presentation quantum chaos with cold atoms in a standing wave laboratoire de physique des lasers...
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Presentation
Quantum chaos with cold atoms in a standing wave Laboratoire de Physique des Lasers Atomes et
Molécules (PhLAM)Villeneuve d’Ascq, France
www.phlam.univ-lille1.fr/atfr/cq
Pascal Szriftgiser Jean RingotHans Lignier
J. C. G.
In collaboration with Dominique Delande LKB - Paris
System
The systemCold atoms in a standing wave
d L/2
V0
dB
H = P2/2 + K cos ( )n (t-n)
Theory: Graham, Schlautman, Zoller PRA 45, 19 (1992) Exp: Moore et al., PRL 73, 2974 (1994); Klappauf et al. PRL 81, 1203 (1998)
Quantum chaos
Classical phase portrait
K = 0.9 K = 2
K = 5 K = 10
Regular dynamics:Periodic orbits
Mixt dynamics:Stability islands
Mixt: Small islands Chaotic:Ergodic diffusion
Quantum chaos
Momentum evolution
P(p) ~ e-|p|/pL
Log P
p
Quantum chaos
Dynamical localization
t
<Ec>
Quantum
ClassicalP(p)
P(p)
P(p)
Theory: Casati, Chirikov, Ford, Izrailev (1979) Experience : Moore, Robinson, Bharucha, Williams, Raizen (1994)
tL ~ tH = h/<E>
Heisenberg time
System
Momentum distribution measurement
Stimulated Raman transitions
R
1 2
hf
Resonant probe
F = 3F = 4
v0 = R/2kL
Mkvv L2
0
vr/10
Quantum chaos
Experiment
Dynamical localization
Momentum distributions
f (kHz)0.001
0.01
0.1
1
-300 -200 -100 0 100 200 300
Final distribution (50 kicks)
-40 -20 0 20 40p/hk
Initial distribution
Exponencial fitGaussian fit
Dynamical localization
Quasi-periodic kicks
f1
f2
f1/19
0
Dynamical localization
Dynamical localization with “two colors”
For « irrational » values of the frequency ratio, the classical diffusive behavior is preserved
0.01
0.1
1
Initial distribution
Freq. ratio = 1.000
Freq. ratio = 1.083
Initial distributionFrequency ratio = 1,0833…
Frequency ratio = 1
Num
ber
of a
tom
s (l
og)
Momentum (hk)-50 0 50
Dynamical localization
Localization measurement
Initial
Localized
Delocalized
Consevation of the atom number The population P(0) of the 0 velocity class is a mesurement of the
degree of localization
Dynamical localization
Localization “spectrum”
1
1/22
3/23/41/3 2/3
4/35/3
5/4
1/4
Loc
aliz
atio
n P
(0)
Frequency ratio0 0.5 1 1.5 2
Breaking the periodicity destroys localization
J. Ringot, P. Szriftgiser, J.C.G., D. Delande, PRL 85, 2341 (2000)
Sub-Fourier
“Sub-Fourier” lines
4.8
4.6
4.4
4.2
4.0
3.8
3.6
1.151.101.051.000.950.900.85
Ato
mic
sig
nal
Frequency ratio r
F12
Experimental F12
The lines ARE NOT FT’s of a temporal signal!
« Sub-Fourier » resolution: f1 f2 for Tmes < 1/|f1-f2|
P. Szriftgiser, J. Ringot, D. Delande, J.C.G., submited to PRL
Resolution ~ 1/Texp
Exp)
F12
137
Sub-Fourier
= 0.02 1/14
= 0.05 1/37
Quantum interference sensitivity ot frequency and phase
Interpretation
Sub-Fourier
Interpretation
0.05
0.10
0.15
0.20
0.25
0 50 100 150 200
0 50 100 150 200Kicks
0.0
5
0.
10
0.
15
0.20
0.25
Wid
th
num
ber
of k
icks
K [1+ A cos(2rt)] n (t-n)
r
Sub-Fourier
Physical mechanism
= P2
2N
r -1r
e -i
N/rN
N ~ 0 periodicity
N ~ /2 quasi-periodicity
Résolution is due to the dynamical spectrum, not to the excitation spectrum
Sub-Fourier
Evolution of the width
4
68
0.01
2
4
68
0.1
2
4
68
1
5 6 7 8 910
2 3 4 5 6 7 8 9100
N
r
N
Fourier limit
K = 14
K = 28
K = 42
1 µs 2 µs3 µs
1/N2
= D N2 N
r -1r
~ N2
Before localization
= PL
2
2 Nr -1r
~ N
After localization
Conclusion
Conclusion
Complex dynamics – unexpected results
Simplicity and versatility
Detailed experimental study of the linewidth
Interpretation – physical mechanisms
New conditions: anomalous diffusion r ~1/N3
Quatum-chaotic signal processing ultrafast frequency locking (?)
Funding
Funding
CNRS
Ministère de la Recherche
Région Nord-Pas de Calais
Comunidade Européia
Sub-Fourier
Resolution (quantitative)
f
f1
f2
F12
r = 0.87
F12
r
F12/2
F12 measures the Fourier resolution
Sub-Fourier
Excitation spectrum
TF
T TF1/T
TF1/
1/
1/T
+1-1
0
-140
-120
-100
-80
-60
-40
-300 0 300
DC Bias 4.6 GHz
Generation of the Raman beams
Master
S+1
S-1
FP
Raman setup Experimental setup
Sub-Fourier
“Sub-Fourier” resonances4.8
4.6
4.4
4.2
4.0
3.8
3.6
1.151.101.051.000.950.900.85
Ato
mic
sig
nal
Frequency ratio r
F12Experimental F12
Experimental)F12
137
P. Szriftgiser, J. Ringot, D. Delande, J.C.G., submetido a Nature
Sub-Fourier
“Sub-Fourier” resonances4.8
4.6
4.4
4.2
4.0
3.8
3.6
1.151.101.051.000.950.900.85
Ato
mic
sig
nal
Frequency ratio r
F12Experimental F12
Experimental)F12
137
P. Szriftgiser, J. Ringot, D. Delande, J.C.G., submetido a Nature
Dynamical localization
Kinetic energy
0 10 20 30 40 50
Kicks
tL
System
Velocity distribution for the sisyphus molasses
-250 -200 -150 -100 -50 0 50 100 150 200
Désaccord Raman (kHz)
0.0
0.5
1.0
1.5
72 kHz
T = 3.3 µK
Raman detuning
Active magnetic field compensation
-100 0 100
ON+ offset
OFF+offset
1 kHz ~ 300 µG ~ vR/8
ON+offset
(kHz)
BW ~ 500 Hz
Magnetic field compensation Experimental setup
Bx
ByBz