 0 from Eq. (86). One can reduce the triple
sum into a single sum if we take into account the degeneracy g(E) of the level
with energy E = h¯ Ω(l + m + n), which is equal to the number of ways to fill the
level. This number can be calculated from the Einstein’s complexion equation,
i.e.,
1 E
E
{(l + m + n) + (3 − 1)}!
=
+2
+1 .
g(E) =
(B1)
(l + m + n)!(3 − 1)!
2 h¯ Ω
h¯ Ω
398
V.V. Kocharovsky et al.
[Appendix B
Now, we have reduced three variables l, m, n to only one. By letting E = s hΩ
¯
where s is integer, one can write
N=
∞
1
2 (s + 2)(s + 1)
1
¯ −1
( + 1)esβ hΩ
s=0 n¯ 0
n¯ 0 +
n¯ 0
S,
(1 + n¯ 0 )
(B2)
where for n¯ 0 1,
S
∞ 1
2 (s + 2)(s + 1)
.
esβ h¯ Ω − 1
(B3)
s=1
The root of the quadratic Eq. (B2) yields
n¯ 0 = −
1
(1 + S − N ) − (1 + S − N )2 + 4N .
2
(B4)
In order to find an analytical expression for S, we write
S
∞
∞
s=0
s=1
1
1 s 2 + 3s
+
.
sβ
h
Ω
sβ
h
Ω
¯
¯
2
e
−1
e
−1
(B5)
Converting the summation into integration by replacing x = sβ hΩ
¯ yields
1
S
a
∞
a
=1−
dx
1
+
ex − 1 2a 3
∞
0
x 2 dx
3
+
ex − 1 2a 2
ζ (3)
1 a
ln e − 1 + 3 +
a
a
π
2a
2
∞
0
x dx
ex − 1
,
(B6)
where a = β hΩ
¯ = h¯ Ω/kB T = (ζ (3)/N)1/3 Tc /T .
Thus, Eq. (B3) gives the following analytical expression for S:
"
3 #
π 2 N 2/3 T 2
T
+
(1 + S − N ) = −N 1 −
Tc
4 ζ (3)
Tc
1/3
T (ζ (3)/N )1/3 Tc /T
N
−
ln e
− 1 + 2.
ζ (3)
Tc
(B7)
Figure 16 compares different approximations for calculating n¯ 0 within the
grand canonical ensemble.
Appendix C] FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES 399
F IG . 16. Grand canonical result for n0 as a function of temperature for N = 200, computed
from analytical expressions, Eqs. (B4) and (B7) (line with circles); semi-analytical expressions given
by Eqs. (B4) and (B3) (solid line); and exact numerical solution of Eq. (86) (dots). Expanded views
show: (b) exact agreement of the semi-analytical approach at low temperature and (c) a small deviation
near Tc . Also, the analytical and the semi-analytical results agree quite well.
Appendix C. Formulas for the Central Moments
of Condensate Fluctuations
By using n0 = N −
k=0 nk
and n0 = N −
k=0 nk ,
0
/
s s
s
n0 − n0 = (−1)
nk − nk ,
we have
(C1)
k=0
which shows that the fluctuations of the condensate particles are proportional to
the fluctuations of the noncondensate particles.
400
V.V. Kocharovsky et al.
[Appendix C
As an example we show how to evaluate the fluctuation for the second-order
moment, or variance,
2 n0 − n0 =
nj nk − nj nk .
(C2)
j,k=0
The numbers of particles in different levels are statistically independent, since
ˆ = Tr{aˆ j† aˆ j ρˆj } Tr{aˆ k† aˆ k ρˆk } = nj nk . Thus, we
nj nk=j = Tr{aˆ j† aˆ j aˆ k† aˆ k ρ}
find
2 n0 − n0 3 n0 − n0 4 n0 − n0 =
n2k − nk 2 ,
(C3)
k=0
=
− n3k + 3 n2k nk − 2nk 3 ,
(C4)
k=0
=
n4k − 4 n3k nk + 6 n2k nk 2 − 3nk 4 .
(C5)
k=0
In the grand canonical approach, nsk can be evaluated using
s
1 s −β(k −μ)nk
n e
,
nk =
Zk n k
(C6)
k
where Zk = (1 − e−β(k −μ) )−1 . An alternative way is to use the formula nsk =
d s Θk
d(iu)s |u=0 derived in Section 5.2. In particular, one can show that in this approach
2
nk = 2nk 2 + nk ,
3
nk = 6nk 3 + 6nk 2 + nk ,
4
nk = 24nk 4 + 36nk 3 + 14nk 2 + nk .
(C7)
(C8)
(C9)
Using Eqs. (C7)–(C9) we obtain
2 n0 − n0 3 n0 − n0 4 n0 − n0 =
nk 2 + nk ,
k=0
=−
2nk 3 + 3nk 2 + nk ,
(C10)
(C11)
k=0
=
9nk 4 + 18nk 3 + 10nk 2 + nk .
k=0
(C12)
Appendix D] FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES 401
Appendix D. Analytical Expression for the Variance
of Condensate Fluctuations
In a spherically symmetric harmonic trap with trap frequency Ω one can convert
the triple sums in Eq. (98)into a single sum
∞by using Eq. (B1), and then do
1
.
.
.
→
integration upon replacing ∞
x=0
0 dx . . . and βE → x, giving
β hΩ
¯
n20 =
∞
1 E
E
+2
+1
2 h¯ Ω
h¯ Ω
E=0
"
1
1
+
×
1
2
[exp(βE)(1 + n¯ 0 ) − 1]
exp(βE)(1 +
2
∞ x
3x
1
dx
+
+2
=
2β h¯ Ω
β h¯ Ω
β h¯ Ω
a
"
1
1
+
×
1
2
[exp(x)(1 + n¯ 0 ) − 1]
exp(x)(1 +
=
1
2a
∞
dx
#
1
n¯ 0 ) − 1
#
1
n¯ 0 ) − 1
2
exp(x)(1 + n¯10 )
x
3x
+
,
+2
a
a
[exp(x)(1 + n¯1 ) − 1]2
(D1)
0
a
where a = β hΩ
¯ = h¯ Ω/kB T = (Tc /T )(ζ (3)/N)1/3 and the density of states
can be written as ρ(E) = 2h¯1Ω [( h¯EΩ )2 + h3E
+2]. Then we integrate by parts using
¯Ω
the identity
exp(x)(1 +
[exp(x)(1 +
1
n¯ 0 )
1
2
n¯ 0 ) − 1]
=−
∂
1
∂x [exp(x)(1 +
1
n¯ 0 ) − 1]
,
arriving at
n20
3
1
=
+ 2
1
a
a[e (1 + n¯ 0 ) − 1] 2a
∞
dx
a
( 2x
a + 3)
[exp(x)(1 +
1
n¯ 0 ) − 1]
The integral in Eq. (D2) can be calculated analytically, using
∞
a
x dx
π2
1
=
− ln2 A + ln Aea − 1 ln A
[exp(x)A − 1]
6
2
a2
+ di log Aea + ,
2
.
(D2)
402
V.V. Kocharovsky et al.
∞
a
[Appendix E
dx
= ln(1 + α) − ln Aea − 1 + a,
[exp(x)A − 1]
where
x
di log(x) =
1
ln(t)
dt.
1−t
As a result, we get
1
1 2
1
1 π2
2
a
n0 = 3
− ln 1 +
+ di log e 1 +
6
n¯ 0
2
n¯ 0
a
1
1
− 1 ln 1 +
+ ln ea 1 +
n¯ 0
n¯ 0
3
n¯ 0 + 1
3
2
+
+ a ln a
+ .
1
a
2
e (n¯ 0 + 1) − n¯ 0
a[e (1 + n¯ ) − 1] a
(D3)
0
Taking into account that a = (Tc /T )(ζ (3)/N )1/3 , we finally obtain
3
N π2
1
T
2
1/3
+ di log exp (Tc /T )(ζ (3)/N )
n0 =
1+
Tc ζ (3) 6
n¯ 0
1
1
− ln2 1 +
2
n¯ 0
1
1
1/3
+ ln exp (Tc /T )(ζ (3)/N)
− 1 ln 1 +
1+
n¯ 0
n¯ 0
2 2/3 3 T
N
n¯ 0 + 1
+
ln
2 Tc
ζ (3)
exp[(Tc /T )(ζ (3)/N )1/3 ](n¯ 0 + 1) − n¯ 0
1/3 T
N
3
+
+2 .
Tc ζ (3)
exp[(Tc /T )(ζ (3)/N )1/3 ](1 + n¯10 ) − 1
(D4)
Appendix E. Single Mode Coupled to a Reservoir of Oscillators
The derivation of the damping Liouvillean proceeds from the Liouville–von Neumann equation
∂
1ˆ
Vsr , ρ(t)
ˆ
,
ρ(t)
ˆ =
∂t
i h¯
(E1)
Appendix E] FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES 403
where Vˆsr is the Hamiltonian in the interaction picture for the system (s) coupled
to a reservoir (r).
We can derive a closed form of the dynamical equation for the reduced density
operator for the system, ρˆs (t) = Trr {ρ(t)}
ˆ
by tracing out the reservoir. This is ac1 t ˆ
ˆ )] dt complished by first integrating Eq. (E1) for ρ(t)
ˆ = ρ(0)+
ˆ
i h¯ 0 [Vsr (t ), ρ(t
and then substituting it back into the right hand side of Eq. (E1), giving
1
∂
ˆ
ρˆs (t) =
Trr Vˆsr , ρ(0)
∂t
i h¯
t
1
+
Vˆsr (t), Vˆsr (t ), ρ(t
Trr
ˆ ) dt .
2
(i h)
¯
(E2)
0
We may repeat this indefinitely, but owing to the weaknesses of the systemreservoir interaction, it is possible to ignore terms higher than 2nd order in Vˆsr .
Furthermore, we assume the system and reservoir are approximately uncorrelated
in the past and the reservoir is so large that it remains practically in thermal equilibrium ρˆrth , so ρ(t
ˆ ) ρˆs (t ) ⊗ ρˆrth and ρ(0)
ˆ
= ρˆs (0) ⊗ ρˆrth .
For a single mode field (f ) coupled to a reservoir of oscillators, one has
Vˆf r = h¯
(E3)
gk bˆk aˆ † ei(ν−νk )t + bˆk† ae
ˆ −i(ν−νk )t .
k
Since Trr {aˆ † ρˆrth } = 0 and Trr {aˆ ρˆrth } = 0, the first term vanishes and by using Eq. (E3) we have 16 terms. But secular approximation reduces the number
of terms by half. We now perform the Markov approximation, ρˆf (t ) ρˆf (t),
stating that the dynamics of the system is independent of the states in the past.
The thermal average of the radiation operators is
t
gk2
k
e−i(ν−νk )(t−t ) Trr ρˆrth bˆk† bˆk dt 0
gk2 πδ(ν − νk )n(ν
¯ k ) = n(ν)G(ν)/2,
¯
k
¯ k − 1)−1 . Thus, we have
where Trr {ρˆrth bˆk† bˆk } = n(ν
¯ k ) = (eβ hν
1 ∂
ρˆf (t) = − C aˆ † aˆ ρˆf (t) − 2aˆ ρˆf (t)aˆ † + ρˆf (t)aˆ † aˆ
∂t
2
1 − D aˆ aˆ † ρˆf (t) − 2aˆ † ρˆf (t)aˆ + ρˆf (t)aˆ aˆ † ,
2
where D = G n¯ and C = G(n¯ + 1).
(E4)
404
V.V. Kocharovsky et al.
[Appendix F
Appendix F. The Saddle-Point Method for Condensed Bose
Gases
The saddle-point method is one of the most essential tools in statistical physics.
Yet, the conventional form of this approximation fails in the case of condensed
ideal Bose gases [13,58]. The point is that in the condensate regime the saddlepoint of the grand canonical partition function approaches the ground-state singularity at z = exp(βε0 ), which is a hallmark of BEC. However, the customary
Gaussian approximation requires that intervals around the saddle-point stay clear
of singularities. Following the original suggestion by Dingle [59], Holthaus and
Kalinowski [60] worked out a natural solution to this problem: One should exempt the ground-state factor of the grand canonical partition function from the
Gaussian expansion and treat that factor exactly, but proceed as usual otherwise.
The success of this refined saddle-point method hinges on the fact that the emerging integrals with singular integrands can be done exactly; they lead directly to
parabolic cylinder functions. Here we discuss the refined saddle-point method in
some detail.
We start from the grand canonical partition function
∞
Ξ (β, z) =
ν=0
1
,
1 − z exp(−βεν )
(F1)
where εν are single-particle energies, β = 1/kB T and z = exp(βμ). The grand
canonical partition function Ξ (β, z) generates the canonical partition functions
ZN (β) by means of the expansion
Ξ (β, z) =
∞
zN ZN (β).
(F2)
N=0
Then we treat z as a complex variable and using Cauchy’s theorem represent
ZN (β) by a contour integral,
'
'
Ξ (β, z)
1
1
dz N+1 =
dz exp −F (z) ,
ZN (β) =
(F3)
2πi
2πi
z
where the path of integration encircles the origin counter-clockwise, and
F (z) = (N + 1) ln z − ln Ξ (β, z) = (N + 1) ln z
∞
+
ln 1 − z exp(−βεν ) .
(F4)
ν=0
The saddle-point z0 is determined by the requirement that F (z) becomes stationary,
∂F (z) = 0,
(F5)
∂z z=z0
Appendix F] FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES 405
giving
N +1=
∞
ν=0
1
.
exp(βεν )/z0 − 1
(F6)
This is just the grand canonical relation between particle number N and fugacity
z0 (apart from the appearance of one extra particle on the left-hand side). Appendix C discusses an approximate analytic solution of such equations.
In the conventional saddle-point method, the whole function F (z) is taken in
the saddle-point approximation
1
F (z) ≈ F (z0 ) + F (z0 )(z − z0 )2 .
2
Doing the remaining Gaussian integral yields
(F7)
exp(−F (z0 ))
ZN (β) ≈ √
.
−2πF (z0 )
(F8)
The canonical occupation number of the ground state, and its mean-square fluctuations, are obtained by differentiating the canonical partition function:
'
∂ ln ZN (β)
1
1
1 ∂Ξ (β, z)
=
dz N+1
,
n¯ 0 =
(F9)
∂(−βε0 )
ZN (β) 2πi
∂(−βε0 )
z
n0 =
∂ 2 ln ZN (β)
1
1
= −n¯ 20 +
2
ZN (β) 2πi
∂(−βε0 )
'
dz
1
zN+1
∂ 2 Ξ (β, z)
. (F10)
∂(−βε0 )2
The saddle-point approximation is then applied to the integrands of Eqs. (F9) and
(F10). Figs. 17 and 18 (dashed curves) show results for n¯ 0 and n0 obtained by
the conventional saddle-point method for a Bose gas with N = 200 atoms in a
harmonic isotropic trap. In the condensate regime there is a substantial deviation
of the saddle-point curves from the “exact” numerical answer obtained by solution
of the recursion equations for the canonical statistics (dots).
The reason for this inaccuracy is that in the condensate region the saddle-point
z0 lies close to the singular point z = exp(βε0 ) of the function F (z). As a result,
the approximation (F7) becomes invalid in the condensate region. To improve the
method, Dingle [59] proposed to treat the potentially dangerous term in (F4) as it
is, and represent ZN (β) as
'
exp(−F1 (z))
1
dz
ZN (β) =
(F11)
,
2πi
1 − z exp(−βε0 )
where
F1 (z) = (N + 1) ln z +
∞
ln 1 − z exp(−βεν )
ν=1
(F12)
406
V.V. Kocharovsky et al.
[Appendix F
F IG . 17. Canonical occupation number of the ground state as a function of temperature for
N = 200 atoms in a harmonic isotropic trap. Dashed and solid curves are obtained by the conventional
and the refined saddle-point method, respectively. Dots are “exact” numerical answers obtained for the
canonical ensemble.
F IG . 18. Variance in the condensate particle number as a function of temperature for N = 200
atoms in a harmonic isotropic trap. Dashed and solid curves are obtained by the conventional and
the refined saddle-point method, respectively. Dots are “exact” numerical answers in the canonical
ensemble.
Appendix F] FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES 407
has no singularity at z = exp(βε0 ). The singular point to be watched now is the
one at z = exp(βε1 ). Since z0 < exp(βε0 ), the saddle-point remains separated
from that singularity by at least the N-independent gap exp(βε1 ) − exp(βε0 ) hω/k
¯
B T . This guarantees that the amputated function F1 (z) remains singularityfree in the required interval around z0 for sufficiently large N . Then the Gaussian
approximation to exp(−F1 (z)) is safe. The subsequently emerging saddle-point
integral for the canonical partition function can be done exactly, yielding [60]
1
η1
ZN (β) ≈ exp βε0 − F1 (z0 ) − 1 + η2 /2 erfc √ ,
(F13)
2
2
√ ∞
where erfc(z) = 2/ π z exp(−t 2 ) dt is the complementary error function, η =
(exp(βε0 ) − z0 ) −F1 (z0 ), and η1 = η − 1/η.
