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VOLUME 90, N UMBER 18
PHYSICA L R EVIEW LET T ERS
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9 MAY 2003
How to Measure the Phase Diffusion Dynamics in the Micromaser
F. Casagrande, A. Ferraro, A. Lulli, and R. Bonifacio
INFM-Dipartimento di Fisica, Universita` di Milano, Via Celoria 16, 20133 Milano, Italy
E. Solano1,2 and H. Walther1
1
2
Max-Planck Institut fu¨ r Quantenoptik, Hans-Kopfermann Strasse 1, 85748 Garching, Germany
Seccio´n Fı´sica, Departamento de Ciencias, Pontificia Universidad Cato´lica del Peru´, Apartado 1761, Lima, Peru
(Received 5 August 2002; published 5 May 2003)
We propose a realistic scheme for measuring the micromaser linewidth by monitoring the phase
diffusion dynamics of the cavity field. Our strategy consists of exciting an initial coherent state with the
same photon number distribution as the micromaser steady-state field, singling out a purely diffusive
process in the system dynamics. After the injection of a counterfield, measurements of the population
statistics of a probe atom allow us to derive the micromaser linewidth in all ranges of the relevant
parameters, establishing experimentally the distinctive features of the micromaser spectrum due to the
discreteness of the electromagnetic field.
DOI: 10.1103/PhysRevLett.90.183601
The micromaser [1] has been the object of continuous
interest as a proper tool for the investigation of fundamental questions in cavity quantum electrodynamics
(QED). For example, it has permitted the generation of
highly pure Fock states [2], the detection of photon
number trapping states [3], and the possibility of testing
new ideas in quantum information [4]. Nevertheless,
some fundamental aspects have remained elusive to the
experiments, such as the measurement of the micromaser
spectrum and its linewidth, whose physical origin is the
decoherence of the cavity field induced by a phase diffusion process. Although different proposals have been
done in the past, measurements related to the micromaser
spectrum remain until now as an experimental challenge
for reasons we discuss thoroughly in this work.
In Refs. [5–7], approximate analytical expressions for
the phase diffusion rate were derived from the micromaser master equation (MME) [8], matching reasonably
the numerical results. When the micromaser operates at
very low temperatures, trapping states of the cavity field
occur [3,9], inducing sharp minima in the mean photon
number. In consequence, as the pumping rate increases,
linewidth oscillations are predicted [5], at strong variance
with the monotonic dependence of the Schawlow-Townes
laser linewidth [10]. In Refs. [5,11] a Ramsey-type interferometric scheme was proposed for an experimental
measurement of the micromaser linewidth. These techniques require atoms in a coherent superposition of the
ground and excited states, which needs further advances
in current technology of very high-Q cavities. Other
treatments approached the spectrum problem by investigating the field and atomic correlation functions [9,12].
Unfortunately, in one way or another, all proposals have
failed in approximating the requirements of a feasible
experiment.
In this Letter, we consider a realistic measurement of
the phase diffusion dynamics of the micromaser and
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0031-9007=03=90(18)=183601(4)$20.00
PACS numbers: 42.50.Ct, 03.65.Yz, 42.50.Pq
propose different strategies for unveiling its special
features.
A micromaser consists of a single quantized mode of a
high-Q cavity driven by excited two-level atoms crossing
the field mode one at a time with pumping rate r. The
atomic levels are long-living Rydberg states that couple
strongly to the cavity microwave field. The cavity is in
contact with a thermal bath producing a mean number of
thermal photons n b and has a linewidth !=Q, where
! is the angular frequency of the cavity field. However,
the decay of the field can be considered as negligible
when an atom is flying through, due to its high speed.
Each one of the pumping atoms interacts with the cavity
field following the Jaynes-Cummings (JC) model [13],
where the evolution of an initial atomic excited state jei
and any field Fock state jni follows:
p
p
jeijni ! cosg n 1jeijni sing n 1jgijn 1i:
(1)
Here, g is the atom-field coupling strength, is the
interaction time, and jgi is the atomic ground state. The
temporal evolution of the density matrix elements n;m t
of the cavity field is ruled by the MME [8]
d
n;m An;m n1;m1 Bn;m n;m Cn;m n1;m1 ;
dt
(2)
where
p
p
p
An;m r sing n sing m n b nm;
p
p
Bn;m r
1 cosg n 1 cosg m 1
n b 1n m n b n m 2;
2
2
p
n b 1 n 1m 1:
Cn;m
 2003 The American Physical Society
(3)
183601-1
1 p
1X
n 1 n;n1 t c:c:
2 n0
(4)
This shows that the dynamics of the electric field is
ruled by the decay of the first off-diagonal elements (m n 1) of the field density matrix, determined by the
MME of Eq. (2). The decay of the elements n;n1 t,
while the photon number distribution remains unchanged, implies the decay of hE^ ti by a phase diffusion
process. The physical picture becomes remarkably simple
if approximately all elements n;n1 t decay exponentially with a similar rate D. Then,
hE^ ti hE^ 0i expDt
(5)
and the micromaser spectrum has a Lorentzian profile
with half width at half maximum (HWHM) D [5–7].
Equations (4) and (5) are at the heart of the phase diffusion model yielding the micromaser spectrum. The
spectrum can be well approximated by a single Lorentzian on a wide range of micromaser parameters. The few
exceptions to this picture happen in the very close vicinity
of certain trapping states [7], allowing us the possibility
of studying the distinctive properties of the micromaser
diffusion process. In principle, on the basis of Eq. (5), it
could be enough to monitor the decay of initial nonvanishing off-diagonal elements n;n1 0 of the cavity
field, created by injecting atoms in a coherent superposition of jgi and jei or preparing the cavity mode in a
coherent state. Unfortunately, this procedure cannot be
implemented, as long as we do not have direct experimental access to the cavity field. One of the features that
makes the micromaser such an interesting device is that
the pumping atoms that are used to build the field serve
also, or could serve, as a quantum probe of it.
The first step of our scheme is the choice of a suitable
initial condition for the cavity field. The micromaser field
approaches a stationary state (ss), independent of the
initial one, described by a diagonal density matrix with
ss
elements ss
n;n pn [8]. If the cavity field is initially
prepared in a coherent state ji, such that n;n 0 pn jhjnij2 pss
n , the photon statistics remains frozen during the micromaser operation. We choose this special
initial condition in an effort for singling out the decay
of the off-diagonal elements due to phase diffusion from
other incoherent processes, allowing us to establish a
nondissipative decoherence dynamics. This requirement
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can be satisfied when the steady-state photon statistics pss
n
has a well defined peak. This convenient scenario happens
to be an excellent approximation in the micromaser
threshold region [8], where the maximum amplification
of the mean photon number occurs, and within a good
approximation in a wide region of the micromaser parameters, including the vicinity of trapping states [14].
We simulate numerically the system dynamics by a
Monte Carlo wave function technique [15], successfully
applied to the investigation of other problems in the
micromaser [16]. In Fig. 1 we show the field density
matrix
n;m 0 of the initial coherent state and also the diagonal
form of the steady state at the end of the diffusion process
ss
with ss
n;m n;m pn , after the progressive vanishing of
field coherences. As an example, we used experimental
parameters corresponding to the region of maximum
amplification of the micromaser.
The expected exponential decay of hE^ ti, Eq. (5), induced by the decay of the coherences n;n1 t, Eq. (4), is
well reproduced in the numerical simulations as shown in
Fig. 2. Consequently, the phase diffusion rate can be
easily derived; in this example D= 0:049, in good
0.15
a)
|ρ(n,m)|
Here, the first terms of An;m and Bn;m are related to the
unitary contribution of the Jaynes-Cummings interaction, after tracing out the atomic degrees of freedom,
and the remaining terms are associated with the incoherent processes.
In the spirit of a seminal work on the micromaser
spectrum [5], based on a successful approach applied to
laser theory [10], we consider the cavity field expectation
value
hE^ ti / Reha^ ti week ending
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PHYSICA L R EVIEW LET T ERS
0
10
m 20
30
20
30
10 n
0.15
b)
|ρ(n,m)|
VOLUME 90, N UMBER 18
0
10
m 20
30
30
10 n
20
FIG. 1. Density matrix n;m of the micromaser cavity field:
(a) initial coherent state ji; (b) diagonal steady-state field
after the phase diffusion process. Dimensionless parameters: 3:0654, g 0:494, g= 12:300, r= 10, and
n b 0:03.
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agreement with approximated analytical expressions [5].
This behavior, however, can be neither directly observed
nor even inferred from measurements on the populations
of outgoing atoms. Actually, from the JC dynamics [13],
exemplified in Eq. (1), the probability pe that any excited
atom, at any moment of the diffusion process, is observed
out of the cavity still in the excited state is
1
X
pe p
2
pss
n cos g n 1:
(6)
n0
In conclusion, no information on field coherences can be
derived from this kind of atomic population measurements, even if the field coherences are nonvanishing and
evolving in time. In order to extract the required information, we suggest a strategy based on the following steps
during the diffusion dynamics: (i) to interrupt the flux
of the pump atoms at any time t in the transient regime,
(ii) to apply a ‘‘counterfield’’ j i, and (iii) to interrogate the backshifted cavity field by a single probe atom.
The controlled injection of an external field (amplitude
and phase) inside a closed cavity, through the same holes
used for the atomic pumping, has already been realized in
the Garching experiments [17].
The application of the counterfield is described by the
^ D
^ y ,
unitary displacement operator [10], D
mixing diagonal and off-diagonal elements of the diffused cavity field. Then, the new photon statistics as a
function of the diffusion time t reads
^ jiihjjD
^ jnii;j t:
hnjD
(7)
Now, an excited probe atom is injected in the cavity, and
the probability of finding it in the excited state after a
transit time p is
1
X
p
~n tcos2 gp n 1:
p
(8)
Re<a>
n0
3
<
2
1
0
0
40
80
γt 120
FIG. 2. Exponential decay of the expectation value of the
cavity field due to phase diffusion Reha^ i vs dimensionless time
t from the MME (same parameters as in Fig. 1).
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~ e t; p Kp expDt p
~e 1; p ;
p
where
Kp 2Re
"
1 X
1
X
(9)
^ jiihi 1jD
^ jni
i;i1 0hnjD
n0 i0
#
p
cos gp n 1
2
(10)
and
~ e 1; p p
1 X
1
X
p
2
2
^
pss
i jhnjDjiij cos gp n 1:
(11)
i;j0
~ e t; p p
Then we are left with a kind of inverse problem in
cavity QED, namely, to extract from the measured atomic
statistics — our scattering data — the information on the
~n t. We
decay of coherences n;n1 t which is hidden in p
indicate two solutions to this problem. The first one ex~n t.
ploits the time dependence of the photon statistics p
~n t reflects the time dependence
Equation (7) shows that p
of the unshifted field density matrix, i;j t. If the probe
atom is injected in the cavity at times t D1 , all
coherences i;ik t for k 2 have already vanished.
~n t in Eq. (7) are given
Hence, the only contributions to p
by the terms with j i, involving the diagonal elements
i;i t pss
i , and the terms with j i 1, involving the
coherences i;i1 t i;i1 0 expDt and their complex conjugates. Hence, for times t D1 , we can rewrite the atomic probability of Eq. (8) as
n0 i0
^ tD
^ jni
~ n t p
~n;n t hnjD
1
X
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~e t; p as a function of t is ruled
Equation (9) shows that p
by the phase diffusion rate D, providing a simple and
direct link between the experimental data and the micro~e 1; p can be
maser linewidth. Notice that Kp and p
calculated directly from the initial conditions and experi~e t; p as obmental parameters. In Fig. 3, we show p
tained from the numerical simulations of the whole
scheme, using the MME (2), with p . In this ex~e t shows a minimum that is followed, for t ample, p
D1 , by an exponential approach to the asymptotic value
~e 1; . Actually, in Fig. 3 we also show that, as prep
~e t; for times t dicted by Eq. (9), the behavior of p
D1 is perfectly fitted by an exponential law. This allows
us to derive the phase diffusion rate D= 0:047, quite
close to the value we calculated from Fig. 2.
As a further example, we estimate the phase diffusion
rate of the micromaser field close to a sharp trapping
state, r= 16 and g 2:985, where the spectrum is
well approximated by a single Lorentzian with HWHM
equal to D= 2:4. Following our method, starting from
a coherent state with 2:402 and using gp 6:676,
~e t; p for t > D1 ,
we obtain D= 2:5 by fitting the p
in good agreement with the exact value [14]. In the
particular cases where the trapping states produce anomalous (non-Lorentzian) spectra [7], Eqs. (5) and (9) contain
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PHYSICA L R EVIEW LET T ERS
VOLUME 90, N UMBER 18
~
p
e
0.75
0.5
0.25
0
25
50
75
γt 100
~e , Eq. (8), that an excited
FIG. 3. Solid line: probability p
probe atom leaves the cavity excited. Dashed line: fit from
~e 1; p 0:551,
Eq. (9) for t > D1 , with Kp 0:283, p
and D= 0:047. Same parameters as in previous figures.
more than one dominant exponential. In those cases the
HWHM criterion provides an average of the different
diffusion rates and our proposal can be simply extended
through a more elaborated fitting. Note that it is in the
trapping regions where the nonclassicality of the micromaser field should manifest and where the departure from
the monotonic Schawlow-Townes linewidth is expected.
Another solution to our problem exploits the fact that
the interaction time p of the probe atom, injected in the
final stage of the measurement, may be different (in fact,
shorter) than the interaction time of the pump atoms.
The crucial point is provided by the expression of the
mean photon number of the backshifted field
^ tD
^ h~N^ ti Tr
a^ y a^ D
2jj2 1 expDt:
(12)
Here, we have used the properties of the trace and of the
^ , with the implications of our
displacement operator D
starting assumptions hN^ ti hN^ 0i jj2 ; ha^ ti ha^ 0i expDt expDt. Equation (12) shows
that the mean photon number of the backshifted field
grows exponentially with the rate D. We observe that,
^ would lead to
in principle, any other displacement D
the same dependence. Our choice, , dictated by
simplicity, implies also a remarkably simple mathematical expression. The remaining step consists of finding a
manifest dependence of the measured atomic statistics,
Eq. (8), on the mean photon number of Eq. (12). We
choose
an interaction time of the probe atom such that
p
2gp n 1 1 for all Rabi frequencies in Eq. (8). In
this case, the probability for the outgoing probe atom to
~g t; p 1 p
~e t; p , can be
be in the ground state, p
approximated by
~ g t; p gp 2 h~N^ it 1:
p
(13)
Replacing h~N^ it of Eq. (12) in Eq. (13) provides another
simple and direct link between measured atomic populations and the phase diffusion rate. For example, using the
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same micromaser parameters as in Figs. 1 and 2, the result
of Eq. (13) turns out to hold [14] for probe atom interaction times p on the range 0 < gp < 0:15. This means
that the velocity of the probe atom should be greater than
the velocity for the micromaser operation (g 0:5), so
that the atomic statistics exhibit an exponential behavior
that will allow us to monitor the phase diffusion dynamics. We remark that the criterion based on Eq. (13) holds
even in the initial stage of the diffusion process, t < D1 ,
where our first criterion failed.
Following two different strategies, we have shown how
the phase diffusion dynamics can be obtained from measurable statistics of probe atoms in all ranges of the
micromaser parameters. An experimental implementation of this scheme is currently planned. We expect that
it helps to unveil the distinctive properties of the micromaser spectrum, when compared with the conventional
Schawlow-Townes analog for the laser, due to the discrete
character of the quantized electromagnetic field.
E. S. is thankful to G. Marchi, T. Becker, and B.-G.
Englert for valuable discussions.
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