Dynamical Localization and Delocalization in a
Quasiperiodic Driven System
Hans Lignier, Jean Claude Garreau, Pascal Szriftgiser
Laboratoire de Physique des Lasers, Atomes et Molécules, PHLAM, Lille, France
Dominique Delande
Laboratoire Kastler-Brossel, Paris, France
FRISNO-8, EIN BOKEK 2005
This work has been
supported by :
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:
V
0
t=T
V  standing wave intensity
0
T < t < 2T
T: kick’s period
Graham, Schlautman, Zoller (1992)
Standing wave intensity v.s. time
Moore, Robinson, Bharucha, Sundaram, Raizen, PRL 75, 4598 (1995)
The kicked rotor classical dynamic
H t ' ,   P 2 / 2  K cos   t 'n 
Pt 1  Pt  K sin  t 1

 t 1   t  Pt
n
t '  t / T ,   2 k L x, P 
K=0
2k LT
p
M
The standard map: B. V. Chirikov, Phys. Rep. 52, 263 (1979)
K = 0.01
p2
p 2  2 Dt
K>>1
Gaussian distribution
time
K~1
K=5
The whole classical dynamic is given by only one parameter: K  8V0 rT / 
: pulse duration ( << T )
Quantized standard map
2
Same Hamiltonian: H t ,   P / 2  K cos   t  n 
Schrödinger equation:
ik

 H
t
n
k  8rT scaled Planck constant
Two parameters: k and K
 i P2 
 i

 n 
Quantization of the map:  n  1  exp   K cos  exp  
 k

 k 2 



 
Kicked Rotor Quantum Dynamics
P(p)
p2
Classical
evolution
p 2  2 Dt
P(p)
2
ploc
Quantum
evolution
P(p)
0
TH: localisation time
* Periodic system: Floquet theorem
* Suppression of classical diffusion
* Exponential localization in the p-space
time
Casati, Chirikov, Ford, Izrailev (1979)
Dynamical Localization
Localisation time:
TH
1K 
 

2 k 
2
Typical experimental
values:
1
0 kicks
10 kicks
10-1
20 kicks
10-2
50 kicks
10  K  20
10-3
1 k  3
10-4
100 kicks
200 kicks
10-5
5  TH  25 Kicks
-600
p / 2k 0
Experiment => atomic velocity measurement
600
A Raman experiment on caesium atoms
200 GHz
Optical
transition
F=
9.2 4GHz
F=
3
Ground state
, detuning ~ kHz
Resonant transition (with a null magnetic field) for:
  2kVatome  Cte
M. Kasevich and S. Chu, Phys. Rev. Lett., 69, 1741 (1992)
Beat power (dBm)
Raman beam generation
-40
-60
-80
-100
-120
-140
-400
FWHM ~ 1 Hz
DC Bias
-200
0
200
Beat frequency: 9 200 996 863 Hz
4.6 GHz
400 Hz
0
-1
FP
S+1
Master
S-1
+1
Experimental Sequence
4
Trap loading
Deeper Sisyphus cooling
Pulse sequence
3
Velocity selection
Cell
Pushing beam
Raman 2
11°
4
Raman 1
3
Repumping
Final probing
Stationary wave beam
Probe beam
Raman 2bis
Trap beams are not shown
Pushing beam
Experimental observation of (one color)
dynamical localization
Initial gaussian distribution
1
0.1
Distribution after 50 kicks
0.01
Gaussian fit
Exponential fit
0.001
f (kHz) -300 -200 -100
p/hk -40
-20
0
0
100 200 300
20
40
Kick’s period: T = 27 µs (36 kHz), 50 pulses of  = 0.5 µs duration.
K~10, k~1.4
B. G. Klappauf, W. H. Oskay, D. A. Steck and M. G. Raizen, Phys. Rev. Lett., 81, 1203 (1998)
Two colours modulation
One colour modulation :
H t ,   P 2 / 2  K cos   t  n 
n
Two colours modulation :


H t ,   P 2 / 2  K cos    t  n     t  n / r   
n
n

r = f1/f2, frequency ratio of two pulse series:
f1
f2
time
-Periodicity breaking and Floquet’s states.
-Relationship between frequency modulation and
effective dimensionality.
-Dynamical localisation and Anderson localisation.
G. Casati, I. Guarneri and D. L. Shepelyansky, Phys. Rev. Lett., 62, 345 (1989)
Two-colours dynamical localization breaking
The population P(0) of the 0 velocity class is a measurement
of the degree of localization
 = 180°
Initial distribution
Localized
1
Delocalized
Standing wave intensity v.s. time
Freq. ratio =
1.083
0.1
Freq. ratio =
1.000
0.01
-60
-40
-20
0
20
40
60
Momentum (recoil units)
For an « irrational » value of the frequency ratio, the classical diffusive behavior is preserved
J. Ringot, P. Szriftgiser, J.C. Garreau and D. Delande, Phys. Rev. Lett., 85, 2741 (2000).
Localization P(0)
« Localization spectrum »
F = 52°
1
1/2
1/4 1/3
2
2/3
3/4
4/3
5/3
3/2
5/4
0
0.5
1
Frequency ratio
1.5
2
Sub-Fourier lines
DExp)
4.8
Experimental
DFT
FT
4.4
FT
Atomic signal
4.6
~ 1
37
4.2
r = 0.87
4.0
f2
f1
3.8
3.6
0.85
0.90
0.95
1.00
1.05
1.10
1.15
FT
Frequency ratio r
Pascal Szriftgiser, Jean Ringot, Dominique Delande, Jean Claude Garreau, PRL, 89, 224101 (2002)
f
First Interpretation
The 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
Experimental points at
N1=10, for  = 1,2,3 µs
Assuming: K  
Resonance width ×N1
1
Fourier limit
8
6
4
2
0.1
1 µs
2 µs
3 µs
4
2
0.01
K = 14
8
6
K = 28
8
6
K = 42
4
5
6
7
8 9
2
10
3
4
5
6
7
8 9
100
Pulse number N1
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
Let’s come back to the periodic case: the Floquet’s States
 i P2 
 i


For a mono-color experiment: n  1  Fn , F  exp   K cos  exp  
k
k
2




An infinity of eigenstates k: F|k> = eie(k) |k>
F: Floquet operator
In the Floquet’s states basis:
0
10
K = 10, k = 2
-1
10
n   F n 0   ck exp  ine k  k
k
-2
|< k |k>|2
10
-3
Only the significant states
are taken into account: |ck|2 > 0.0001
10
-4
10
-5
10
-6
10
-250
ck  k 0
-200
-150
-100
-50
0
50
Momentum
100
150
200
250
The non periodic case: Dynamic of the Floquet’s States
K
k
Only the significant states are plotted (|ck|2 > 0.0001):
K+K
kk
time
0
10
K = 10, k = 2
-1
10
-2
10
-3
10
-4
10
Avoided
crossings
-5
10
-6
10
-250
-200
-150
-100
-50
0
50
Momentum
100
150
200
250
C
H. Lignier, J. C. Garreau, P. Szriftgiser, D. Delande,
Europhys. Lett., 69, 327 (2005)
Partial Reversibility in DL Destruction
1
0
1.5
1.5
1
1
1.5
0.5
0.5
1.4
0
0
1.7
1.6
-40
1.3
-20
0
20
40 -40
-20
40
50
0
20
40
1.2
1.1
1
P = 0 W
0.9
0.8
P = 50 W
0
10
20
30
60
Kicks number (first series)
70
Conclusion

Dynamical localization destruction

Complex dynamics – unexpected results

Observation of a partial reconstruction of DL
n   F n 0   ck exp  ine k  k
k
ck  k 0
p 2 nT    ck ck ' exp  ine k  e k '  k ' p 2 k
*
k ,k '
At long time (i.e. after localization time), the interference terms
will on the average cancel out:
2
2
2
p
  ck
k p k
k
Adiabatic case:
Different state + random phase
Intermediate case:
Diabatic case:
Same state + random phase