Calculation of occupation numbers and their fluctuations deals with integrals
from derivatives of Ξ (β, z) with respect to −βε0 . In such expressions the factors
singular at z = exp(βε0 ) should be taken exactly. This leads to the integrals of
the following form:
'
1
exp[−f1 (z) − (σ − 1)βε0 ]
dz
2πi
(1 − z exp(−βε0 ))σ
(σ −1)/2
1 −f1 (z∗ )
≈√
2π
× exp βε0 − f1 (z∗ ) − σ + η2 /2 − η12 /4 D−σ (η1 ),
(F14)
where η = (exp(βε0 ) − z∗ ) −f1 (z∗ ), η1 = η − σ/η, and z∗ is a saddle-point of
the function
f (z) = f1 (z) + (σ − 1)βε0 + σ ln 1 − z exp(−βε0 ) ;
(F15)
D−σ (z) is a parabolic cylinder function, which can be expressed in terms of hypergeometric functions as
Ds (z) = 2s/2 e−z
2 /4
√
π
1 F1 (−s/2, 1/2, z
2 /2)
#[(1 − s)/2]
√
2z ·1 F1 ((1 − s)/2, 3/2, z2 /2)
−
.
#[−s/2]
(F16)
Figures 17 and 18 (solid curves) show n¯ 0 (T ) and n0 (T ) obtained by the refined
saddle-point method. These results are in remarkable agreement with the exact
dots. Figure 19 demonstrates that this refined method also provides good accuracy
for the third central moment of the number-of-condensed-atoms fluctuations.
408
V.V. Kocharovsky et al.
[10
F IG . 19. The third central moment (n0 − n¯ 0 )3 for N = 200 atoms in a harmonic isotropic trap
obtained using the refined saddle-point method (solid curve). Exact numerical results for the canonical
ensemble are shown by dots.
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ADVANCES IN ATOMIC, MOLECULAR AND OPTICAL PHYSICS, VOL. 53
LIDAR-MONITORING OF THE AIR WITH
FEMTOSECOND PLASMA CHANNELS
LUDGER WÖSTE1 , STEFFEN FREY2 AND JEAN-PIERRE WOLF3
1 Physics Department, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany
2 MIT, Department of Earth, Atmospheric, and Planetary Sciences, 77 Massachusetts Avenue,
Cambridge, MA 02139, USA
3 GAP-Biophotonics, University of Geneva, 20, rue de l’Ecole de Médecine,
1211 Geneva 4, Switzerland
1.
2.
3.
4.
5.
6.
7.
8.
9.
Introduction . . . . . . . . . . . . . . . . . . . . . . .
Conventional LIDAR Measurements . . . . . . . . .
The Femtosecond-LIDAR Experiment . . . . . . . .
Nonlinear Propagation of Ultra-Intense Laser Pulses
White Light Femtosecond LIDAR Measurements . .
Nonlinear Interactions with Aerosols . . . . . . . . .
Conclusion . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgements . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . .
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413
415
419
421
427
433
437
438
439
Abstract
LIDAR (Light Detection and Ranging) is the only remote sensing technique that
is capable to provide 3-dimensional range resolved measurements of atmospheric
constituents like pollutants, humidity or the aerosol, and of atmospheric parameters
like temperature or wind. Further perspectives arise from the advent of ultra-fast
high-power laser sources which are capable to generate extended plasma channels in
the air, so called filaments. Their extraordinary properties, like backwards-enhanced
white light emission, plasma generation along their trajectories, and their electrical
conductivity provide further fascinating perspectives for applications in atmospheric
research and beyond. Examples are remote multi-component analyses of the air and
the aerosol, bio-aerosol detection, hard target analysis and even lightning control.
1. Introduction
Observing and controlling the earth atmosphere is the most important issue for
research. The total population on earth has reached a density where air pollu413
© 2006 Elsevier Inc. All rights reserved
ISSN 1049-250X
DOI 10.1016/S1049-250X(06)53011-3
414
L. Wöste et al.
[1
tion on local and global scales has severe consequences for mankind, aggravating
catastrophes like hurricanes, floods and smog. The destruction of the ozone layer
and the global warming are two prominent examples for anthropogenic changes
that occurred to the complex dynamic system of the atmosphere during the industrial age. Since the change occurs fast, its consequences for the weather, the
climate and the air composition cannot be predicted at the required precision.
This makes the problem so dangerous. Large congested areas, so called megacities, are growing all around the globe and lead into a spiral of more pollution
associated with a deterioration of living conditions and health for hundreds of
million of people. Nowadays fear of the release of deadly toxic substances by
industry, research, military, and terrorists provides further reasons for the development of reliable observation techniques. Precise and reliable measurements are,
last but not least, essential for improved weather prediction and support of traffic
and aviation.
Optical remote sensing techniques have a long tradition when it comes to atmospheric observations. Sky observation by naked eye may be the oldest method
for weather prediction. Some vital discoveries, among them the surprising detection of the ozone hole in the stratosphere, are results obtained with optical
remote sensing instruments. Its discovery is historically of special interest, because it demonstrates the importance of reliably operating observation platforms.
Satellites for global ozone monitoring where already in place (Stolarski et al.,
1986), but the unexpected catastrophe was discovered with the proven Brewerspectrometer (Farman et al., 1985). The observation was confirmed later by a
re-evaluation of the existing satellite data. Prior to that the scientists did not yet
sufficiently trust the accuracy of these data and might have ignored the ozone hole
completely.
Modern optical remote sensing instruments can be divided in active and passive devices. The former have their own light source whereas the latter utilize
naturally existent sources like the sun, the moon, the earth, or stars. Passive devices are much easier to build and to operate. Prominent examples of this species
are ground based, air borne, and space borne spectrometer or photometer. The
measurement is an integral over the whole light path and therefore the instrument
cannot provide range resolution. As a result, only models can emphasize corrections for errors resulting from inhomogeneous distributions. These instruments
commonly use lamps as a light source, which need to be placed at a suitable
distance from the detector (bi-static system). Alternatively the emitted light can
be reflected back to the place of origin (mono-static system). Depending on the
bandwidth of the light source another distinction of optical remote sensing instruments can be made. Long path differential absorption techniques, like DOAS
(differential optical absorption spectrometer) and FTIR (Fourier transform infrared spectrometer) use broadband sources. They are able to quantify a large
number of atmospheric compounds simultaneously with a very high sensitivity
2]
LIDAR-MONITORING OF THE AIR
415
(Platt et al., 1979). Mounted on space or airborne platforms they provide measurements on regional and global scales. Those advantages explain the widespread
use of long-path absorption techniques for studies in atmospheric chemistry and
global long term monitoring. Again, however, the instrumentation is only capable to measure atmospheric constituents along a defined optical path. It does not
provide 3-dimensionally resolved concentration distributions, which are most important in order to understand the dynamics of atmospheric reactions and transport
processes.
2. Conventional LIDAR Measurements
3-dimensional range-resolved optical measurements can only be achieved by employing the LIDAR techniques (Light Detection and Ranging). In such systems
a pulsed laser is commonly emitted into the atmosphere. There—after some
distance—the light is scattered, e.g., by means of Rayleigh- or Mie-scattering.
A portion of the backscattered light is then re-collected at the site of the emitter
with a telescope. A time-resolved detection scheme allows to determine the length
of the light path from the emitter to the scatter site and back to the receiver. The
second and third dimension of the measurement can be obtained by horizontally
or vertically turning either the entire system around, or by deflecting the emitted
and received beams with appropriate mirrors. In order to discriminate particular
atmospheric pollutants, the Differential Absorption LIDAR technique (DIAL) is
commonly used: It employs a set of two (or more) not too different wavelengths,
which are collinearly emitted into the air. If one of those wavelengths is more absorbed by a specific pollutant than the other, a data analysis allows to retrieve the
concentration distribution of this pollutant along the light path. A pioneer of this
method is Herbert Walther, who—in the early days—established one of the first
DIAL systems, which was used for realistic pollution monitoring purposes. One
of the results is depicted in Fig. 1 (Rothe et al., 1974); it shows an example of a
remote obtained concentration map of NO2 in the area of a chemical plant. The
result clearly indicates the location of the NO2 emission source.
Results as shown in Fig. 1 convincingly demonstrated the great monitoring power of the differential absorption LIDAR techniques. So we decided to
construct—supported by the Swiss National Science Foundation—a similar system which was mobile. This allowed us to perform DIAL measurements at sites
where most relevant results could be expected. In the Rhone valley we measured,
for example, the pollutant distributions of NO2 and SO2 . The results convincingly demonstrated—especially during inversion weather situations—the environmental damage caused by industrial emissions in such mountainous regions
(Beniston et al., 1990). We also performed DIAL measurements of traffic-caused
NO-concentrations over urban areas like Lyon and Geneva (Kölsch et al., 1992).
416
L. Wöste et al.
[2
F IG . 1. Measurement of the distribution of NO2 emission from a chemical plant by the differential
absorption technique (Rothe et al., 1974).
The results clearly indicated the architectural need of sufficient ventilation space
to avoid intolerable traffic smog formation. At that time one city was—due to is
multiple gradient situation—of greatest interest: the still divided city of Berlin:
One side of the city was mainly heated by local brown coal, the other one by
oil; one side was mainly motorized by two-stroke vehicles, the other one by
four-stroke engines, etc. We therefore brought our Swiss-licensed LIDAR system across the transit way to West Berlin and performed there in winter SO2 measurements across the city. One of the obtained results is presented in Fig. 2.
It shows an example of unpurified emission from “Kraftwerk Mitte” which—at
that time still—burnt brown coal. From there the plume spread—unhindered by
the Berlin wall—over the entire city, contributing to the creation of dangerous
winter-smog (Kölsch et al., 1994).
2]
LIDAR-MONITORING OF THE AIR
417
F IG . 2. DIAL-measurement of a SO2 -plume from a power station in former “East”-Berlin taken
in winter 1988.
Today, fortunately, the winter-smog situation in Berlin has much improved,
because the unpurified emission of industrially burnt local brown coal, which is
strongly contaminated with sulphur, is not allowed any longer. Another air pollution problem, however, the so-called photo-dynamical, or summer-smog, still
remains; due to the constantly increasing traffic density it even increases. The
cause of ozone formation results from sunlight photo-dissociation of NO2 to NO
and O. Then the radical oxygen reacts with O2 forming the ozone. In the boundary
layer near ground ozone is—due to its high toxicity—the most important ingredient of summer-smog. In order to monitor the development, we established on the
top of the centrally located Charité building in the middle of Berlin a LIDAR observation station. A typical ozone-concentration map taken over Berlin on a summer day at 100 m above ground is shown in Fig. 3. Surprisingly the distribution
reaches its highest values in the recreational area of the “Tiergarten”, where no
traffic passes (Stein et al., 1996). This clearly indicates the subtle equilibrium of
ozone-formation and -destruction mechanisms due to the Leighton-relationship.
Even more threatening than the ozone increase in the troposphere is its depletion in the stratosphere: The O3 amount of the undisturbed stratosphere is so
high that no summer smog- related increase can compensate the O3 loss in the
stratosphere and its consequences with regard to the UV-radiation shield. Again
Herbert Walther was among the first, who performed from the mountain station
“Zugspitze” DIAL measurements of stratospheric ozone concentrations (Werner
et al., 1983), see Fig. 4. Responsible for the ozone depletion process is anthropogenic CFC, which after being released into the air, slowly migrates over years
to the stratosphere. There it gets photo-dissociated by the hard UV-radiation forming active Cl. This destroys in a cyclic photo-catalytic process, e.g., on the particle
surface in natural polar stratospheric clouds (PSC) huge amounts of the ozone
(Crutzen and Arnold, 1986). The presence of such PSC’s, is therefore most important with regard to the stratospheric O3 depletion process. For this reason we
418
L. Wöste et al.
[2
F IG . 3. Horizontal distribution of the ozone concentration over Berlin taken 100 m above ground
in summer 1997.
routinely performed PSC-LIDAR measurements in Sodankyla in North Finland,
which is well located for observing the rim of the arctic vortex. Occasionally we
even detected PSC’s from the Charité station above Berlin. For that purpose we
had developed a 4-wavelength backscatter LIDAR, which was capable to provide
altitude profiles of aerosol abundances, and their size distribution. By performing
depolarization measurements we were even able to distinguish between liquid and
solid particles (Stefanutti et al., 1992).
The application of the above-mentioned “conventional” optical remote sensing
methods allowed us also to spot their limits: DOAS and FTIR detect simultaneously a large variety of air constituents and pollutants, but only along a fixed
optical path. Further, they do not provide detailed information about the aerosol.
DIAL provides up to 3-dimensional-resolved concentration maps of individual
pollutants. The amount of simultaneously detectable substances, however, is limited; usually it is just one substance at a time. Moreover in the IR-region the
method is not very sensitive, mainly because of the rather poor backscatter crosssection there. The multiple-wavelength backscatter LIDAR provides aerosol distributions at a high spatial resolution. Particle compositions, however, cannot be
determined by this method. Therefore a method was required, which does not
exhibit these disadvantages.
3]
LIDAR-MONITORING OF THE AIR
419
F IG . 4. Monthly averages of stratospheric ozone concentrations for February 1983 (lower curve)
and June 1983 (upper curve) (Werner et al., 1983).
3. The Femtosecond-LIDAR Experiment
New perspectives for optical remote sensing arose from the development of ultrafast laser sources. They open an intensity regime, in which entirely new optical
phenomena occur, and where the classical laws of optics are generally no longer
valid. Already 10 years ago our group performed encouraging laboratory experiments for the characterization of aerosol particles like water micro-droplets by
means of nonlinear femtosecond (fs) laser spectroscopy (Kasparian et al., 1997).
Then, with amplified fs-laser systems a power regime was made available, which
seemed to allow the realization of an old dream: A white light emitting artificial
star, which could freely be moved across the sky. In the laboratory this can simply be achieved by tightly focusing (centimeters) a powerful nanosecond laser
into air, which then creates the well-known plasma focus. Performing the experiment at larger distances (kilometers) is more demanding, because it requires a
significantly higher power. With the advent of high-power fs-laser sources such
420
L. Wöste et al.
[3
F IG . 5. Experimental setup of the femtosecond LIDAR.
a power level, could be reached even at large focusing lengths, provided that the
group velocity dispersion (GVD) of the spectrally broad fs-pulse was appropriately compensated.
First experiments were carried out in 1996 at the laboratory of R. Sauerbrey in
Jena, where a 100 fs-laser with 4 terawatt peak power had just been installed. For
the first experiments the laser was slightly focused with a 30 m focusing lens. The
required compensation of the group velocity dispersion of the spectrally broad
laser pulse was achieved by negatively pre-chirping the time structure of the laser
pulse, so that its differently fast propagating spectral components converged after
30 m propagation in air at the site of the focus (temporal lens). As Fig. 5 shows,
this was achieved by accordingly detuning the pulse compressor of the laser behind its last amplification stage. Then the slightly pre-focused and pre-chirped
laser beam was fired out of the laboratory into the night time sky. Identical to the
setup of a conventional LIDAR system a telescope was directed along the emitted
laser beam in order to collect the backscattered light and feed it into a spectrometer. This allowed a spectrally and temporally resolved detection of the return
signals. Surprisingly, however, the expected artificial star did not appear. Instead
4]
LIDAR-MONITORING OF THE AIR
421
F IG . 6. Photo of the first fs-LIDAR experiment in Jena: A clearly visible, white plasma channel
emerges from the emitted IR-laser beam.
of that a clearly visible, extended white light channel emerged—as shown in the
photography of Fig. 6—from the almost invisible titanium–sapphire laser pulse
(λ = 800 nm) (Wöste et al., 1997). Time resolved measurements indicated white
light signals coming from altitudes up to 12 kilometers. Their spectrum exceeded
the entire visible range. Later it was shown in the laboratory, that these amazing
plasma channels were even electrically conductive (Schillinger and Sauerbrey,
1999).
4. Nonlinear Propagation of Ultra-Intense Laser Pulses
The same effect of white light plasma-channel formation in high-intensity fs-laser
beams had briefly before also been seen in the laboratory (Braun et al., 1995).
Soon after the underlying physical principles were also described: High power
laser pulses propagating in transparent media—like air—undergo nonlinear propagation. For high intensities I , the refractive index n of the air is modified by the
Kerr effect (Kelley, 1965):
n(I ) = n0 + n2 · I,
(1)
422
L. Wöste et al.
[4
F IG . 7. Kerr self-focusing (a) and plasma defocusing (b) of a high power laser beam.
where n2 is the nonlinear refractive index of the air (n2 = 3.10–19 cm2 /W). As
the intensity in a cross-section of the laser beam is not uniform, the refractive
index increase in the center of the beam is larger than on the edge (Fig. 7a).
This induces a radial refractive index gradient equivalent to a lens (called ‘Kerr
lens’) of focal length f (I ). The beam is focused by this lens, which leads to an
intensity increase resulting in turn to a shorter focal length lens, so the whole
beam collapses on itself. Kerr self-focusing should therefore prevent propagation
of high power lasers in air. The Kerr effect becomes significant when the selffocusing effect is larger than natural diffraction, i.e. over a critical power Pcrit :
Pcrit =
λ2
.
4π · n2
(2)
If the laser pulse intensity reaches 1013 –1014 W/cm2 , higher order nonlinear
processes occur such as multi-photon ionization (MPI). At 800 nm, 8 to 10 photons are needed to ionize N2 and O2 molecules giving rise to plasma (Talebpour
et al., 1999). The ionization process can involve tunneling as well, because of the
very high electric field carried by the laser pulse. However, following Keldysh’s
4]
LIDAR-MONITORING OF THE AIR
423
F IG . 8. Filamentation of a high power laser beam as it propagates in air (Berge and Couairon,
2001).
theory (Keldysh, 1965), MPI dominates for intensities lower than 1014 W/cm2 .
In contrast to longer pulses, fs-pulses combine high ionization efficiency due
to their very high intensity, with a limited overall energy, so that the generated
electron densities (1016 –1017 cm−3 ) are far from saturation. Losses by inverse
Bremsstrahlung are therefore negligible, in contrast to a plasma generated by a
nanosecond laser pulse. However, the electron density ρ induces a negative variation of the refractive index and, because of the radial intensity profile of the laser
beam, a negative refractive index gradient. This acts as a negative lens, which
defocuses the laser beam, as schematically shown in figure Fig. 7b.
Kerr self-focusing and plasma defocusing should thus prevent long distance
propagation of high power laser beams. However, a remarkable behavior occurs in air, where both effects exactly compensate and give rise to a self-guided
quasi-solitonic propagation (Berge and Couairon, 2001). The laser beam is first
self-focused by the Kerr effect. This focusing then increases the beam intensity
and generates a plasma by MPI, which in turns defocuses the beam. The intensity then decreases and plasma generation stops, which allows Kerr re-focusing
to take over again. This dynamic balance between Kerr effect and plasma generation leads to the formation of stable structures called “filaments” (Fig. 8). These
light filaments have remarkable properties. In particular, they can propagate over
several hundreds of meters, although their diameter is only 100–200 µm, which
widely beats the normal diffraction limit. Their intensity (∼1014 W/cm2 ), their
energy (∼1 mJ), their diameter, and their electron density (1016 –1017 cm−3 ) are
almost constant.
The laser pulse propagation is governed by the Maxwell wave equation:
1 ∂ 2E
∂E
∂ 2P
(3)
·
=
μ
·
σ
·
·
,
+
μ
0
0
∂t
c2 ∂t 2
∂t 2
where σ is the conductivity and accounting for the losses, and P is the polarization of the medium. In contrast to the linear wave propagation equation, P now
contains a self-induced nonlinear contribution corresponding to Kerr focusing and
∇ 2E −
424
L. Wöste et al.
[4
plasma generation:
P = PL + PNL = ε0 · (χL + χNL ) · E,
(4)
where χL and χNL are the linear and nonlinear susceptibilities, respectively. Considering a radially symmetric pulse propagating along the z-axis in a reference
frame moving at the group velocity vg yields the following nonlinear Schrödinger
equation (NLSE) (Berge and Couairon, 2001):
ρ
∂ε
2
ε + 2i k
+ 2k 2 n2 · |ε|2 · ε − k 2 · ε = 0,
∇⊥
(5)
∂z
ρc
where ε = ε(r, z, t) is the pulse envelope of the electric field and ρc the critical
electron density (1.8 × 1021 cm−3 at 800 nm). ε is assumed to vary slowly as
compared to the carrier oscillation and to have a smooth radial decrease. In this
first order treatment, group velocity dispersion (GVD) and losses due to multiphoton and plasma absorption are neglected (σ = 0). In (5), the Laplacian models
wave diffraction in the transverse plane, while the two last terms are the nonlinear
contributions: Kerr focusing and plasma defocusing (notice the opposite signs).
The electronic density ρ(r, z, t) is computed using the rate equation (6) in a selfconsistent way with (5):
∂ρ
− γ |ε|2α (ρn − ρ) = 0,
(6)
∂t
where ρn is the neutral molecular concentration in air, γ the MPI efficiency, and α
the number of photons needed to ionize an air molecule (typ. α = 10 (Talebpour
et al., 1999)). Numerically solving the NLSE equation leads to the evolution of
the pulse intensity
I = |ε|2
as a function of propagation distance, as shown in Fig. 8. Initial Kerr lens selffocusing and subsequent stabilization by the MPI-generated plasma are well reproduced by these simulations. Notice that the filamentary structure of the beam,
although only 100 µm in diameter, is sustained over 60 m. Numerical instability related to the high nonlinearity of the NLSE prevents simulations over longer
distances.
For higher laser powers P Pcrit the beam breaks up into several localized
filaments. The intensity in each filament is indeed clamped at 1013 –1014 W/cm2
corresponding to a few mJ, so that an increase in power leads to the formation
of more filaments. Figure 9 shows a cross-section of laser beams undergoing
mono-filamentation (Fig. 9a, 5 mJ) and multi-filamentation (Fig. 9b, 400 mJ).
The stability of this quasi-solitonic structure is remarkable: filaments have been
observed to propagate over more than 300 m. However no direct measurements
could be performed on longer distances yet, because of experimental constraints.
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F IG . 9. (a) Mono-filamentation (5 mJ) and (b) multi-filamentation (400 mJ) of high power laser
beams. Scale: The size of the filament in panel (a) (200 µm) is similar to one of the many filaments on
panel (b). The rings around the filaments are colorful and correspond to conical emission.
The spectral content of the emitted light is of particular importance for LIDAR
applications. Nonlinear propagation of high intensity laser pulses not only provides self-guiding of the light but also an extraordinary broad continuum spanning
from the UV to the IR. This super-continuum is generated by self-phase modulation as the high intensity pulse propagates. As depicted above, Kerr effect leads,
because of the spatial intensity gradient, to self-focusing of the laser beam. However, the intensity also varies with time, and the instantaneous refractive index of
the air is modified as:
n(t) = n0 + n2 · I (t).
(7)
This gives rise to a time dependent phase shift dφ = −n2 I (t)ω0 z/c, where ω0
is the carrier frequency, which generates additional frequencies ω in the spectrum
ω = ω0 + dφ/dt. The smooth temporal envelope of the pulse induces thus a
strong spectral broadening of the pulse around ω0 . Figure 10 shows the spectrum
emitted by filaments that were created by the propagation of a 2 TW pulse in the
laboratory. The super-continuum spans from 400 nm to over 4 µm, which covers
absorption bands of many trace gases in the atmosphere (methane, VOCs, CO2 ,
NO2 , H2 O, etc.). Recent measurements showed that the super-continuum extends
in the UV down to 230 nm (see below), due to efficient third harmonic generation
(THG) and frequency mixing (Akozbek et al., 2002; Yang et al., 2003). These
results open further multi-spectral LIDAR applications for detecting aromatics,
SO2 , and ozone.
Most of the filamentation studies showed that white light was generated in the
filamentary structure, and leaking due to coupling with the plasma in form of
a narrow cone in the forward direction (called “conical emission”, see Fig. 9a)
(Kosareva et al., 1997; Nibbering et al., 1996). This cone spans from the longer
wavelengths in the center to the shorter wavelengths at the edge, with a typical
half-angle of 0.12◦ .
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F IG . 10. Super-continuum generation in air: the dots are measurements in the laboratory after
a few meters propagation while the lines exhibit further broadening after having propagated across
kilometer distances (Mejean et al., 2003).
However, an important aspect for LIDAR-applications is the angular distribution of the white light continuum in the near backward direction. Already in
the first fs-LIDAR experiments, a pronounced backscattering component of the
emitted white light was observed (see Fig. 6). For this reason angular resolved
scattering experiments have been performed. The emission close to the backward
direction of the super-continuum from light filaments was found to be significantly enhanced as compared to linear Rayleigh–Mie scattering (Yu et al., 2001).
Figure 11 shows the comparison of the linearly backscattered light (Rayleigh–
Mie) from a weak laser beam and the nonlinear emission from a filament, for
both s- (left part) and p- (right part) polarizations. At 179◦ the backward enhancement extends an order of magnitude. An even greater enhancement is expected at
180◦ (limited by the experimental apparatus). The enhancement may qualitatively
be attributed to a copropagating, self-generated longitudinal index gradient due
to plasma generation, inducing a back-reflection. Combined with self-guiding,
which drastically reduces beam divergence, this aspect is extremely important for
LIDAR experiments: Unlike backscatter-based, conventional LIDAR-systems a
significantly larger portion of the emitted white light is, therefore, collected by
the fs-LIDAR receiver.
To summarize, nonlinear propagation of TW-laser pulses exhibit outstanding
properties for multi-spectral LIDAR measurements: extremely broadband coher-
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F IG . 11. Angular emission of the emitted white light compared to linearly scattered light.
ent light emission (“white light laser”), confined in a self-guided beam, and an
increased back-reflection to the emitter as the laser pulse propagates.
5. White Light Femtosecond LIDAR Measurements
Since filamentation counteracts diffraction over long distances, it allows to deliver high laser intensities at high altitudes and over long ranges. This contrasts
with linear propagation, in which the laser intensity is always decreasing while
propagating away from the source, unless focusing optics such as large-aperture
telescopes with adaptative optics are used to reach focal lengths of the order of
hundreds of meters.
The distance R0 at which high powers are reached and thus where filamentation
starts, can be controlled by the following laser parameters: the initial laser diameter and divergence, and the pulse duration and chirp. The geometrical parameters
are set by the transmitting telescope while the temporal parameters, leading to
“temporal focusing” are determined by the grating compressor. These parameters
are used to control the power and the intensity of the beam while propagating.
A particular aspect is temporal focusing using an initial chirp, as it can be used
together with the air GVD, to obtain the shortest pulse duration and thus the onset of filamentation at the desired location R0 . The compressor is then aligned in
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such a way that a negatively chirped pulse is launched into the atmosphere, i.e. the
blue component of the broad laser spectrum precedes its red component. In the
near infrared, air is normally dispersive, and the red components of the laser spectrum propagate faster than the blue ones. Therefore, while propagating, the pulse
shortens temporally and its intensity increases. At the pre-selected altitude R0 , the
filamentation process starts and white light is generated.
The extraordinary properties of the white light emitting plasma channels convinced our funding agencies DFG and CNRS to establish the French/German
Cooperation project “Teramobile”, which allowed us to construct a mobile femtosecond LIDAR-system (Wille et al., 2002). Its basic setup is according to the
one shown in Fig. 5: the fs-laser pulse—after passing the compressor set as a
chirp generator—passes an emission telescope and then vertically sent into the
atmosphere. The back-scattered portion of the white light generated in the atmosphere is then collected and spectrally resolved by the LIDAR receiver. The
system was installed in a mobile, self-contained standard container, so it could
easily be moved. This allowed us to perform fs-white-light LIDAR measurements
at different, relevant sites. Most rewarding in this regard was a campaign, which
we performed at the Thüringer Landessternwarte Tautenburg (Germany). There
we could make use of the detection power of its 2 m diameter telescope, which
we operated at the high-resolution imaging mode. During these experiments the
laser was launched into the atmosphere, and the backscattered light was imaged
through the telescope of the observatory.
Figure 12a shows a typical image at the fundamental wavelength of the laser
pulse (λ = 800 nm), over an altitude range from 3 to 20 km. In this picture, strong
scattering is observed from a haze layer at an altitude of 9 km and a weaker one
around 4 km. In some cases, scattered signal could be detected from distances up
to 20 km. Tuning the same observation to the blue–green band (385–485 nm), i.e.
observing the white light super-continuum, leads to the images shown in Fig. 12b
and c. As observed, filamentation and white light generation strongly depends on
the initial chirp of the laser pulse, i.e. white light signal can only be observed
for adequate GVD pre-compensation (Fig. 12b). With optimal chirp parameters,
the white light channel could be imaged over more than 9 km. It should also be
pointed out that, as presented above, the angular distribution of the emitted white
light from filaments is strongly peaked in the backward direction, and that most
of the light is not collected in this imaging configuration.
Under some initial laser parameter settings, conical emission due to leakage out
of the plasma channel could also be imaged on a haze layer, as shown in Fig. 12d.
Since conical emission is emitted side-wards over the whole channel length, the
visible rings indicate that under these experimental conditions the channel was
restricted to a shorter length at low altitude.
Conventional LIDAR systems are based on the use of backscatter processes
like Rayleigh-, fluorescence or Mie-scattering, which only return a small fraction
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429
F IG . 12. Long-distance filamentation and control of nonlinear optical processes in the atmosphere.
Pictures of the beam propagating vertically, imaged by the CCD camera of a 2-m telescope.
(a) Fundamental wavelength, visible up to 25 km through 2 aerosol layers. (b)–(d) Super-continuum
(390–490 nm band) generated by 600-fs pulses with respectively negative (GVD pre-compensating),
positive, and slightly negative chirp. On picture (d), conical emission appears as a ring on the
high-altitude haze layer.
of the emitted light back to the observer. This leads necessarily to an unfavorable
1/R 2 -dependency of the received light, where R is the distance from the scatter
location to the observer. When spectrally dispersed, this usually leaves too small
signals on the receiver, as arc-lamp-based LIDAR experiments have shown in the
past (Strong and Jones, 1995). Unlike these linear processes, the femtosecond
white light plasma channel presents an almost ideal source for LIDAR applications: Its strong backward emission allows high spectral resolution of the observed
signals, even at large distances. As a result, highly resolved spectral fingerprints
of atmospheric absorbers along the light path can be retrieved.
Figure 13a shows examples of spectrally filtered white light LIDAR returns in
three different spectral regions (visible at 600 nm and UV around 300 and 270 nm,
1000 shots average). These profiles of in situ generated white light show scatter
features of the planetary boundary layer. The much faster decrease of the UV
signal is due to the stronger Rayleigh scattering at shorter wavelengths, and, the
absorption at 270 nm (compared to 300 nm) due to high ozone concentration. The
white light spectrum generated over long distances in the atmosphere showed sig-
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[5
F IG . 13. (a) White light LIDAR returns. The O3 absorption at 270 nm is clearly visible; (b) white
light LIDAR resolved spectrum returned from 4.5 km altitude.
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nificant differences with respect to the spectrum of Fig. 10, previously recorded in
the laboratory (Kasparian et al., 2000). Figure 13b displays the white light spectrum backscattered from an altitude of 4.5 km (Mejean et al., 2003). The infrared
part of the spectrum (recorded with filters) is significantly stronger (full line, typ. 2
orders of magnitude higher) than in the laboratory, which is very encouraging for
future multi-VOC detection. A quantitative explanation of this IR-enhancement
requires the precise knowledge of the nonlinear propagation of the terawatt laser
pulse, which cannot be simulated with the present numerical codes. However,
it qualitatively indicates that the pulse shortens and/or splits while propagating,
causing broader spectral components.
On the other end of the spectrum (not shown) it was observed that the supercontinuum extends continuously down to 230 nm (limited by the spectrometer).
This UV-part of the super-continuum is the result of efficient third harmonic generation in air (Akozbek et al., 2002; Yang et al., 2003) and mixing with different
components of the Vis-IR part of the spectrum. This opens very attractive applications, such as multi-aromatics (Benzene, Toluene, Xylene, etc.) detection
without interference, NOx and SO2 multi-DIAL detection, and O3 measurements,
for which the aerosol interference can be subtracted due to the available broadband UV detection.
Very rich features arise from the white light backscattered signal, when it is
recorded across a high-resolution spectrometer, as shown in Fig. 13b. The spectrum, which was detected from an altitude of 4.5 km with an intensified charge
coupled device (ICCD), shows a wealth of atmospheric absorption lines at high
resolution (0.01 cm−1 ). The excellent signal to noise ratio (2000 shots average)
demonstrates the advantages of using a high-brightness white light channel for
multi-component LIDAR detection. The well-known water vapor bands around
720 nm, 830 nm and 930 nm are observed simultaneously. Depending on the altitude (i.e. the water vapor concentration), the use of stronger or weaker absorption
bands can be selected. Figure 14 (upper) shows a higher resolution picture of
the spectrum around 815 nm of the water vapor ((000) → (211) transition) and
Fig. 14 (lower) of the X → A transition of molecular oxygen. A fit using the
HITRAN-database is shown in both cases. It leads to a mean water vapor concentration of 0.4%. Notice the excellent agreement with the database, demonstrating
that no nonlinear effects or saturation are perturbing the absorption spectrum.
This is explained by the fact that the white light returned to the LIDAR receiver
is not intense enough to induce saturation, and that the volume occupied by the
filaments (the white light sources) is very small compared to the investigated volume.
The spectrum used to retrieve the water vapor concentration contains about
700 data points. The use of so many wavelengths allows a major improvement in sensitivity as compared to the usual 2-wavelengths-DIAL. A systematic
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F IG . 14. Upper: The water (000) → (211) transition. Lower: high-resolution spectrum of the
X → A transition of molecular oxygen. The comparison with calculated results using the HITRAN-database shows excellent agreement in both cases. It leads to a mean water vapor concentration
of 0.4%.
study is in progress to quantify this gain, connected to the obtained signal-tonoise-ratio in each spectral element. Using a gated ICCD, the spectrum of the
atmosphere can be recorded at different altitudes, yielding range resolved measurements.
Information about atmospheric temperature (and/or pressure) could be retrieved from the line-shapes of the absorption lines. Another possibility is to
measure the intensities of the hot bands, in order to address the ground state population. As the molecular oxygen spectrum is very well known, O2 is particularly
well suited for this purpose. The access of the whole spectrum should again allow to obtain significantly better precision than in former DIAL investigations
(Megie and Menzies, 1980). The spectrum covered by the white light in Fig. 13b
gives access to many bands to measure the H2 O concentration, and to 2 bands
of O2 ((0) → (0) and (0) → (1) sequences of the X → A transition, around
760 nm and 690 nm, respectively) to determine the atmospheric temperature. The
combination of both information with good precision could allow to construct an
efficient “relative humidity LIDAR profiler” in the future.
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6. Nonlinear Interactions with Aerosols
Particles are present in the atmosphere as a broad distribution of size (from 10 nm
to 100 µm), shape (spherical, fractal, crystals, aggregates, etc.), and composition (water, soot, mineral, bio-agents, e.g., bacteria or viruses, etc.). The LIDAR
technique has shown remarkable capabilities in fast 3D-mapping of aerosols, but
mainly qualitatively through the measurement of statistical average backscattering and extinction coefficients. The most advanced methods use several wavelengths, usually provided by the fixed outputs of standard lasers. The set of
LIDAR equations derived from the obtained multi-wavelength LIDAR data is
subsequently inverted using sophisticated algorithms or multi-parametric fits of
pre-defined size distributions with some assumptions about the size range and
complex refractive indices. In order to obtain quantitative mappings of aerosols,
complementary local data (obtained with, e.g., laser particle counters, or multistage impactors to identify the sizes and composition) are often used together
with the LIDAR measurements. The determination of the size distribution and
composition using standard methods must, however, be taken cautiously as complementary data, because of its local character in both, time and space. Nonlinear
spectroscopy using ultra-short laser pulses appears as a promising new technique
for a quantitative aerosol detection. In this section, we describe important nonlinear interactions that exhibit a real potential to simultaneously measure size
and composition of aerosol mixtures, and to identify a particular type of particle within an ensemble.
The first approach of using ultra-intense laser pulses in a LIDAR arrangement
to characterize aerosol particles is a direct extension of the multi-wavelength scattering technique. The extraordinary broadness of the super-continuum, spanning
from the UV to the mid-IR, opens new perspectives in this respect. Instead of
some discrete wavelengths, backscattered and extinction coefficients can now be
obtained range-resolved over the whole continuum. This is particularly advantageous for mixtures of unidentified particles, where a wide range of size parameters
x = 2πr/λ has to be addressed. The data inversion can follow the stream of
the already developed sophisticated multi-wavelength algorithms, which will be
very powerful using such a wide and continuous spectrum. Besides the experiments performed with the Teramobile system (Kasparian et al., 2003), some
recent LIDAR measurements of aerosols using the femtosecond super-continuum
generated in rare gas before transmitting were reported (Galvez et al., 2002). Part
of the spectrum might also be analyzed at higher resolution, in order to detect variations in the imaginary part of the refractive index, characterizing the absorption
process and thus provide some further insight on the particles composition.
A key parameter in these femtosecond LIDAR experiments will be the location
of the onset and the end of filaments. In particular, if the pulses are shaped in such
a way (negative chirp, see above) that filamentation occurs at short distances and
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lasts only for some hundreds of meters, the light scattered back from longer distances can be considered as linearly scattered. The data inversion can thus safely
be performed with the usual linear LIDAR algorithms. Conversely, if the laser
pulses are initially shaped in a way that high intensity and filaments are present
in the investigated volume, nonlinear effects are induced directly in the aerosols,
and new inversion algorithms have to be developed (Faye et al., 2001; Kasparian
and Wolf, 1998). Examples of these nonlinear processes induced in the aerosol
particles are presented below.
Femtosecond laser pulses are able to provide very high pulse intensity at low
energy, which allows to induce nonlinear processes in particles without thermal
deformation effects. The most prominent feature of nonlinear processes in aerosol
particles, and in particular in spherical micro-droplets, is the strong localization
of the emitting molecules within the particle, and the subsequent backward enhancement of the emitted light. This unexpected behavior is extremely attractive
for LIDAR applications. For homogeneous spherical micro-particles, molecules
in certain regions are indeed more excited than others because of the focusing
properties of the spherical micro-resonator. Further localization is achieved by the
nonlinear processes, which typically involve the nth power of the internal intensity I n (r) (r denotes the position inside the droplet). Because the droplet acts as
a spherical lens, the re-emission from these internal focal points occurs predominantly in the backward direction. The backward enhancement can be explained by
the reciprocity principle (Boutou et al., 2002; Hill et al., 2000): Reemission from
regions of high I (r) tends to return toward the illuminating source by essentially
retracing the direction of the incident beam that gave rise to the focal point.
We investigated, both theoretically and experimentally, incoherent multiphoton processes involving n = 1 to 5 photons (Boutou et al., 2002; Favre et al.,
2002; Hill et al., 2000). For n = 1, 2, 3, we focused on multi-photon-excited fluorescence (MPEF) of fluorophors- or amino acids-containing droplets. For n = 5
(or more) photons we examined laser-induced breakdown (LIB) in water microdroplets, initiated by multi-photon ionization (MPI). The ionization potential of
water molecules is Eion = 6.5 eV (Noack and Vogel, 1999; Williams et al.,
1976), so that 5 photons are required at a laser wavelength of 800 nm to initiate
the process of plasma formation. The growth of the plasma is also a nonlinear
function of I (r). We showed that both localization and backward enhancement
strongly increases with the order n of the multi-photon process. Both MPEF and
LIB have the potential of providing information about the aerosol composition.
The strongly anisotropic spontaneous emission of MPEF in a micro-droplet
was demonstrated on Coumarin 510 doped ethanol (Hill et al., 2000) droplets with
sizes ranging from 10 to 50 µm. The directionality of the emission is dependent
on the increase of n, because the excitation process involves the nth power of the
intensity I n (r). The ratio Rf = U (180◦ )/U (90◦ ) increases from 1.8 to 9 when
n changes from 1 to 3. For 3-photon MPEF (3PEF), fluorescence from aerosol
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435
micro-particles is, therefore, mainly backwards emitted, which is ideal for LIDAR
experiments.
The backward enhancement also depends on the particle relative refractive index m: the higher m the higher Rf . When excited by one photon at 266 nm,
Rf from dye-doped polystyrene latex (PSL) micro-spheres (m = 1.59, typical
diameter = 22.1 µm, fluorescence peaked at 375 nm) reaches 3.2 instead of
1.8 for Coumarin 510 doped ethanol droplet. Such enhancement effect is also
observed for nonspherical transparent particles such as clusters of small (diameter smaller than 2 µm) PSL-spheres (Yong-Le et al., 2002). Another remarkable
property is that the backward enhancement is insensitive to the size if the droplet
diameter exceeds some micrometers. This was shown by both the calculations
for liquid spherical droplets and by experiments on clusters of PSL-spheres, for
which the equivalent diameter was changed from 2 to 10 µm. However, although
the Rf ratio is not sensitive to the particle shape at least for a one-photon excitation process, the high resolution 2D-angular pattern in the near backward
direction might be specific of its morphology.
Laser-induced breakdown (LIB) experiments were performed in pure and
saline water droplets. The white light (500±35 nm) angular distribution was measured in the scattering plane for an incident intensity of 1.8 × 1012 W/cm2 . The
observed far-field emission is strongly enhanced in the backward direction, and
exhibits a secondary narrow lobe near 150◦ . The agreement between the experimental results and our Lorentz–Mie calculations (Favre et al., 2002) is excellent.
LIB then takes place only at the internal hot spot of the droplet, and generates
a plasma of nanometric dimensions because of the I 5 (r) dependency of multiphoton ionization (MPI). The white light emitted by the nanoplasma has a ratio
Rp = Up (180◦ )/Up (90◦ ) that exceeds 35, i.e. 3× higher than for 3PEF.
The spectrum and the related plasma temperature have been measured by using
an optical multi-channel analyzer (OMA). The broadband visible emission was
recorded in the backward direction from pure and saline droplets with various
incident intensities. In Fig. 15a we show that in the case of saline droplets and
for an incident intensity Iinc = 1.6 × 1012 W/cm2 , the spectrum can be fitted
by a Maxwell–Planck law, in agreement with laser heated plasma emission. Also,
when the incident intensity is gradually increased to 1013 W/cm2 (curves (b)
and (c)), the emission spectrum shifts towards the blue consistent with an increase
of the plasma temperature from 5000 to 7000 K. Similar behavior has been observed for pure water droplets (Fig. 15d) but unexpected and unidentified atomic
or molecular lines appear in the spectrum. The shift of the emission maximum is
correlated to the change in the angular distribution.
Dye-doped micro-droplets, because of their high multi-photon absorption
cross-sections, are good test cases to demonstrate the advantage of combining
MPEF and LIDAR techniques in order to identify the presence of fluorescing
aerosols. An attractive application of the combined techniques is bio-aerosol de-
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F IG . 15. Broadband emission spectra of saline water droplets (a), (b), (c) irradiated at increasing
intensities from 1.6 × 1012 W/cm2 (curve (a)) to 1013 W/cm2 (curve (c)). The spectra exhibit besides
the Na D lines continua that can be fitted to the Maxwell–Planck law, yielding plasma temperatures
from 5000 to 7000 K. Pure water droplets (d) show a similar behavior.
tection in the atmosphere. For this purpose, we performed the first multi-photon
excited fluorescence LIDAR detection of biological aerosols. The particles, consisting of water droplets containing 0.03 g/l riboflavin (a characteristic tracer of
bio-agents (Hill et al., 2001; Pan et al., 2001), were generated at a distance of 50 m
from the Teramobile system. The size distribution peaked around 1 µm, a typical
size of airborne bacteria. Riboflavin was excited with two photons at 800 nm and
emitted a broad fluorescence around 540 nm. This experiment is the first demonstration of remote detection of bio-aerosols using a 2PEF-femtosecond LIDAR
(Fig. 16) (Kasparian et al., 2003). The broad fluorescence signature is clearly observed from the particle cloud (typ. 104 p/cm3 ), with a range resolution of a few
meters. As a comparison, droplets of pure water did not exhibit any parasitic fluorescence in this spectral range. However, a background is observed for both types
of particles, arising from the scattering of white light generated by the filaments
in air. Competition between super-continuum generation before the laser reaches
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F IG . 16. Two-photon excited fluorescence (2PEF)-LIDAR detection of bio-aerosols compared to
water-aerosols.
the particles and 2PEF within the particles appeared critical. A possible solution to this problem is to adapt the initial pulse duration, chirp, and geometrical
characteristics of the laser such that the needed high intensity is reached exactly
at the target location. The use of tailored pulses is under investigation to solve
this problem; they will also be used to investigate possible simultaneous size and
composition measurements in a pump–probe frame.
MPEF might be advantageous as compared to linear laser-induced fluorescence
(LIF) for the following reasons: (1) MPEF is enhanced in the backward direction
and (2) the transmission of the atmosphere is much higher for longer wavelengths.
For example, if we consider the detection of tryptophan (another typical bio-tracer
that can be excited with 3 photons of 810 nm), the transmission of the atmosphere
is typically 0.6 km−1 at 270 nm, whereas it is 3 × 10−3 km−1 at 810 nm (for a
clear atmosphere, depending on the background ozone concentration). This might
compensate the lower 3-PEF cross-section compared to the 1-PEF cross-section
at longer distances. The most attractive feature is however the possibility of using
pump–probe techniques to measure both, composition and size.
7. Conclusion
The nonlinear propagation of ultra-short ultra-intense laser pulses provides
unique features for LIDAR applications: a coherent white light emitting super-
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continuum, which is self-guided and back-reflected towards the source. Backward enhancement also occurs for multi-photon-excited fluorescence (MPEF)
and laser-induced breakdown (LIB) processes in aerosol particles. These characteristics open new perspectives for LIDAR measurements in the atmosphere:
multi-component detection, reduced spectral interference, better precision due
to more absorption lines, improved IR-LIDAR measurements in aerosol-free
atmospheres, and remote measurement of aerosols size distributions and compositions. The wide spread of the technique needs further characterization of the
propagation of laser pulses, in order to foresee the onset and the length of the
filaments, and to better control the intensity at each location along the laser path.
Beyond LIDAR applications the observed plasma channels exhibit fascinating
further perspectives; one of them is remote laser-induced breakdown spectroscopy
(LIBS): By irradiating, for example, copper and iron plates with fs-plasma channels, we have generated and identified their plasma lines over distances of 100 meters. Another exciting application is in the field of lightning control. The filaments
are electrically conductive; they may therefore be used as a laser lightning rod. In
first exploratory experiments, which we performed at a high-voltage facility in
Berlin, we could show that high-voltage discharges could indeed be triggered and
guided—so far still over distances of some meters (Kasparian et al., 2003). Another application concerns the triggered nucleation of water droplets. In a supersaturated atmosphere the laser-induced charges act—like in fog chambers—as
condensation germs for droplet formation. The phenomenon allows the remote
detection of super-saturation in the atmosphere.
As presented above, the potential applications of femtosecond plasma channels
in air are exciting, numerous and wide spread. It all resulted from an unsuccessful
attempt:
The creation of an artificial star!
8. Acknowledgements
The authors wish to thank the entire Teramobile team for the numerous hard working nights, during which the results presented here were achieved. We owe particular thanks to Roland Sauerbrey, André Mysyrowicz, Jerome Kasparian, Estelle
Salmon, Jin Yu, Miguel Rodriguez, Holger Wille, Yves Bernard Andre, Michel
Franco, Bernard Prade, Stelios Tzortzakis, Guillaume Mejean, Didier Mondelain
and Riad Bourayou. Also we want to acknowledge the financial support of CNRS
and DFG. One of the authors (L.W.) wishes to thank Professor Herbert Walther
for having filled him with enthusiasm for spectroscopy, lasers and LIDAR.
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Beniston, M., Beniston-Rebetez, M., Kölsch, H.J., Rairoux, P., Wolf, J.P., Wöste, L. (1990). Use of
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Index
Abbe’s theory of microscope, 106
– and wavefront coding, 111–15
Ablation, 238–9
– from silicon, 239–40
– – recoil pressure, 240–2
– – surface morphology, 242–6
– see also Coulomb explosion; Phase explosion
Ablation regime, 234
Ablation threshold, 234
Acousto-optic modulators (AOMs), 95, 96
Aerosols, 433–7
Ambiguity functions, 110
Amplitude object transmission function, 112
Ancilla modes, nature, 144–7
Anderson localization, 46
Annihilation operators, 259, 261, 360
Anti-bunching, 7, 14, 255
Arbitrary power-law trap, 351–3
– cross-excitation parameter, 352–3
– ideal gas BEC statistics in, 364–70
– with interacting Bose gas, 381–2
– single-particle energy spectrum, 351, 364
– see also Box; Harmonic trap; Isotropic harmonic trap
Artificial atoms see Quantum dots
Aspheres, manufacture, 129–30
Asymmetry coefficient, 382
Atmosphere
– parameters measurement, 432
– super-saturation detection, 438
Atom-field interaction, 191–4
Atom laser
– linewidth problem, 359
– phase fluctuations of matter beam, 394
Atom Trap Trace Analysis, 79
Atomic conductance, fluctuations, 42–6
Atomic photo cross sections, Ericson
fluctuations in, 49–51
Atomic quasimomentum, 66
Atomic transport, directed, due to interactioninduced quantum chaos, 55–9
ATTA method, 79
Autocorrelation function, 17
– of average detection probability, 276–7, 285
– of pupil function, 113
Avalanche processes, 238
Average detection probability, 259–60, 271,
273–4, 279–80
Barcode reading system, 109–10, 111
BB84 protocol, 23, 26–7
BCS state, momentum distribution of molecules from, 180–1
Beam splitters, 146, 147, 261–2
BEC
– counting statistics of molecules, 168–70
– discovery, 307–8
– experimental demonstration, 297
– momentum distribution of molecules,
177–8
– spinor-BECs, in spin-dependent optical lattices, 223
– wave-packet motion, 223
BEC-BCS crossover, 152, 173
BEC fluctuations
– future research, 394
– ground-state occupation distribution, 346–7
– in ideal Bose gas, 323–8, 399–400, 401–2
– in interacting Bose gas see Interacting Bose
gas
– relations between statistics in various ensembles, 316–20
– see also Condensate master equation approach; Quasiparticle approach; Systems with broken continuous symmetry
Beliaev–Popov approximation, 394
Benand–Marangoni instability, 247
Berlin, 416–17
Biexcitons, 5, 16
Bifurcations, 246
Bilinear transfer function, 115, 116
Binomial distribution, 348
Binomial theorem, generalized, 396
Bio-aerosol detection, 435–7
443
444
Blackbody radiation, 299
Bloch oscillations, fermionic, 57
Bloch vector model, 87
Blood oxygenization determination, 123–5
Bloodflow measurement, 125–7
Blue-detuned lattices see Gray lattices
Bogoliubov canonical transformation, 373
Bogoliubov coupling, 373–4
Bogoliubov–Popov energy spectrum, 373,
379
Bogoliubov’s 1/k 2 theorem, 385, 389
Bohr–Sommerfeld quantization rule, 300
Born–Oppenheimer approximation, 213
Bose, Satyendranath, 293–4, 298–9
Bose–Einstein condensation see BEC
Bose–Einstein distribution
– history, 298–315
– – analysis, 306–14
– – Bose contribution, 299–303
– – comparison between microstate counting
ways, 314–15, 395–7
– – Einstein contribution, 303–6
Bose–Hubbard Hamiltonian, 36, 55–6
Boson–fermion model Hamiltonian, 163–4
Bosonic bath, 56–9
Bosonic commutation relation, 140
Box
– BEC fluctuations in ideal gas in, 369–70
– temperature scaling of BEC fluctuations,
376, 382–3
Breathing-mode wave-packets, 197, 200
Broad resonance, 176
Caesium atom, 77–8, 80–1, 84–6, 88–9, 254
– see also Neutral atoms; Single atoms
Cahn–Hilliard type equation, 247
Calcium ion, single photon generation, 254
Canonical ensemble, 335
– counting statistics, 317
– N -particle constraint, 316, 357
– quasiparticle approach see Quasiparticle
approach
Canonical partition functions, 318, 346, 404
– contour integral representation, 404
– for one-dimensional harmonic trap, 347
Cataract surgery, 127
Cavity-QED, 77, 84, 94, 95, 99
Central limit theorem, 376
Central retinal artery, 125, 126
CFC, 417
Index
Chaotic cavities, photonic transport in, 51–5
Choroid, 125, 126, 127
Chromium atom, feedback scheme in MOT,
82
Clebsch–Gordan coefficients, 192
Coherence function, 111–12
– second-order, 7, 17
Coherence theory, 175
Coherence time, 255–6, 286
Coherent backscattering, 55
Coherent pumping, 156
Coincidence probability see Joint detection
probability
Cold atom physics
– quantum optics and, 152
– see also Ultra-cold molecules
Cold collisions, 81, 84, 100–1
Computer numerical controlled (CNC)
machines, 129, 130
Condensate master equation approach, 297
– BEC statistics, 350–5
– derivation of equation, 335–41
– laser phase-transition analogy, 334–5, 342
– low temperature approximation, 342–3
– mesoscopic and dynamical effects in BEC,
355–7
– physical interpretation of coefficients, 340
– quasithermal approximation for noncondensate occupations, 344–5
– solution of equation, 345–9
Conical emission, 425, 428, 429
Cooling coefficient, 338–9, 340
– low temperature approximation, 342
– quasithermal approximation, 345
Cooper instability, 172
Cooper pairs, 180
– coherent conversion, 173
– detection, 174–5
Corkscrew cooling, 206, 207
Corneal reflex, 128
Correlation function
– connected four-point, 387
– with jitter, 270, 272, 274
– longitudinal, 386
– second-order, 263, 264–6
– transverse, 386
Corrugation, variation of, 248
Coulomb explosion, 238, 239, 241, 242, 246
Counting statistics, 168–73, 317
Coupled atom–molecule system, 155–8
Index
– see also Molecular micromaser
Coupled fermion–molecule system, 175–7
Creation operators, 257, 259, 261, 360
Critical power, 422
Critical temperature, 356–7, 383
Cross-correlation function, 16, 18, 20
Cubic phaseplates, 107–8, 117–20
Cycling transition, 192
Decoherence, suppression, 86
Degenerate approximation, 164–5
Delbrück, M., 307, 313
Density operator
– photon pairs with jitter, 270
– single-photon light field, 258
Dephasing, suppression, 86
Depth of focus (DOF), 106
Depth resolution, lateral resolution and, 106
Desorption dynamics, 234–8
Diabatic potentials, 193
DIAL, 415, 431, 432
Dicke superradiance, 152, 164
Differential optical absorption spectrometer
(DOAS), 414, 418
Diffusive dynamics, 80–1
Dipole trap (DT)
– controlling atoms’ positions, 94–9
– loading multiple atoms into, 89–91
– optical conveyor belt, 95, 96
– preparing single atoms in, 82–4
DOAS, 414, 418
Dressed atom approach, 228
Drift-mode ToF spectra, 238–9
Dynamical localization, 46
Dynamical master equation see Condensate
master equation
Effective nonlinear σ model, 385–8
Einstein, Albert, 294, 303, 307
Ekert protocol, 26
Electro-optic feedforward amplifier, 148
Electro-optic modulators (EOMs), 27
Elliptic islands, 59–61
Emission-time jitter, 273–5, 285–6, 287
Energy coupling, 229–33, 239
Energy gap, in trap, 377, 379, 380
Energy spectrum, effective, 380
Entanglement schemes, four-photon, 100
Equilibrium entropy, of ideal gas, 306
Ericson fluctuations, 49–51
445
Ethanol, 434–5
Excess coefficient, 382
Excitons, 5, 16
Eye, length measurement, 127–9
Fermi gas with superfluid component, 171–3
Fermions, interaction with bosonic bath,
56–9
Filamentation, control of start, 427–8
Filaments, 423
Finite negative binomial distribution, 347–8
Flatness, 383
Floquet–Bloch operator, 36–8
Floquet operator, 38–40
Fluctuation–dissipation theorem, 53, 385
Flux operator, 258, 259
Fock regime, 161–2
Fokker–Planck equation, 342
Fourier transform infrared spectrometer
(FTIR), 414, 418
Free carrier absorption, 233, 239
Frequency jitter, 271–3, 285–6, 287
Frozen planet configuration, 64–5
FTIR, 414, 418
Fundus
– bloodflow measurement, 125–7
– imaging systems, 123
– – numerical aperture, 106
Fundus reflex, 128–9
Gamow factor, 198–9
Gauge potential, 213
Generalized binomial theorem, 396
Generalized Zeta functions, 371, 388
Generating cumulants, 362
Ginzburg–Landau type free energy, 342
Giorgini–Pitaevskii–Stringari result, defense,
392–3
Girardeau–Arnowitt operators, 357
Glauber coherent field, 169
Glaucoma, 123
Goldstone fields, 385, 386
Good quantum numbers, 35
Grand canonical ensemble, 321–8, 335,
389–90
– chemical potential, 325
– condensate fluctuations in ideal Bose gas,
323–8, 399–400, 401–2
– condensate order parameter for ideal Bose
gas, 390
446
– counting statistics, 317
– mean condensate particle number in ideal
Bose gas, 321–3, 397–9
– mean noncondensate occupation, 325
Grand canonical partition function, 318, 404
Gravitational constant, high precision measurements, 67
Gray lattices, 200–4
– influence of magnetic fields on tunneling,
213–19
– periodic well-to-well tunneling in, 208–13
– see also Sloshing-type wave-packets
Group velocity dispersion (GVD), 424
Hadamard gates, 94
Hanbury Brown–Twiss setup, 8, 14–15, 17
Harmonic trap, 350–1, 369
– see also Isotropic harmonic trap
He II superfluid, 297
Heating coefficient, 338–9, 340
– low temperature approximation, 342
– quasithermal approximation, 345
Helium, liquid, Lambda point, 295
Helium atom, 63–5
Helium spectrum, semiclassical elucidation,
35
Hemoglobin, absorption coefficients, 123
Hohenberg–Mermin–Wagner theorem, 387
Homogeneous broadening limit, 165
Husimi phase space projections, 46, 48
Hydrogen atom, microwave driven, 41, 44–6
Ideal Bose gas
– central moments, 348–9
– condensate fluctuations, 323–8, 399–400,
401–2
– exact recursion relation for number of condensed atoms, 320–1
– master equation, 336–7
– mean condensate particle number, 321–3
– – analytical expression for, 397–9
Ideal gas + thermal reservoir model, 336
Image intensity distribution, partial coherent
illumination, 115, 116
Impact ionization, 233
Incubation, 229–30, 236
Inhomogeneously broadened Tavis–
Cummings model, 164, 170, 171
InP quantum dots, 11–13
Intensity modulation amplifier, 147
Index
Interacting Bose gas
– BEC fluctuations as anomalously large and
non-Gaussian, 375–9
– – cumulants, 374–5
– canonical-ensemble quasiparticles in
Bogoliubov approximation, 372–4
– characteristic function for total number of
atoms, 374
– crossover between ideal and interactiondominated BEC, 379–83
– mesoscopic effects, 383
– pair correlation effect, 380–1
– see also Systems with broken continuous
symmetry
Interaction energy, 377, 379
Interaction volume, 242
Interaction-induced, quantum chaos, 55–9
Interferogram, spatial modulation frequency,
114–15
Ion beam erosion, 247
Ion etching configurations, 246
Ion traps, selective addressing in, 91
Ionization yield, of one electron Rydberg
states under microwave driving, 42–4
Irregular level dynamics, 35–6
Isotropic harmonic trap
– condensate particle number, 322–3, 405–8
– cross-excitation parameter, 351
– with interactive Bose gas, 382
Jaynes–Cummings interaction, 156, 157
Jitter, 270–7, 285–7
Joint coherence operators, 165
Joint detection probability
– with emission-time jitter, 274–5
– with frequency jitter, 272, 273
– interference without time resolution, 264,
266–7
– for perpendicular polarized photon pairs,
276, 277
– time-resolved interference, 264, 267–70,
280–5
Josephson regime, 161, 162
KAM theorem, 40, 60–1
Kerr effect, 421–2
Kerr self-focusing, 423, 424
Kicked cold atoms, 66–7
Kicked harmonic oscillator
– Floquet operator, 38–40
Index
– mean energy, 47
– web-assisted transport in, 46–9
Klystron, 330–2
Kolmogorov–Arnold–Moser (KAM)
theorem, 40, 60–1
KPZ type equation, 247–8
Kuramoto–Sivashinsky type equation, 247
Lamb–Dicke effect, 196, 200
Lamb–Dicke parameter, 40, 46, 200
Lambda point, in liquid helium, 295
Landau–Zener transitions, 193
Laser(s)
– in chaotic resonators, 52
– emission properties, 54
– quantum theory of, 329–34, 402–3
– see also Random lasers
Laser cooling, types, 190–1, 206–8
Laser-induced breakdown (LIB), 434, 435
Laser-induced breakdown spectroscopy
(LIBS), 438
Laser-induced fluorescence (LIF), linear, 437
Laser interaction with solid surfaces, 227–49
– discussion, 246–9
– energy coupling, 229–33, 239
– secondary processes, 233–46
– – see also Ablation; Desorption dynamics
Laser master equation, 332–4
Laser pulse propagation, 423
Laserscanning microscope, 112–13
Lateral resolution, depth resolution and, 106
Lateral resolution limit, 106
LIB, 434, 435
LIBS, 438
LIDAR (Light Detection and Ranging)
– applications, 425, 438
– conventional, 415–19, 428–9
– Differential Absorption (DIAL), 415, 431,
432
– femtosecond, experimental setup, 419–21
– nonlinear interactions with aerosols, 433–7
– nonlinear propagation of ultra-intense laser
pulses, 421–7
– tailored pulses, 437
– white light femtosecond, 427–32
Light fields, 190–1
Light quanta
– indistinguishability, 301, 303, 307
– momentum, 299
Lightning control, 438
447
Lin-perp-lin configuration, 197
Linear optical coherence tomography
(LOCT), 120, 130–3
Linear optical device, general linear input–
output transformation, 140–1
Linear optical quantum computation
(LOQC), 29, 254, 255
Linear optical system theory, 106
LOCT, 120, 130–3
Long wavelength phase fluctuations, 384–5
Longitudinal correlation function, 386
Longitudinal susceptibility, 386, 389
LOQC, 29, 254, 255
Lorentz–Mie calculations, 435
Luggage identification, 110, 111
Lyapunov exponent, 50, 51
Macular degeneration, 123
Magnetic field gradients method, 92–3
Magnetic-field-induced laser cooling
(MILC), 204
Magnetic-field-induced lattices, 204–6
Magnetization, 385, 386, 387
Magneto-optical trap see MOT
Mandel Q-parameter, 159
Many-body Hamiltonian, excitation spectrum, 56
Matter waves, many-body, 313
Maxwell–Planck law, 435
Maxwell wave equation, 423
Maxwell’s demon ensemble, 324, 325, 326,
363
Mean-field Popov approximation, 364
Metal–Organic Vapor Phase Epitaxy
(MOVPE), 6, 11
Metrology, industrial, 106
Michelson add/drop filter, 22–6
– application to quantum key distribution,
26–9
Michelson interferometer, 14–15
Microcanonical ensemble, 311, 335
– counting statistics, 317
– energy conservation, 316
– ground-state fluctuations in onedimensional harmonic trap, 317
– ground-state occupation probability, 321
– particle number conservation, 316
– partition function, 321
Microcanonical partition function, 318, 346
Micro-photoluminescence, 9–10
448
Microscope
– laserscanning, 112–13
– theory of, 106, 111–15
Microstate counting
– Bose way, 301, 303, 307, 308
– classical way, 310
– comparison between Bose and Einstein
ways, 314–15, 395–7
MILC-type lattices, 204–6
Mode function, 257
– see also Jitter
Modulation transfer function (MTF), 107
– of focused diffraction limited system, 108,
109
– of laserscanning microscope, 113
Molecular Beam Epitaxy (MBE), 6
Molecular damping, 156
Molecular fields, counting statistics, 168–73
Molecular formation, passage time statistics,
163–8
Molecular micromaser, 153–63
– model, 154–8
– results, 158–63
– see also Coupled atom–molecule system
Mono-filamentation, 424, 425
MOT, 77
– single atoms in, 77–82
– transfer of atoms to dipole trap, 82–4
– – modified procedure, 90–1
MOVPE, 6, 11
MPEF, 434, 435, 437
MPI see Multi-photon ionization
MTF see Modulation transfer function
Multi-filamentation, 424, 425
Multinomial theorem, 396
Multi-photon cascades, 16–22
Multi-photon-excited fluorescence (MPEF),
434, 435, 437
Multi-photon ionization (MPI), 231–3,
422–3, 424, 434, 435
Multiplexed quantum cryptography, 22–9
Multiplexing, 22
Multi-wavelength algorithms, 433
Nanocrystals, 3–4
Narrow resonance, 176
Nernst’s theorem, 310
Neutral atoms, 76–7
– Bose–Einstein condensation with, 76
– entanglement, 99–101
Index
– localization in space, 76
– position control, 94–9
– single see Single atoms
– see also Dipole trap
No-cloning theorem, 23, 26
NO2 emission distribution, 415–16
Nondegenerate Tavis–Cummings model,
164, 170, 171
Nondispersive wave packets, 60–1
– in kicked cold atoms, 66–7
– in one particle dynamics, 61–3
– in three body Coulomb problem, 63–5
Nonequilibrium Keldysh diagram technique,
391, 394
Nonlinear Schrödinger equation (NLSE), 424
Normal Fermi gas (NFG)
– counting statistics of molecules from,
170–1
– momentum distribution of molecules from,
179
Numerical aperture (NA), 106
Occupation number operator, 359
OCT see Optical coherence tomography
One-dimensional lattice configurations, 196–
208
Optical amplifiers, 139–48
Optical attenuators, 140, 141, 147
Optical cavity-QED, 77, 84, 94, 95, 99
Optical coherence tomography (OCT), 107,
120–36
Optical communication, 148
Optical conveyor belt, 95, 96
Optical high-finesse resonator, 99–100
Optical lattices, 187–223
– applications, 188
– atom-field interaction, 191–4
– future research, 223
– light fields, 190–1
– quantum Monte-Carlo wave-function simulations see QMCWF simulations
– see also Gray lattices; One-dimensional
lattice configurations; Sloshing-type
wave-packets
Optical manufacturing, 129–30
Optical multi-channel analyzer (OMA), 435
Optical pumping, 80
Optical remote sensing instruments, 414–15
Optical transfer function (OTF), 107, 113,
117
Index
– inverse, 108
Optical tweezers, 98
Ozone, 414, 417–18, 437
P -representation, 342
Parametric amplifiers, 140, 144
Partial coherent illumination, 115–17
Partial coherent imaging, theory of, 115
Passage time statistics, of molecular formation, 163–8
Paul trap, 76
Pauli Exclusion Principle, 152
Pegg–Barnett phase states, 162
Perturbation depth, 247
Perturbation Theory, 232
Petermann factor, 53
Phase explosion, 239, 242
Phase insensitive amplifiers, 139–40, 141–3
– multimode, 143–4
Phase insensitivity, 142
Phase sensitive amplifiers, 140
Phase space structure, mixed regular chaotic,
40–1
Phaseplates, 107–8, 117–20
Photo-association, 155, 163–8
Photo-dissociation, 166–8
Photography, wavefront coding in, 136
Photon detection, theory of, 177
Photon ensemble, temporal envelope, 255
Photonic localization length, 44
π -pulses, 86
Planck’s law of radiation, 311
Plasma channels, 421, 438
Plasma defocusing, 423, 424
Point spread function (PSF), 108, 110, 116
Polar stratospheric clouds (PSC), 417–18
Polishing robot, 119, 129
Politzer asymptotic approximation, 326, 327
Polystyrene latex (PSL), 435
Porter–Thomas distribution, 54
Posterior ciliary artery, 125, 126
Power exchange, 209, 220
PSF see Point spread function
Pupil function, 107, 113
Purcell effect, 22
QMCWF simulations, 194–6, 199, 211
– influence of magnetic fields on tunneling,
214, 215, 216
449
– sloshing-type wave-packet motion, 220,
221
Quadrature-phase amplitudes, 381
Quantum accelerator modes, 66–7
Quantum chaos, 34–68
– applications, 34, 68
– control through, 59–67
– – see also Nondispersive wave packets
– interaction-induced, directed atomic transport due to, 55–9
– spectral properties, 34–41
– see also Quantum transport
Quantum cloning, 140
Quantum cryptography see Multiplexed
quantum cryptography
Quantum dots, 4–7, 11–13
Quantum gates, 84, 89, 99–101
Quantum information processing, 2, 68, 84,
188
– see also LOQC; Quantum gates; Quantum
registers
Quantum key distribution, 26–9
Quantum memory, 68
Quantum Monte-Carlo wave-function
(QMCWF) simulations, 194–6, 199,
211
– influence of magnetic fields on tunneling,
214, 215, 216
– sloshing-type wave-packet motion, 220,
221
Quantum numbers, good, 35
Quantum optics, 34
– cold atom physics and, 152
– of ultra-cold molecules see Ultra-cold
molecules
Quantum phase gate, 100
Quantum registers, 91–4
Quantum resonances, 66–7
Quantum states control
– by rapid adiabatic passage, 93–4
– magnetic field gradients method, 92–3
Quantum transport, 41–59
– atomic conductance fluctuations, 42–6
– directed atomic transport due to interactioninduced quantum chaos, 55–9
– Ericson fluctuations in atomic photo cross
sections, 49–51
– photonic transport in chaotic cavities, 51–5
– web-assisted, in kicked harmonic oscillator,
46–9
450
Quantum walks, 100
Quasiparticle approach, 357–71
– in atom-number-conserving Bogoliubov
approximation, 372–4
– cumulants of BEC fluctuations in ideal
Bose gas, 361–4
– – equivalent formulation in terms of poles
of generalized Zeta function, 370–1
– grand canonical approximation for quasiparticle occupations, 363
– ideal gas BEC statistics in arbitrary powerlaw traps, 364–70
– in reduced Hilbert space, 359–61
– see also Interacting Bose gas
Qubits, 84
Rabi oscillations, 80, 88–9
Rabi regime, 161, 162
Radiation field, interference fluctuations, 312
Radiative escape processes, 81
Raman amplifier, 148
Raman photo-association, two-photon, 155
Raman–Nath approximation, 165
Ramsey spectroscopy, 87–8
Random lasers, 51–5
Rayleigh scattering, 429
Reciprocity principle, 434
Recoil pressure, 240–2
Red-detuned lattices, 197–200
Regular level dynamics, 35
Relative humidity LIDAR profiler, 432
Relaxation times, of atoms in dipole trap,
87–9
Resonances, 49, 176
Retina
– bloodflow measurement, 125–7
– tomographic imaging, 123–7
Retinal reflex, 128–9
Riboflavin, 436
Riemann Zeta functions, 371
Ripples, 242–6
– orientation of structures, 248
Rubidium
– one-dimensional lattice structures see Onedimensional lattice configurations
– Rydberg states, 50, 51
– single photon generation, 254
Rydberg electrons, continuum decay, 49–51
Saddle-point method
Index
– conventional, 404, 405, 406
– refined, 328, 404–8
Sag function, 107
Saturation photon numbers, 54
SDOCT see Spectral domain optical coherence tomography
Second-order coherence function, 7, 17
Semiclassical limit, 34–5
Semiconductor superlattices, 49
Sensor technology, 136
Shannon entropy, 304
Shift invariant function, 112
Silicon
– ablation from, 239–40
– – recoil pressure, 240–2
– – surface morphology, 242–6
Single atoms
– in MOT, 77–82
– position control, 94–5
– preparing in dipole trap, 82–4
– quantum state detection, 85–6
– quantum state preparation, 84–5
– Stern–Gerlach experiment, 100
– superposition states, 86–9
– see also Dipole trap
Single photon(s)
– add/drop filter, 22–6
– – application to quantum key distribution,
26–9
– characterization using two-photon interference see two-photon interference
– detection, 259–60
– duration
– – lower limit, 286
– – measurement, 255
– generation, 7–13
– light fields, 257–8
– as particle and wave, 13–16
– source realization, 254
Sisyphus cooling, 190, 197, 199, 200
Skewness, 383
Sloshing-type wave-packets, 197, 200,
219–22
SO2 emission distribution, 415–17
Spatial correlations, molecules as probes of,
173–81
Spectral domain optical coherence tomography (SDOCT), 133–6
– noise comparison with TDOCT, 134–6
Spin stiffness, 386
Index
Spontaneous emission, 141
Spontaneous magnetization, 385, 386, 387
Squeezing
– noise, 298, 381
– two-mode, 298, 374, 381
Stark manifold, 61
Stern–Gerlach experiment, single atom, 100
Stimulated emission amplifiers, 144–7
STIRAP, 254, 278
Stop function, 107
Stranski–Krastanov growth, 5
Strong localization, 46
Super-continuum generation, 425, 426
Superfluid(s)
– description in effective nonlinear σ model,
386
– static susceptibility, 385
– universal scaling of condensate fluctuations
in, 388
Superfluid-Mott insulator phase transition,
163
Superfluidity, detection, 168, 174
Superradiance, 163
– Dicke, 152, 164
Surface morphology, 242–6
Surface roughening, 247
Surface smoothening, 247
Systems with broken continuous symmetry,
383–90
Tavis–Cummings Hamiltonian, 165
Tavis–Cummings model, inhomogeneously
broadened, 164, 170, 171
TDOCT see Time domain optical coherence
tomography
Temporal focusing, 427
Teramobile system, 428, 433, 436
Thomas–Fermi approximation, 177
Thomas–Fermi regime, 377–9
Threshold inversion, 355–6
Time domain optical coherence tomography
(TDOCT), 120–30
– blood oxygenization determination with,
123–5
– bloodflow measurement and, 125–7
– drawbacks, 130–1
– eye length measurement with, 127–9
– of fundus of eye, 122
– interference signal, 121–2
– noise comparison with SDOCT, 134–6
451
– in optical manufacturing, 129–30
Time evolution operator, 42
Time-of-Flight (ToF) spectroscopy, 236,
238–9
Transverse correlation function, 386
Transverse susceptibility, 386, 389
Trapping states, 159
Triexcitons, 16, 17–18, 19
Tryptophan, 437
Tunneling
– influence of magnetic fields on, 213–19
– periodic well-to-well, 208–13
Tunneling ionization, 233
Two-mode noiseless amplifier, 142
Two-photon interference, 260–70
– correlation function, 263, 264–6
– experimental investigation, 277–86
– with jitter see Jitter
– principle, 262–3
– quantum description of beam splitter,
261–2
– single photon duration, 286
– temporal aspects, 263–4
– time-resolved, 267–8, 280–5
– without time resolution, 266–7
Two-photon Raman photo-association, 155
Uhlenbeck, George, 294
Uhlenbeck dilemma, 294, 355
Ultra-cold molecules, 151–82
– counting statistics of molecular fields,
168–73
– passage time statistics of molecule formation, 163–8
– – see also Coupled atom–molecule system;
Molecular micromaser; Spatial correlations
van Zittert–Zernike theorem, 111
Velocity-selective coherent population trapping (VSCPT), 206–8
Virial theorem, 199
Waiting time distribution, 9
Water
– droplets, 435, 436, 438
– ionization potential of molecules, 434
– vapor bands, 431–2
Wave mechanics, formulation, 313
Wave–particle duality, 13
452
Wavefront, laserscanning microscope, 114
Wavefront coding, 107–20
– Abbe’s theory of microscope and, 111–15
– limiting factors, 136
– new applications, 136
– partial coherent illumination and, 115–17
– with variable phaseplates, 117–20
Index
Web-assisted transport, in kicked harmonic
oscillator, 46–9
White light laser, 427
Wien’s law of radiation, 311
Zeta functions, 371, 388
CONTENTS OF VOLUMES IN
THIS SERIAL
Volume 1
Molecular Orbital Theory of the Spin
Properties of Conjugated Molecules,
G.G. Hall and A.T. Amos
Electron Affinities of Atoms and
Molecules, B.L. Moiseiwitsch
Atomic Rearrangement Collisions,
B.H. Bransden
The Production of Rotational and
Vibrational Transitions in Encounters
between Molecules, K. Takayanagi
The Study of Intermolecular Potentials
with Molecular Beams at Thermal
Energies, H. Pauly and J.P. Toennies
High-Intensity and High-Energy Molecular
Beams, J.B. Anderson, R.P. Anders and
J.B. Fen
Volume 2
The Calculation of van der Waals
Interactions, A. Dalgarno and
W.D. Davison
Thermal Diffusion in Gases, E.A. Mason,
R.J. Munn and Francis J. Smith
Spectroscopy in the Vacuum Ultraviolet,
W.R.S. Garton
The Measurement of the Photoionization
Cross Sections of the Atomic Gases,
James A.R. Samson
The Theory of Electron–Atom Collisions,
R. Peterkop and V. Veldre
Experimental Studies of Excitation in
Collisions between Atomic and Ionic
Systems, F.J. de Heer
Mass Spectrometry of Free Radicals,
S.N. Foner
Volume 3
The Quantal Calculation of Photoionization
Cross Sections, A.L. Stewart
Radiofrequency Spectroscopy of Stored
Ions I: Storage, H.G. Dehmelt
Optical Pumping Methods in Atomic
Spectroscopy, B. Budick
Energy Transfer in Organic Molecular
Crystals: A Survey of Experiments,
H.C. Wolf
Atomic and Molecular Scattering from
Solid Surfaces, Robert E. Stickney
Quantum, Mechanics in Gas
Crystal-Surface van der Waals
Scattering, E. Chanoch Beder
Reactive Collisions between Gas and
Surface Atoms, Henry Wise and Bernard
J. Wood
Volume 4
H.S.W. Massey—A Sixtieth Birthday
Tribute, E.H.S. Burhop
Electronic Eigenenergies of the Hydrogen
Molecular Ion, D.R. Bates and
R.H.G. Reid
Applications of Quantum Theory to the
Viscosity of Dilute Gases,
R.A. Buckingham and E. Gal
Positrons and Positronium in Gases,
P.A. Fraser
Classical Theory of Atomic Scattering,
A. Burgess and I.C. Percival
Born Expansions, A.R. Holt and
B.L. Moiseiwitsch
Resonances in Electron Scattering by
Atoms and Molecules, P.G. Burke
453
454
CONTENTS OF VOLUMES IN THIS SERIAL
Relativistic Inner Shell Ionizations,
C.B.O. Mohr
Recent Measurements on Charge Transfer,
J.B. Hasted
Measurements of Electron Excitation
Functions, D.W.O. Heddle and
R.G.W. Keesing
Some New Experimental Methods in
Collision Physics, R.F. Stebbings
Atomic Collision Processes in Gaseous
Nebulae, M.J. Seaton
Collisions in the Ionosphere, A. Dalgarno
The Direct Study of Ionization in Space,
R.L.F. Boyd
Volume 5
Flowing Afterglow Measurements of
Ion-Neutral Reactions, E.E. Ferguson,
F.C. Fehsenfeld and A.L. Schmeltekopf
Experiments with Merging Beams, Roy
H. Neynaber
Radiofrequency Spectroscopy of Stored
Ions II: Spectroscopy, H.G. Dehmelt
The Spectra of Molecular Solids,
O. Schnepp
The Meaning of Collision Broadening of
Spectral Lines: The Classical Oscillator
Analog, A. Ben-Reuven
The Calculation of Atomic Transition
Probabilities, R.J.S. Crossley
Tables of One- and Two-Particle
Coefficients of Fractional Parentage for
Configurations s λ s tu p q ,
C.D.H. Chisholm, A. Dalgarno and
F.R. Innes
Relativistic Z-Dependent Corrections to
Atomic Energy Levels, Holly Thomis
Doyle
Volume 6
Dissociative Recombination, J.N. Bardsley
and M.A. Biondi
Analysis of the Velocity Field in Plasmas
from the Doppler Broadening of Spectral
Emission Lines, A.S. Kaufman
The Rotational Excitation of Molecules by
Slow Electrons, Kazuo Takayanagi and
Yukikazu Itikawa
The Diffusion of Atoms and Molecules,
E.A. Mason and T.R. Marrero
Theory and Application of Sturmian
Functions, Manuel Rotenberg
Use of Classical Mechanics in the
Treatment of Collisions between
Massive Systems, D.R. Bates and
A.E. Kingston
Volume 7
Physics of the Hydrogen Maser, C. Audoin,
J.P. Schermann and P. Grivet
Molecular Wave Functions: Calculations
and Use in Atomic and Molecular
Process, J.C. Browne
Localized Molecular Orbitals, Harel
Weinstein, Ruben Pauncz and Maurice
Cohen
General Theory of Spin-Coupled Wave
Functions for Atoms and Molecules,
J. Gerratt
Diabatic States of Molecules—QuasiStationary Electronic States, Thomas F.
O’Malley
Selection Rules within Atomic Shells,
B.R. Judd
Green’s Function Technique in Atomic and
Molecular Physics, Gy. Csanak,
H.S. Taylor and Robert Yaris
A Review of Pseudo-Potentials with
Emphasis on Their Application to Liquid
Metals, Nathan Wiser and
A.J. Greenfield
Volume 8
Interstellar Molecules: Their Formation
and Destruction, D. McNally
Monte Carlo Trajectory Calculations of
Atomic and Molecular Excitation in
Thermal Systems, James C. Keck
CONTENTS OF VOLUMES IN THIS SERIAL
Nonrelativistic Off-Shell Two-Body
Coulomb Amplitudes, Joseph C.Y. Chen
and Augustine C. Chen
Photoionization with Molecular Beams,
R.B. Cairns, Halstead Harrison and
R.I. Schoen
The Auger Effect, E.H.S. Burhop and
W.N. Asaad
Volume 9
Correlation in Excited States of Atoms,
A.W. Weiss
The Calculation of Electron–Atom
Excitation Cross Section, M.R.H. Rudge
Collision-Induced Transitions between
Rotational Levels, Takeshi Oka
The Differential Cross Section of
Low-Energy Electron–Atom Collisions,
D. Andrick
Molecular Beam Electric Resonance
Spectroscopy, Jens C. Zorn and Thomas
C. English
Atomic and Molecular Processes in the
Martian Atmosphere, Michael
B. McElroy
Volume 10
Relativistic Effects in the Many-Electron
Atom, Lloyd Armstrong Jr. and Serge
Feneuille
The First Born Approximation, K.L. Bell
and A.E. Kingston
Photoelectron Spectroscopy, W.C. Price
Dye Lasers in Atomic Spectroscopy,
W. Lange, J. Luther and A. Steudel
Recent Progress in the Classification of the
Spectra of Highly Ionized Atoms,
B.C. Fawcett
A Review of Jovian Ionospheric Chemistry,
Wesley T. Huntress Jr.
Volume 11
The Theory of Collisions between Charged
Particles and Highly Excited Atoms,
I.C. Percival and D. Richards
455
Electron Impact Excitation of Positive
Ions, M.J. Seaton
The R-Matrix Theory of Atomic Process,
P.G. Burke and W.D. Robb
Role of Energy in Reactive Molecular
Scattering: An Information-Theoretic
Approach, R.B. Bernstein and
R.D. Levine
Inner Shell Ionization by Incident Nuclei,
Johannes M. Hansteen
Stark Broadening, Hans R. Griem
Chemiluminescence in Gases, M.F. Golde
and B.A. Thrush
Volume 12
Nonadiabatic Transitions between Ionic
and Covalent States, R.K. Janev
Recent Progress in the Theory of Atomic
Isotope Shift, J. Bauche and
R.-J. Champeau
Topics on Multiphoton Processes in Atoms,
P. Lambropoulos
Optical Pumping of Molecules, M. Broyer,
G. Goudedard, J.C. Lehmann and
J. Vigué
Highly Ionized Ions, Ivan A. Sellin
Time-of-Flight Scattering Spectroscopy,
Wilhelm Raith
Ion Chemistry in the D Region, George
C. Reid
Volume 13
Atomic and Molecular Polarizabilities—
Review of Recent Advances, Thomas
M. Miller and Benjamin Bederson
Study of Collisions by Laser Spectroscopy,
Paul R. Berman
Collision Experiments with Laser-Excited
Atoms in Crossed Beams, I.V. Hertel and
W. Stoll
Scattering Studies of Rotational and
Vibrational Excitation of Molecules,
Manfred Faubel and J. Peter Toennies
456
CONTENTS OF VOLUMES IN THIS SERIAL
Low-Energy Electron Scattering by
Complex Atoms: Theory and
Calculations, R.K. Nesbet
Microwave Transitions of Interstellar
Atoms and Molecules, W.B. Somerville
Volume 14
Resonances in Electron Atom and
Molecule Scattering, D.E. Golden
The Accurate Calculation of Atomic
Properties by Numerical Methods, Brain
C. Webster, Michael J. Jamieson and
Ronald F. Stewart
(e, 2e) Collisions, Erich Weigold and Ian
E. McCarthy
Forbidden Transitions in One- and
Two-Electron Atoms, Richard Marrus
and Peter J. Mohr
Semiclassical Effects in Heavy-Particle
Collisions, M.S. Child
Atomic Physics Tests of the Basic
Concepts in Quantum Mechanics,
Francies M. Pipkin
Quasi-Molecular Interference Effects in
Ion–Atom Collisions, S.V. Bobashev
Rydberg Atoms, S.A. Edelstein and
T.F. Gallagher
UV and X-Ray Spectroscopy in
Astrophysics, A.K. Dupree
Volume 15
Negative Ions, H.S.W. Massey
Atomic Physics from Atmospheric and
Astrophysical, A. Dalgarno
Collisions of Highly Excited Atoms,
R.F. Stebbings
Theoretical Aspects of Positron Collisions
in Gases, J.W. Humberston
Experimental Aspects of Positron
Collisions in Gases, T.C. Griffith
Reactive Scattering: Recent Advances in
Theory and Experiment, Richard
B. Bernstein
Ion–Atom Charge Transfer Collisions at
Low Energies, J.B. Hasted
Aspects of Recombination, D.R. Bates
The Theory of Fast Heavy Particle
Collisions, B.H. Bransden
Atomic Collision Processes in Controlled
Thermonuclear Fusion Research,
H.B. Gilbody
Inner-Shell Ionization, E.H.S. Burhop
Excitation of Atoms by Electron Impact,
D.W.O. Heddle
Coherence and Correlation in Atomic
Collisions, H. Kleinpoppen
Theory of Low Energy Electron–Molecule
Collisions, P.O. Burke
Volume 16
Atomic Hartree–Fock Theory, M. Cohen
and R.P. McEachran
Experiments and Model Calculations to
Determine Interatomic Potentials,
R. Düren
Sources of Polarized Electrons,
R.J. Celotta and D.T. Pierce
Theory of Atomic Processes in Strong
Resonant Electromagnetic Fields,
S. Swain
Spectroscopy of Laser-Produced Plasmas,
M.H. Key and R.J. Hutcheon
Relativistic Effects in Atomic Collisions
Theory, B.L. Moiseiwitsch
Parity Nonconservation in Atoms: Status of
Theory and Experiment, E.N. Fortson
and L. Wilets
Volume 17
Collective Effects in Photoionization of
Atoms, M.Ya. Amusia
Nonadiabatic Charge Transfer,
D.S.F. Crothers
Atomic Rydberg States, Serge Feneuille
and Pierre Jacquinot
Superfluorescence, M.F.H. Schuurmans,
Q.H.F. Vrehen, D. Polder and
H.M. Gibbs
Applications of Resonance Ionization
Spectroscopy in Atomic and Molecular
CONTENTS OF VOLUMES IN THIS SERIAL
Physics, M.G. Payne, C.H. Chen,
G.S. Hurst and G.W. Foltz
Inner-Shell Vacancy Production in
Ion–Atom Collisions, C.D. Lin and
Patrick Richard
Atomic Processes in the Sun, P.L. Dufton
and A.E. Kingston
Volume 18
Theory of Electron–Atom Scattering in a
Radiation Field, Leonard Rosenberg
Positron–Gas Scattering Experiments,
Talbert S. Stein and Walter E. Kaupplia
Nonresonant Multiphoton Ionization of
Atoms, J. Morellec, D. Normand and
G. Petite
Classical and Semiclassical Methods in
Inelastic Heavy-Particle Collisions,
A.S. Dickinson and D. Richards
Recent Computational Developments in the
Use of Complex Scaling in Resonance
Phenomena, B.R. Junker
Direct Excitation in Atomic Collisions:
Studies of Quasi-One-Electron Systems,
N. Andersen and S.E. Nielsen
Model Potentials in Atomic Structure,
A. Hibbert
Recent Developments in the Theory of
Electron Scattering by Highly Polar
Molecules, D.W. Norcross and
L.A. Collins
Quantum Electrodynamic Effects in
Few-Electron Atomic Systems,
G.W.F. Drake
Volume 19
Electron Capture in Collisions of Hydrogen
Atoms with Fully Stripped Ions,
B.H. Bransden and R.K. Janev
Interactions of Simple Ion Atom Systems,
J.T. Park
High-Resolution Spectroscopy of Stored
Ions, D.J. Wineland, Wayne M. Itano and
R.S. Van Dyck Jr.
457
Spin-Dependent Phenomena in Inelastic
Electron–Atom Collisions, K. Blum and
H. Kleinpoppen
The Reduced Potential Curve Method for
Diatomic Molecules and Its
Applications, F. Jenˇc
The Vibrational Excitation of Molecules by
Electron Impact, D.G. Thompson
Vibrational and Rotational Excitation in
Molecular Collisions, Manfred Faubel
Spin Polarization of Atomic and Molecular
Photoelectrons, N.A. Cherepkov
Volume 20
Ion–Ion Recombination in an Ambient
Gas, D.R. Bates
Atomic Charges within Molecules,
G.G. Hall
Experimental Studies on Cluster Ions,
T.D. Mark and A.W. Castleman Jr.
Nuclear Reaction Effects on Atomic
Inner-Shell Ionization, W.E. Meyerhof
and J.-F. Chemin
Numerical Calculations on Electron-Impact
Ionization, Christopher Bottcher
Electron and Ion Mobilities, Gordon
R. Freeman and David A. Armstrong
On the Problem of Extreme UV and X-Ray
Lasers, I.I. Sobel’man and
A.V. Vinogradov
Radiative Properties of Rydberg States in
Resonant Cavities, S. Haroche and
J.M. Raimond
Rydberg Atoms: High-Resolution
Spectroscopy and Radiation
Interaction—Rydberg Molecules,
J.A.C. Gallas, G. Leuchs, H. Walther,
and H. Figger
Volume 21
Subnatural Linewidths in Atomic
Spectroscopy, Dennis P. O’Brien, Pierre
Meystre and Herbert Walther
Molecular Applications of Quantum Defect
Theory, Chris H. Greene and Ch. Jungen
458
CONTENTS OF VOLUMES IN THIS SERIAL
Theory of Dielectronic Recombination,
Yukap Hahn
Recent Developments in Semiclassical
Floquet Theories for Intense-Field
Multiphoton Processes, Shih-I Chu
Scattering in Strong Magnetic Fields,
M.R.C. McDowell and M. Zarcone
Pressure Ionization, Resonances and the
Continuity of Bound and Free States,
R.M. More
Volume 22
Positronium—Its Formation and
Interaction with Simple Systems,
J.W. Humberston
Experimental Aspects of Positron and
Positronium Physics, T.C. Griffith
Doubly Excited States, Including New
Classification Schemes, C.D. Lin
Measurements of Charge Transfer and
Ionization in Collisions Involving
Hydrogen Atoms, H.B. Gilbody
Electron Ion and Ion–Ion Collisions with
Intersecting Beams, K. Dolder and
B. Peart
Electron Capture by Simple Ions, Edward
Pollack and Yukap Hahn
Relativistic Heavy-Ion–Atom Collisions,
R. Anholt and Harvey Gould
Continued-Fraction Methods in Atomic
Physics, S. Swain
Volume 23
Vacuum Ultraviolet Laser Spectroscopy of
Small Molecules, C.R. Vidal
Foundations of the Relativistic Theory of
Atomic and Molecular Structure, Ian
P. Grant and Harry M. Quiney
Point-Charge Models for Molecules
Derived from Least-Squares Fitting of
the Electric Potential, D.E. Williams and
Ji-Min Yan
Transition Arrays in the Spectra of Ionized
Atoms, J. Bauche, C. Bauche-Arnoult
and M. Klapisch
Photoionization and Collisional Ionization
of Excited Atoms Using Synchrotron
and Laser Radiation, F.J. Wuilleumier,
D.L. Ederer and J.L. Picqué
Volume 24
The Selected Ion Flow Tube (SIDT):
Studies of Ion-Neutral Reactions,
D. Smith and N.G. Adams
Near-Threshold Electron–Molecule
Scattering, Michael A. Morrison
Angular Correlation in Multiphoton
Ionization of Atoms, S.J. Smith and
G. Leuchs
Optical Pumping and Spin Exchange in
Gas Cells, R.J. Knize, Z. Wu and
W. Happer
Correlations in Electron–Atom Scattering,
A. Crowe
Volume 25
Alexander Dalgarno: Life and Personality,
David R. Bates and George A. Victor
Alexander Dalgarno: Contributions to
Atomic and Molecular Physics, Neal
Lane
Alexander Dalgarno: Contributions to
Aeronomy, Michael B. McElroy
Alexander Dalgarno: Contributions to
Astrophysics, David A. Williams
Dipole Polarizability Measurements,
Thomas M. Miller and Benjamin
Bederson
Flow Tube Studies of Ion–Molecule
Reactions, Eldon Ferguson
Differential Scattering in He–He and
He+ –He Collisions at keV Energies,
R.F. Stebbings
Atomic Excitation in Dense Plasmas, Jon
C. Weisheit
Pressure Broadening and Laser-Induced
Spectral Line Shapes, Kenneth M. Sando
and Shih-I. Chu
Model-Potential Methods, C. Laughlin and
G.A. Victor
CONTENTS OF VOLUMES IN THIS SERIAL
Z-Expansion Methods, M. Cohen
Schwinger Variational Methods, Deborah
Kay Watson
Fine-Structure Transitions in Proton–Ion
Collisions, R.H.G. Reid
Electron Impact Excitation, R.J.W. Henry
and A.E. Kingston
Recent Advances in the Numerical
Calculation of Ionization Amplitudes,
Christopher Bottcher
The Numerical Solution of the Equations
of Molecular Scattering, A.C. Allison
High Energy Charge Transfer,
B.H. Bransden and D.P. Dewangan
Relativistic Random-Phase Approximation,
W.R. Johnson
Relativistic Sturmian and Finite Basis Set
Methods in Atomic Physics,
G.W.F. Drake and S.P. Goldman
Dissociation Dynamics of Polyatomic
Molecules, T. Uzer
Photodissociation Processes in Diatomic
Molecules of Astrophysical Interest,
Kate P. Kirby and Ewine F. van Dishoeck
The Abundances and Excitation of
Interstellar Molecules, John H. Black
Volume 26
Comparisons of Positrons and Electron
Scattering by Gases, Walter E. Kauppila
and Talbert S. Stein
Electron Capture at Relativistic Energies,
B.L. Moiseiwitsch
The Low-Energy, Heavy Particle
Collisions—A Close-Coupling
Treatment, Mineo Kimura and Neal
F. Lane
Vibronic Phenomena in Collisions of
Atomic and Molecular Species, V. Sidis
Associative Ionization: Experiments,
Potentials and Dynamics, John Weiner
Françoise Masnou-Seeuws and Annick
Giusti-Suzor
On the β Decay of 187 Re: An Interface of
Atomic and Nuclear Physics and
459
Cosmochronology, Zonghau Chen,
Leonard Rosenberg and Larry Spruch
Progress in Low Pressure Mercury-Rare
Gas Discharge Research, J. Maya and
R. Lagushenko
Volume 27
Negative Ions: Structure and Spectra,
David R. Bates
Electron Polarization Phenomena in
Electron–Atom Collisions, Joachim
Kessler
Electron–Atom Scattering, I.E. McCarthy
and E. Weigold
Electron–Atom Ionization, I.E. McCarthy
and E. Weigold
Role of Autoionizing States in Multiphoton
Ionization of Complex Atoms,
V.I. Lengyel and M.I. Haysak
Multiphoton Ionization of Atomic
Hydrogen Using Perturbation Theory,
E. Karule
Volume 28
The Theory of Fast Ion–Atom Collisions,
J.S. Briggs and J.H. Macek
Some Recent Developments in the
Fundamental Theory of Light, Peter
W. Milonni and Surendra Singh
Squeezed States of the Radiation Field,
Khalid Zaheer and M. Suhail Zubairy
Cavity Quantum Electrodynamics,
E.A. Hinds
Volume 29
Studies of Electron Excitation of Rare-Gas
Atoms into and out of Metastable Levels
Using Optical and Laser Techniques,
Chun C. Lin and L.W. Anderson
Cross Sections for Direct Multiphoton
Ionization of Atoms, M.V. Ammosov,
N.B. Delone, M.Yu. Ivanov, I.I. Bandar
and A.V. Masalov
Collision-Induced Coherences in Optical
Physics, G.S. Agarwal
460
CONTENTS OF VOLUMES IN THIS SERIAL
Muon-Catalyzed Fusion, Johann Rafelski
and Helga E. Rafelski
Cooperative Effects in Atomic Physics,
J.P. Connerade
Multiple Electron Excitation, Ionization,
and Transfer in High-Velocity Atomic
and Molecular Collisions, J.H. McGuire
Volume 30
Differential Cross Sections for Excitation
of Helium Atoms and Helium-Like Ions
by Electron Impact, Shinobu Nakazaki
Cross-Section Measurements for Electron
Impact on Excited Atomic Species,
S. Trajmar and J.C. Nickel
The Dissociative Ionization of Simple
Molecules by Fast Ions, Colin J. Latimer
Theory of Collisions between Laser Cooled
Atoms, P.S. Julienne, A.M. Smith and
K. Burnett
Light-Induced Drift, E.R. Eliel
Continuum Distorted Wave Methods in
Ion–Atom Collisions, Derrick
S.F. Crothers and Louis J. Dube
Volume 31
Energies and Asymptotic Analysis for
Helium Rydberg States, G.W.F. Drake
Spectroscopy of Trapped Ions,
R.C. Thompson
Phase Transitions of Stored Laser-Cooled
Ions, H. Walther
Selection of Electronic States in Atomic
Beams with Lasers, Jacques Baudon,
Rudalf Dülren and Jacques Robert
Atomic Physics and Non-Maxwellian
Plasmas, Michèle Lamoureux
Volume 32
Photoionization of Atomic Oxygen and
Atomic Nitrogen, K.L. Bell and
A.E. Kingston
Positronium Formation by Positron Impact
on Atoms at Intermediate Energies,
B.H. Bransden and C.J. Noble
Electron–Atom Scattering Theory and
Calculations, P.G. Burke
Terrestrial and Extraterrestrial H+
3,
Alexander Dalgarno
Indirect Ionization of Positive Atomic Ions,
K. Dolder
Quantum Defect Theory and Analysis of
High-Precision Helium Term Energies,
G.W.F. Drake
Electron–Ion and Ion–Ion Recombination
Processes, M.R. Flannery
Studies of State-Selective Electron Capture
in Atomic Hydrogen by Translational
Energy Spectroscopy, H.B. Gilbody
Relativistic Electronic Structure of Atoms
and Molecules, I.P. Grant
The Chemistry of Stellar Environments,
D.A. Howe, J.M.C. Rawlings and
D.A. Williams
Positron and Positronium Scattering at
Low Energies, J.W. Humberston
How Perfect are Complete Atomic
Collision Experiments?, H. Kleinpoppen
and H. Handy
Adiabatic Expansions and Nonadiabatic
Effects, R. McCarroll and
D.S.F. Crothers
Electron Capture to the Continuum,
B.L. Moiseiwitsch
How Opaque Is a Star?, M.T. Seaton
Studies of Electron Attachment at Thermal
Energies Using the Flowing
Afterglow–Langmuir Technique, David
Smith and Patrik Španˇel
Exact and Approximate Rate Equations in
Atom–Field Interactions, S. Swain
Atoms in Cavities and Traps, H. Walther
Some Recent Advances in Electron-Impact
Excitation of n = 3 States of Atomic
Hydrogen and Helium, J.F. Williams and
J.B. Wang
Volume 33
Principles and Methods for Measurement
of Electron Impact Excitation Cross
CONTENTS OF VOLUMES IN THIS SERIAL
Sections for Atoms and Molecules by
Optical Techniques, A.R. Filippelli,
Chun C. Lin, L.W. Andersen and
J.W. McConkey
Benchmark Measurements of Cross
Sections for Electron Collisions:
Analysis of Scattered Electrons,
S. Trajmar and J.W. McConkey
Benchmark Measurements of Cross
Sections for Electron Collisions:
Electron Swarm Methods,
R.W. Crompton
Some Benchmark Measurements of Cross
Sections for Collisions of Simple Heavy
Particles, H.B. Gilbody
The Role of Theory in the Evaluation and
Interpretation of Cross-Section Data,
Barry I. Schneider
Analytic Representation of Cross-Section
Data, Mitio Inokuti, Mineo Kimura,
M.A. Dillon, Isao Shimamura
Electron Collisions with N2 , O2 and O:
What We Do and Do Not Know,
Yukikazu Itikawa
Need for Cross Sections in Fusion Plasma
Research, Hugh P. Summers
Need for Cross Sections in Plasma
Chemistry, M. Capitelli, R. Celiberto
and M. Cacciatore
Guide for Users of Data Resources, Jean
W. Gallagher
Guide to Bibliographies, Books, Reviews
and Compendia of Data on Atomic
Collisions, E.W. McDaniel and
E.J. Mansky
Volume 34
Atom Interferometry, C.S. Adams,
O. Carnal and J. Mlynek
Optical Tests of Quantum Mechanics,
R.Y. Chiao, P.G. Kwiat and
A.M. Steinberg
Classical and Quantum Chaos in Atomic
Systems, Dominique Delande and
Andreas Buchleitner
461
Measurements of Collisions between
Laser-Cooled Atoms, Thad Walker and
Paul Feng
The Measurement and Analysis of Electric
Fields in Glow Discharge Plasmas,
J.E. Lawler and D.A. Doughty
Polarization and Orientation Phenomena in
Photoionization of Molecules,
N.A. Cherepkov
Role of Two-Center Electron–Electron
Interaction in Projectile Electron
Excitation and Loss, E.C. Montenegro,
W.E. Meyerhof and J.H. McGuire
Indirect Processes in Electron Impact
Ionization of Positive Ions, D.L. Moores
and K.J. Reed
Dissociative Recombination: Crossing and
Tunneling Modes, David R. Bates
Volume 35
Laser Manipulation of Atoms,
K. Sengstock and W. Ertmer
Advances in Ultracold Collisions:
Experiment and Theory, J. Weiner
Ionization Dynamics in Strong Laser
Fields, L.F. DiMauro and P. Agostini
Infrared Spectroscopy of Size Selected
Molecular Clusters, U. Buck
Fermosecond Spectroscopy of Molecules
and Clusters, T. Baumer and G. Gerber
Calculation of Electron Scattering on
Hydrogenic Targets, I. Bray and
A.T. Stelbovics
Relativistic Calculations of Transition
Amplitudes in the Helium Isoelectronic
Sequence, W.R. Johnson, D.R. Plante
and J. Sapirstein
Rotational Energy Transfer in Small
Polyatomic Molecules, H.O. Everitt and
F.C. De Lucia
Volume 36
Complete Experiments in Electron–Atom
Collisions, Nils Overgaard Andersen and
Klaus Bartschat
462
CONTENTS OF VOLUMES IN THIS SERIAL
Stimulated Rayleigh Resonances and
Recoil-Induced Effects, J.-Y. Courtois
and G. Grynberg
Precision Laser Spectroscopy Using
Acousto-Optic Modulators, W.A. van
Mijngaanden
Highly Parallel Computational Techniques
for Electron–Molecule Collisions, Carl
Winstead and Vincent McKoy
Quantum Field Theory of Atoms and
Photons, Maciej Lewenstein and Li You
Volume 37
Evanescent Light-Wave Atom Mirrors,
Resonators, Waveguides, and Traps,
Jonathan P. Dowling and Julio
Gea-Banacloche
Optical Lattices, P.S. Jessen and
I.H. Deutsch
Channeling Heavy Ions through Crystalline
Lattices, Herbert F. Krause and Sheldon
Datz
Evaporative Cooling of Trapped Atoms,
Wolfgang Ketterle and N.J. van Druten
Nonclassical States of Motion in Ion Traps,
J.I. Cirac, A.S. Parkins, R. Blatt and
P. Zoller
The Physics of Highly-Charged Heavy Ions
Revealed by Storage/Cooler Rings,
P.H. Mokler and Th. Stöhlker
Volume 38
Electronic Wavepackets, Robert R. Jones
and L.D. Noordam
Chiral Effects in Electron Scattering by
Molecules, K. Blum and D.G. Thompson
Optical and Magneto-Optical Spectroscopy
of Point Defects in Condensed Helium,
Serguei I. Kanorsky and Antoine Weis
Rydberg Ionization: From Field to Photon,
G.M. Lankhuijzen and L.D. Noordam
Studies of Negative Ions in Storage Rings,
L.H. Andersen, T. Andersen and
P. Hvelplund
Single-Molecule Spectroscopy and
Quantum Optics in Solids, W.E. Moerner,
R.M. Dickson and D.J. Norris
Volume 39
Author and Subject Cumulative Index
Volumes 1–38
Author Index
Subject Index
Appendix: Tables of Contents of Volumes
1–38 and Supplements
Volume 40
Electric Dipole Moments of Leptons,
Eugene D. Commins
High-Precision Calculations for the
Ground and Excited States of the
Lithium Atom, Frederick W. King
Storage Ring Laser Spectroscopy, Thomas
U. Kühl
Laser Cooling of Solids, Carl E. Mangan
and Timothy R. Gosnell
Optical Pattern Formation, L.A. Lugiato,
M. Brambilla and A. Gatti
Volume 41
Two-Photon Entanglement and Quantum
Reality, Yanhua Shih
Quantum Chaos with Cold Atoms, Mark
G. Raizen
Study of the Spatial and Temporal
Coherence of High-Order Harmonics,
Pascal Salières, Ann L’Huillier, Philippe
Antoine and Maciej Lewenstein
Atom Optics in Quantized Light Fields,
Matthias Freyburger, Alois
M. Herkommer, Daniel S. Krähmer,
Erwin Mayr and Wolfgang P. Schleich
Atom Waveguides, Victor I. Balykin
Atomic Matter Wave Amplification by
Optical Pumping, Ulf Janicke and
Martin Wikens
CONTENTS OF VOLUMES IN THIS SERIAL
Volume 42
Fundamental Tests of Quantum Mechanics,
Edward S. Fry and Thomas Walther
Wave-Particle Duality in an Atom
Interferometer, Stephan Dürr and
Gerhard Rempe
Atom Holography, Fujio Shimizu
Optical Dipole Traps for Neutral Atoms,
Rudolf Grimm, Matthias Weidemüller
and Yurii B. Ovchinnikov
Formation of Cold (T ≤ 1 K) Molecules,
J.T. Bahns, P.L. Gould and W.C. Stwalley
High-Intensity Laser-Atom Physics,
C.J. Joachain, M. Dorr and N.J. Kylstra
Coherent Control of Atomic, Molecular
and Electronic Processes, Moshe Shapiro
and Paul Brumer
Resonant Nonlinear Optics in Phase
Coherent Media, M.D. Lukin, P. Hemmer
and M.O. Scully
The Characterization of Liquid and Solid
Surfaces with Metastable Helium Atoms,
H. Morgner
Quantum Communication with Entangled
Photons, Herald Weinfurter
Volume 43
Plasma Processing of Materials and
Atomic, Molecular, and Optical Physics:
An Introduction, Hiroshi Tanaka and
Mitio Inokuti
The Boltzmann Equation and Transport
Coefficients of Electrons in Weakly
Ionized Plasmas, R. Winkler
Electron Collision Data for Plasma
Chemistry Modeling, W.L. Morgan
Electron–Molecule Collisions in
Low-Temperature Plasmas: The Role of
Theory, Carl Winstead and Vincent
McKoy
Electron Impact Ionization of Organic
Silicon Compounds, Ralf Basner, Kurt
Becker, Hans Deutsch and Martin
Schmidt
463
Kinetic Energy Dependence of
Ion–Molecule Reactions Related to
Plasma Chemistry, P.B. Armentrout
Physicochemical Aspects of Atomic and
Molecular Processes in Reactive
Plasmas, Yoshihiko Hatano
Ion–Molecule Reactions, Werner
Lindinger, Armin Hansel and Zdenek
Herman
Uses of High-Sensitivity White-Light
Absorption Spectroscopy in Chemical
Vapor Deposition and Plasma
Processing, L.W. Anderson, A.N. Goyette
and J.E. Lawler
Fundamental Processes of Plasma–Surface
Interactions, Rainer Hippler
Recent Applications of Gaseous
Discharges: Dusty Plasmas and
Upward-Directed Lightning, Ara
Chutjian
Opportunities and Challenges for Atomic,
Molecular and Optical Physics in Plasma
Chemistry, Kurl Becker Hans Deutsch
and Mitio Inokuti
Volume 44
Mechanisms of Electron Transport in
Electrical Discharges and Electron
Collision Cross Sections, Hiroshi Tanaka
and Osamu Sueoka
Theoretical Consideration of
Plasma-Processing Processes, Mineo
Kimura
Electron Collision Data for
Plasma-Processing Gases, Loucas G.
Christophorou and James K. Olthoff
Radical Measurements in Plasma
Processing, Toshio Goto
Radio-Frequency Plasma Modeling for
Low-Temperature Processing, Toshiaki
Makabe
Electron Interactions with Excited Atoms
and Molecules, Loucas G.
Christophorou and James K. Olthoff
464
CONTENTS OF VOLUMES IN THIS SERIAL
Volume 45
Comparing the Antiproton and Proton, and
Opening the Way to Cold Antihydrogen,
G. Gabrielse
Medical Imaging with Laser-Polarized
Noble Gases, Timothy Chupp and Scott
Swanson
Polarization and Coherence Analysis of the
Optical Two-Photon Radiation from the
Metastable 22 Si1/2 State of Atomic
Hydrogen, Alan J. Duncan, Hans
Kleinpoppen and Marian O. Scully
Laser Spectroscopy of Small Molecules,
W. Demtröder, M. Keil and H. Wenz
Coulomb Explosion Imaging of Molecules,
Z. Vager
Volume 46
Femtosecond Quantum Control, T. Brixner,
N.H. Damrauer and G. Gerber
Coherent Manipulation of Atoms and
Molecules by Sequential Laser Pulses,
N.V. Vitanov, M. Fleischhauer,
B.W. Shore and K. Bergmann
Slow, Ultraslow, Stored, and Frozen Light,
Andrey B. Matsko, Olga Kocharovskaya,
Yuri Rostovtsev George R. Welch,
Alexander S. Zibrov and Marlan
O. Scully
Longitudinal Interferometry with Atomic
Beams, S. Gupta, D.A. Kokorowski,
R.A. Rubenstein, and W.W. Smith
Volume 47
Nonlinear Optics of de Broglie Waves,
P. Meystre
Formation of Ultracold Molecules
(T ≤ 200 μK) via Photoassociation in a
Gas of Laser-Cooled Atoms, Françoise
Masnou-Seeuws and Pierre Pillet
Molecular Emissions from the
Atmospheres of Giant Planets and
Comets: Needs for Spectroscopic and
Collision Data, Yukikazu Itikawa, Sang
Joon Kim, Yong Ha Kim and Y.C. Minh
Studies of Electron-Excited Targets Using
Recoil Momentum Spectroscopy with
Laser Probing of the Excited State,
Andrew James Murray and Peter
Hammond
Quantum Noise of Small Lasers,
J.P. Woerdman, N.J. van Druten and
M.P. van Exter
Volume 48
Multiple Ionization in Strong Laser Fields,
R. Dörner Th. Weber, M. Weckenbrock,
A. Staudte, M. Hattass, R. Moshammer,
J. Ullrich and H. Schmidt-Böcking
Above-Threshold Ionization: From
Classical Features to Quantum Effects,
W. Becker, F. Grasbon, R. Kapold,
D.B. Miloševi´c, G.G. Paulus and
H. Walther
Dark Optical Traps for Cold Atoms, Nir
Friedman, Ariel Kaplan and Nir
Davidson
Manipulation of Cold Atoms in Hollow
Laser Beams, Heung-Ryoul Noh, Xenye
Xu and Wonho Jhe
Continuous Stern–Gerlach Effect on
Atomic Ions, Günther Werth, Hartmut
Haffner and Wolfgang Quint
The Chirality of Biomolecules, Robert
N. Compton and Richard M. Pagni
Microscopic Atom Optics: From Wires to
an Atom Chip, Ron Folman, Peter
Krüger, Jörg Schmiedmayer, Johannes
Denschlag and Carsten Henkel
Methods of Measuring Electron–Atom
Collision Cross Sections with an Atom
Trap, R.S. Schappe, M.L. Keeler,
T.A. Zimmerman, M. Larsen, P. Feng,
R.C. Nesnidal, J.B. Boffard, T.G. Walker,
L.W. Anderson and C.C. Lin
Volume 49
Applications of Optical Cavities in Modern
Atomic, Molecular, and Optical Physics,
Jun Ye and Theresa W. Lynn
CONTENTS OF VOLUMES IN THIS SERIAL
Resonance and Threshold Phenomena in
Low-Energy Electron Collisions with
Molecules and Clusters, H. Hotop,
M.-W. Ruf, M. Allan and I.I. Fabrikant
Coherence Analysis and Tensor
Polarization Parameters of (γ , eγ )
Photoionization Processes in Atomic
Coincidence Measurements,
B. Lohmann, B. Zimmermann,
H. Kleinpoppen and U. Becker
Quantum Measurements and New
Concepts for Experiments with Trapped
Ions, Ch. Wunderlich and Ch. Balzer
Scattering and Reaction Processes in
Powerful Laser Fields, Dejan
B. Miloševi´c and Fritz Ehlotzky
Hot Atoms in the Terrestrial Atmosphere,
Vijay Kumar and E. Krishnakumar
Volume 50
Assessment of the Ozone Isotope Effect,
K. Mauersberger, D. Krankowsky,
C. Janssen and R. Schinke
Atom Optics, Guided Atoms, and Atom
Interferometry, J. Arlt, G. Birkl, E. Rasel
and W. Ertmet
Atom–Wall Interaction, D. Bloch and
M. Ducloy
Atoms Made Entirely of Antimatter: Two
Methods Produce Slow Antihydrogen,
G. Gabrielse
Ultrafast Excitation, Ionization, and
Fragmentation of C60 , I.V. Hertel,
T. Laarmann and C.P. Schulz
Volume 51
Introduction, Henry H. Stroke
Appreciation of Ben Bederson as Editor of
Advances in Atomic, Molecular, and
Optical Physics
Benjamin Bederson Curriculum Vitae
Research Publications of Benjamin
Bederson
A Proper Homage to Our Ben, H. Lustig
Benjamin Bederson in the Army, World
War II, Val L. Fitch
465
Physics Needs Heroes Too, C. Duncan Rice
Two Civic Scientists—Benjamin Bederson
and the other Benjamin, Neal Lane
An Editor Par Excellence, Eugen
Merzbacher
Ben as APS Editor, Bernd Crasemann
Ben Bederson: Physicist–Historian,
Roger H. Stuewer
Pedagogical Notes on Classical Casimir
Effects, Larry Spruch
Polarizabilities of 3 P Atoms and van der
Waals Coefficients for Their Interaction
with