1. 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 This work has been supported by : FRISNO-8, EIN BOKEK 2005

2. 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?

3. 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 standing wave intensity

4. The kicked rotor classical dynamic K = 0K = 0.01K ~ 1 K = 5 The standard map: B. V. Chirikov, Phys. Rep. 52, 263 (1979) The whole classical dynamic is given by only one parameter:  : pulse duration ( << T ) time Gaussian distribution K>>1

5. Quantized standard map Two parameters:  and K Quantization of the map: Same Hamiltonian: Schrödinger equation: scaled Planck constant

6. Kicked Rotor Quantum Dynamics time Classical evolution Casati, Chirikov, Ford, Izrailev (1979) * Periodic system: Floquet theorem * Exponential localization in the p-space * Suppression of classical diffusion P(p)P(p) P(p)P(p) 0 Quantum evolution P(p)P(p) T H : localisation time

7. Dynamical Localization 1 10 -5 10 -4 10 -3 10 -2 10 -1 -6006000 0 kicks 10 kicks 20 kicks 50 kicks 100 kicks 200 kicks Localisation time: Kicks Experiment => atomic velocity measurement Typical experimental values:

8. Ground state Optical transition F= 4 F= 3 9.2 GHz 200 GHz , detuning ~ kHz Resonant transition (with a null magnetic field) for: M. Kasevich and S. Chu, Phys. Rev. Lett., 69, 1741 (1992) A Raman experiment on caesium atoms

9. Raman beam generation DC Bias4.6 GHz Master S +1 S -1 FP -100 -80 -60 -40 Beat power (dBm) FWHM ~ 1 Hz -140 -120 -400-2000200400 Beat frequency: 9 200 996 863 Hz Hz

10. Deeper Sisyphus coolingTrap loadingPulse sequence Raman 2bis Raman 2 Raman 1 Stationary wave beam Probe beam Pushing beam 11° Cell Trap beams are not shown Experimental Sequence Velocity selectionRepumping Final probing Pushing beam 4 3 4 3

12. Experimental observation of (one color) dynamical localization f (kHz) 0.001 0.01 0.1 1 -300-200-1000100200300 Distribution after 50 kicks -40 -20 0 2040 p/hk Initial gaussian distribution Exponential fit Gaussian 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

13. Two colours modulation -Periodicity breaking and Floquet’s states. -Relationship between frequency modulation and effective 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 = f 1 /f 2, frequency ratio of two pulse series: f1f1 f2f2 time

14. 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 -60-40-200204060 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

15. « Localization spectrum » 1 1/2 2 3/2 3/4 1/3 2/3 4/3 5/3 5/4 1/4 Localization P(0) Frequency ratio 00.5 1 1.5 2  = 52°

16. Sub-Fourier lines Atomic signal Frequency ratio r FTFT  Exp) FTFT  1 37 FTFT Experimental Pascal Szriftgiser, Jean Ringot, Dominique Delande, Jean Claude Garreau, PRL, 89, 224101 (2002) f f1f1 f2f2 FTFT r = 0.87

17. 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/T excitation 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 Resonance width ×N 1 Fourier limit K = 14 K = 28 K = 42 1 µs 2 µs 3 µs Pulse number N 1 Experimental points at N 1 =10, for  = 1,2,3 µs Assuming:

18. Let’s come back to the periodic case: the Floquet’s States F: Floquet operator For a mono-color experiment: K = 10,  = 2 An infinity of eigenstates  k : F|  k > = e i  (k) |  k > Only the significant states are taken into account: |c k | 2 > 0.0001 | | 2 In the Floquet’s states basis:

19. The non periodic case: Dynamic of the Floquet’s States H. Lignier, J. C. Garreau, P. Szriftgiser, D. Delande, Europhys. Lett., 69, 327 (2005) K = 10,  = 2 Avoided crossings Only the significant states are plotted (|c k | 2 > 0.0001): time KK K+  K  C

20. Partial Reversibility in DL Destruction Momentum distribution Kicks number Kicks number (first series)

21. Conclusion  Complex dynamics – unexpected results  Dynamical localization destruction  Observation of a partial reconstruction of DL

22. At long time (i.e. after localization time), the interference terms will on the average cancel out: Adiabatic case:Different state + random phase Diabatic case: Same state + random phase Intermediate case: