Dynamical Localization and Delocalization in a Quasiperiodic Driven System

Hans Lignier, Jean Claude Garreau, Pascal SzriftgiserLaboratoire de Physique des Lasers, Atomes et Molécules, PHLAM, Lille, France

Dominique DelandeLaboratoire Kastler-Brossel, Paris, France

This work has been supported by :

FRISNO-8, EIN BOKEK 2005

The Quantum Chaos Project:

- An experimental realization of an atomic kicked rotor

-The observation of the « Dynamical Localization » Phenomenon, and its destruction induced by time periodicity breaking

- Observation of sub-Fourier resonances

- Is DL’s destruction reversible?

The atomic kicked rotor

Free evolving atoms… 0 < t < T

… periodically kicked by a far detuned laser standing wave:

T < t < 2T

Graham, Schlautman, Zoller (1992)

Moore, Robinson, Bharucha, Sundaram, Raizen, PRL 75, 4598 (1995)

T: kick’s period

Standing wave intensity v.s. time

t = T 0V

0Vstanding wave intensity

The kicked rotor classical dynamic

K = 0 K = 0.01

K ~ 1 K = 5

The standard map: B. V. Chirikov, Phys. Rep. 52, 263 (1979)

ttt

ttt

P

KPP

1

11 sin

pM

TkPxkTtt

ntKPtH

LL

n

2,2,/'

'cos2/,' 2

The whole classical dynamic is given by only one parameter: /8 0 TVK r: pulse duration ( << T )

2p

time

Dtp 22

Gaussian distribution

K>>1

Quantized standard map

Two parameters: and K

Quantization of the map:

n

PiK

in

2expcosexp1

2

ntKPtHn

cos2/, 2Same Hamiltonian:

Schrödinger equation:

Ht

i scaled Planck constantTr 8

Kicked Rotor Quantum Dynamics

2p

time

Dtp 22

Classicalevolution

Casati, Chirikov, Ford, Izrailev (1979)* Periodic system: Floquet theorem

* Exponential localization in the p-space* Suppression of classical diffusion

P(p)

P(p)

0

Quantumevolution

P(p)

TH: localisation time

2locp

Dynamical Localization

1

10-5

10-4

10-3

10-2

10-1

kp 2/-600 6000

0 kicks

10 kicks

20 kicks

50 kicks

100 kicks

200 kicks

Localisation time:2

2

1

K

TH

255 HT Kicks

Experiment => atomic velocity measurement

Typical experimental values:

2010 K

31

Ground state

Optical transition

F=4F=3

9.2 GHz

200 GHz

, detuning ~ kHz

Resonant transition (with a null magnetic field) for:

2kVatome Cte

M. Kasevich and S. Chu, Phys. Rev. Lett., 69, 1741 (1992)

A Raman experiment on caesium atoms

Raman beam generation

DC Bias 4.6 GHz

Master

S+1

S-1

FP

-100

-80

-60

-40

Bea

t pow

er (

dBm

)

FWHM ~ 1 Hz

-140

-120

-400 -200 0 200 400Beat frequency: 9 200 996 863 Hz

Hz

+1-1

0

Deeper Sisyphus coolingTrap loading Pulse sequence

Raman 2bis

Raman 2

Raman 1

Stationary wave beam

Probe beam

Pushing beam

11°

Cell

Trap beams are not shown

Experimental Sequence

Velocity selection

Repumping Final probing

Pushing beam

4

3

4

3

Experimental observation of (one color)dynamical localization

f (kHz)0.001

0.01

0.1

1

-300 -200 -100 0 100 200 300

Distribution after 50 kicks

-40 -20 0 20 40p/hk

Initial gaussian distribution

Exponential fitGaussian fit

B. G. Klappauf, W. H. Oskay, D. A. Steck and M. G. Raizen, Phys. Rev. Lett., 81, 1203 (1998)

Kick’s period: T = 27 µs (36 kHz), 50 pulses of = 0.5 µs duration. K~10, ~1.4

Two colours modulation

-Periodicity breaking and Floquet’s states.-Relationship between frequency modulation andeffective dimensionality. -Dynamical localisation and Anderson localisation.

