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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: