“Modern Trends in Atomic Physics II“
Symposium in honour of Professor Ingvar Lindgren,
Chalmers University of Technology,
Göteborg, Sweden, June 2-3, 2006
______________________________________________________________________
A single quantum system:
Evolution, prediction, observation
Peter E. Toschek
Universität Hamburg, Institut für Laser-Physik
Support by Deutsche Forschungsgemeinschaft
Entanglement I
Two correlated spins:
1
01
0
1
1
2
or
( 01 + 10 )
10
0
A. Einstein, B. Podolsky, N. Rosen, Phys.Rev. 45, 777, 1935
"... spooky action on a distance ... "
A. Einstein
Entanglement II
Electronic excitation correlated with ionic vibration:
n+1
n + 1 1〉
n
n+1
1
cosθ n 0 + sinθ n +11 )
(
2
n 0 〉
n
B. Appasamy, Y. Stalgies, P.E. Toschek, Phys. Rev.Lett. 80,
2805, 1998 .,
Application to QIP: J.I. Cirac, P. Zoller, PRL 74, 4091, 1995
With spin excitation: C. Monroe, D.M. Meekhof, B.E. King, W.M. Itano, D.J. Wineland, P.R.L. 75, 4714,
1995
Observation of laser cooling:
W Neuhauser, M Hohenstatt, H Dehmelt, PE Toschek,
Phys.Rev.Lett. 41, 233 (1978).
Trapping of individual ion:
W Neuhauser, M Hohenstatt, H Dehmelt, PE Toschek,
Phys.Rev. A 22, 1137 (1980).
Entanglement II
Electronic excitation correlated with ionic vibration:
n+1
n + 1 1〉
n
n+1
1
cosθ n 0 + sinθ n +11 )
(
2
n 0 〉
n
B. Appasamy, Y. Stalgies, P.E. Toschek, Phys. Rev.Lett. 80,
2805, 1998 .,
Application to QIP: J.I. Cirac, P. Zoller, PRL 74, 4091, 1995
With spin excitation: C. Monroe, D.M. Meekhof, B.E. King, W.M. Itano, D.J. Wineland, P.R.L. 75, 4714,
1995
Quantum Measurement
Quantum system correlated with signal
signal on
coherent
preparation
of superposition state:
detection
or off
observed
eigenstate:
or
back
back action
action ??
1
2
ensemble of
detections
confirms
(cos0 0 on + sin0 1 off )
Detected Signal
••••
•••
••
••
••••••
•••• •••••
•••
Number of Measurement, Time
•
1.Temporal evolution of a quantummechanical system
2.The quantum-mechanical measurement
3.The emergence of the classical world
out of the quantum micro-cosmos
Evolution of a Quantum System is Frustrated
by Repeated Measurement
L.A. Khalfin, Pis'ma Zh.Eksp.Teor.Fiz. 8, 106 (1968) [JETP
Lett. 8, 65 (1968)].
L. Fonda et al., Nuovo Cimento A 15, 689 (1973).
B. Misra and E.C.G. Sudarshan, J. Math.Phys. (N.Y.) 18, 756
(1977).
"Quantum Zeno Effect"
A. Beige and G.C. Hegerfeld, Phys.Rev. A 53, 53 (1996);
J.Phys. A 30, 1323 (1997)
(many more references).
L/2
L/2
θ
P1
θ
2
P2
P0
θ
2
Frustrated Rotation of Light Polarization
Medium of length L rotates light polarization
by angle θ, as shown by analyser 1.
Additional n analysers (2, ...), at distances
L/n of each other, represent n additional
"measurements"; they yield
Total rotational angle:
θ/n → 0
(n→ ∞)
Loss per measurement: θ2/n2
Total loss:
2
θ /n → 0
(n→ ∞)
z
complete polarisation
partial polarisation
y>0
x
y<0
Configuration space of light polarisation:
Poincaré sphere
Probability of driven transition
|0〉 → |1〉 after n additional light
pulses considered measurements:
P01(1) = sin2θ ≅ Ω2(∆t)2
Configuration space
|1〉
P01(n) = Ω2n(∆t)2
cos θ
P01(n) ∝ T2/n → 0 (n → ∞)
Probability of surviving in same state:
P00(n) = 1/2 [1 + cosn (π/n)] → 1
RJ Cook, Phys.Scr. T 21, 49 (1988).
θ
θ
(n → ∞)
sinθ
∆t = T/n
|0〉
Experiment on 5000 Be+ ions in electromagnetic trap
("Penning trap")
approved predictions of quantum mechanics:
expectation values
W.M. Itano, D.J. Heinzen, J.J. Bollinger, and D.J. Wineland, Phys.Rev. A 41,
2295 (1990); loc. cit. A 43, 5168 (1991).
Strategy of Measurement
on Ion Cloud
Photon
Counter
Drive Light
Probe
Light
Probe
Light
Time
Preparation
θ=
π
Detection:
Size of Signal ?
Experiments on ensembles: What do they reveal ?
Problems and criticism:
1. Interpretation by projection postulate and state reduction said inadequate
2. Interpretation as QZE said inadequate because state reduction not required
3. Recording the net probability of finding the ion in same state, after n light
pulses, is said not to establish the interactions of these pulses as measurements
H. Nakazato, M. Namiki, S. Pascazio, H. Rauch, Phys.Lett. A 217, 203 (1996).
4. Each observation qualifying for QZE is said to require a non-local effect from
a null result
D. Home, M.A.B. Whitaker, Ann.Phys. (N.Y.) 258, 237 (1997).
5. With an ensemble, the effect of a measurement is indistinguishable from the
effect of dynamic phase perturbation on the wave function
V.B. Braginski, F.Ya.. Khalili, Quantum Measurement (Cambridge University
Press, Cambridge, MA, 1992).
T.P. Spiller, Phys.Lett. A 192, 163 (1994).
O. Alter, Y. Yamamoto, Phys.Rev. A 55, R2499 (1997).
Preparation and addressing of an
individual quantum system
system:
ensemble
individual
quantum system
application
_________________________________________________________________________________________________________________________________________________
quantity:
expectation value eigenvalue
Oi
〈 O〉
perturbation:
interaction with
neighbours
no
frequency standard
observed:
macro state,
average
micro state
quantum information processing
result:
deterministic
stochastic (?)
cryptography
ensemble
single atom
measurement
Preparation and
result
measurement
measurement
of quantum system
actual
potential actual
z
| 1〉
with two
eigenstates
of energy (SU2)
potential
or
P1
expectation
value
θ
P0
| 00 〉
time
0
eigenvalue
or
time
Evolution of an individual quantum system
impeded by measurement
Chr Balzer, R Huesmann, W Neuhauser, PE Toschek,
Opt.Communic. 180, 115 (2000).
PE Toschek and Chr Balzer, Laser Physics 2, 253 (2002).
Chr Balzer, Th Hannemann, Chr Wunderlich, W Neuhauser, PE Toschek,
Opt.Communic. 211, 235 (2002).
Instantaneous Electric Potential in Paul Trap
Two 138Ba+ Ions
Recordings: Klaus Abich, Universität Hamburg
„ ... the essence of the quantum Zeno effect is that
it is a nonlocal negative-result effect between a
microscopic system and a macroscopically
separated macroscopic measuring device ...“
D. Home, M.A.B. Whitaker, Annals of Physics 258, 237 (1997).
171Yb+
P1/2
F=0
photon
counter PC
probe
369 nm
F=1
drive
12,6 GHz
PC
S1/2
F=0
π
n
prep
π
n
π
n
π
n
π
n
time
counts
probe
••••
•••
••
••
••••
••••
•••
••••
••
number of measurement
Nutational Dynamics of a Spin-like Quantum System
SU2 symmetry, no relaxation, θ = Ωt :
Probability of survival:
Transition probability:
2
θ/2
2
θ/2
p0 = p1 = cos
p01 = p10 = sin
*
Probability U(q) for q successive equal results (all "on" or all
"off"):
U( q ) = U( 1 ) ⋅ V( q − 1 ),
where
V(q) = p iq ( i = 0 or 1)
is the conditional
probability of the system surviving in same eigenstate.
F=0
0
θ=
log V(q–1)
-2
2(π - 0.1) - 0.1
(π - 0.1) / 5
-4
-6
(π - 0.1) / 2
(π-0.1)
-8
0
30
tprobe= 2ms
Trf = 4.9ms
60
90
length of sequence q – 1
120
N = 2000
150
Perturbations and decoherence?
• No energy dissipation with quantum object!
Only probe light scattered from quantum probe is
dissipated
• No phase perturbation of quantum object's wave
function !
Driving light's phase has standard deviation δϕ << 2π
• Recoil on quantum object ?
Driving is recoil-free! ("strong" trapping)
Probing exerts
– recoil, when "on" results (but too small for effect)
– no recoil, when "off" results.
QND measurements!
PC
a)
prep
probe
rf π-pulse
time
PC
b)
prep
π
n
π
n
π
n
π
n
π
n
π
n
π
n
probe
time
PC
c)
prep
π
n
π
n
π
n
π
n
π
n
π
n
π
n
probe
time
Driving by n-fold fractionated π-pulses with
interposed probing
Probability of survival when
• series
with intermediate “on“ rejected (“selective“
measurements):
(s)
00
2n
P (n) = cos (π / 2n)
→ proves QZP
• results of intermediate probing ignored (“non-selective“
measurements):
( ns )
00
P
1
(n) = (1 + cos n (π / n) )
2
→ cannot prove QZE or QZP
no “state reduction“
experiment
Summary
•
Quantum-mechanical evolution of individual atom (ion) delayed
or impeded by measurement:
Quantum Zeno Effect and even Quantum Zeno Paradox on
electronic and hyperfine resonances
•
No dynamic perturbation of quantum state
•
No "Heisenberg microscope"-type back action
•
Quantum system entangled with signal
•
Coherent rotation ("one-qubit gate") demonstrated on electronic
resonance of individual ion: Precondition for application to
quantum information processing
On vibrational resonance entangled with microwave transition:
D.M. Meekhof, C. Monroe, B.E. King, W.M. Itano, D.J.Wineland,
Phys.Rev.Lett. 76, 1796 (1996).
On vibrational resonance entangled with electronic transition:
B. Appasamy, Y. Stalgies, and P.E. Toschek, Phys.Rev.Lett. 80, 2805
(1998).
Was können wir wissen ?
Was sollen wir tun ?
Was dürfen wir hoffen ?
Immanuel Kant (1724-1804)
So long as anything is in a space equal to itself,
it is at rest.
An arrow is in a space equal to itself at every
moment of its flight, and therefore also
during the whole of its flight.
Thus the flying arrow is at rest.
Zeno of Elea
about 495-430 B.C.
Quotation after F. Cajori, Am.Math.Mon. 22, 292 (1915).