One colour modulation :

Two colours modulation :

G. Casati, I. Guarneri and D. L. Shepelyansky, Phys. Rev. Lett., 62, 345 (1989)

r = f1/f2, frequency ratio of two pulse series:

ntKPtHn

cos2/, 2

rntntKPtH

nn

/cos2/, 2

f1

f2

time

Two-colours dynamical localization breaking

J. Ringot, P. Szriftgiser, J.C. Garreau and D. Delande, Phys. Rev. Lett., 85, 2741 (2000).

Initial distribution

0.01

0.1

1

-60 -40 -20 0 20 40 60

Momentum (recoil units)

= 180°

Freq. ratio = 1.000

Standing wave intensity v.s. time Freq. ratio = 1.083

For an « irrational » value of the frequency ratio, the classical diffusive behavior is preserved

The population P(0) of the 0 velocity class is a measurement of the degree of localization

Localized

Delocalized

« Localization spectrum »

1

1/2 2

3/23/41/3 2/3 4/3

5/3

5/4

1/4

Loc

aliz

atio

n P

(0)

Frequency ratio0 0.5 1 1.5 2

= 52°

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

FT

Exp)

FT 1

37FT

Experimental

Pascal Szriftgiser, Jean Ringot, Dominique Delande, Jean Claude Garreau, PRL, 89, 224101 (2002)

f

f1

f2

FT

r = 0.87

First InterpretationThe higher harmonics in the excitation spectrum are responsible of the higher resolution:

(1) The resonance’s width is independent of the kick’s strength K

(2) If the pulse width is increased => the resonance’s width should increase as well

(3) The resonance’s width decay as 1/Texcitation sequence

Numerical evaluation of the resonance’s width as a function of time.The resonance width shrinks faster than the reciprocal length of the excitation time

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

Res

onan

ce w

idth

×N

1

Fourier limit

K = 14

K = 28

K = 42

1 µs 2 µs3 µs

Pulse number N1

Experimental points at N1=10, for = 1,2,3 µs

KAssuming:

Let’s come back to the periodic case: the Floquet’s States

F: Floquet operator

2expcosexp,1

2PiK

iFnFn

For a mono-color experiment:

-250 -200 -150 -100 -50 0 50 100 150 200 25010

-6

10-5

10-4

10-3

10-2

10-1

100

Momentum

K = 10, = 2

An infinity of eigenstates k: F|k> = ei(k) |k>

Only the significant statesare taken into account: |ck|2 > 0.0001|<

k |

k>

|2

In the Floquet’s states basis:

kk

kkn incFn exp0

0 kkc

The non periodic case: Dynamic of the Floquet’s States

H. Lignier, J. C. Garreau, P. Szriftgiser, D. Delande, Europhys. Lett., 69, 327 (2005)

-250 -200 -150 -100 -50 0 50 100 150 200 25010

-6

10-5

10-4

10-3

10-2

10-1

100

Momentum

K = 10, = 2

Avoidedcrossings

Only the significant states are plotted (|ck|2 > 0.0001):

time

K

K+K

C

Partial Reversibility in DL Destruction

0 10 20 30 40 50 60 700.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

0

1

-40 -20 0 20 40

0

0.5

1

1.5

-40 -20 0 20 40

0

0.5

1

1.5

P = 0 W

P = 50 WMomentum distribution

Kicks number

Kicks number (first series)

Conclusion

Complex dynamics – unexpected results

Dynamical localization destruction

Observation of a partial reconstruction of DL

kk

kkn incFn exp0 0 kkc

kkkk

kkkk pinccnTp 2'

','

*'

2 exp

At long time (i.e. after localization time), the interference termswill on the average cancel out:

kkk

k pcp 222

Adiabatic case: Different state + random phase

Diabatic case:Same state + random phase

Intermediate case: