Dynamical Localization and Delocalization in a Quasiperiodic Driven System

1. 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 Dynamical Localization and Delocalization in a Quasiperiodic Driven System FRISNO-8, EIN BOKEK 2005 This work has been supported by :

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. Free evolving atoms… 0 < t < T t = T Standing wave intensity v.s. time standing wave intensity T < t < 2T The atomic kicked rotor … periodically kicked by a far detuned laser standing wave: T: kick’s period Graham, Schlautman, Zoller (1992) Moore, Robinson, Bharucha, Sundaram, Raizen, PRL 75, 4598 (1995)

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

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

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

7. 1 0 kicks 10 kicks 10-1 Typical experimental values: 20 kicks Localisation time: 10-2 50 kicks 10-3 100 kicks 10-4 200 kicks Kicks 10-5 -600 0 600 Dynamical Localization Experiment => atomic velocity measurement

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

9. -40 -60 Beat power (dBm) FWHM ~ 1 Hz -80 -100 -120 DC Bias 4.6 GHz -140 Hz -400 -200 0 200 400 Beat frequency: 9 200 996 863 Hz Raman beam generation FP S+1 Master S-1

10. 4 4 Trap loading Deeper Sisyphus cooling Pulse sequence 3 3 Velocity selection Repumping Final probing Experimental Sequence Pushing beam Raman 2 Cell 11° Raman 1 Stationary wave beam Probe beam Trap beams are not shown Raman 2bis Pushing beam

12. Initial gaussian distribution Distribution after 50 kicks 1 0.1 Gaussian fit Exponential fit 0.01 f (kHz) -300 -200 -100 0 100 200 300 0.001 -40 p/hk -20 20 40 0 Experimental observation of (one color)dynamical localization Kick’s period: T = 27 µs (36 kHz), 50 pulses of t = 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)

13. f2 f1 Two colours modulation One colour modulation : Two colours modulation : r = f1/f2, frequency ratio of two pulse series: 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)

14. The population P(0) of the 0 velocity class is a measurement of the degree of localization f = 180° Localized Freq. ratio = 1.083 1 Delocalized Standing wave intensity v.s. time 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 Two-colours dynamical localization breaking Initial distribution J. Ringot, P. Szriftgiser, J.C. Garreau and D. Delande, Phys. Rev. Lett., 85, 2741 (2000).

15. 1 Localization P(0) 1/2 2 1/4 5/3 1/3 2/3 4/3 3/2 3/4 5/4 0 0.5 1 2 1.5 Frequency ratio « Localization spectrum » F = 52°

16. f2 f1 FT D(Exp) 1 ~ 37 DFT r = 0.87 f FT Sub-Fourier lines Experimental FT Atomic signal Frequency ratio r Pascal Szriftgiser, Jean Ringot, Dominique Delande, Jean Claude Garreau, PRL, 89, 224101 (2002)

17. Fourier limit 1 µs K = 14 Resonance width ×N1 2 µs 3 µs K = 28 K = 42 Experimental points at N1=10, for t = 1,2,3 µs Assuming: 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 Pulse number N1 FirstInterpretation • 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

18. For a mono-color experiment: K = 10, k = 2 Let’s come back to the periodic case:the Floquet’s States F: Floquet operator An infinity of eigenstatesfk: F|fk> = eie(k) |fk> In the Floquet’s states basis: |< fk |fk>|2 Only the significant states are taken into account: |ck|2> 0.0001

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

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

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

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