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