Dynamical Localization and Delocalization in a Quasiperiodic Driven System

Transcription

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 This work has been supported by : Dominique Delande Laboratoire Kastler-Brossel, Paris, France FRISNO-8, EIN BOKEK 005

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: V 0 V 0 t = T standing wave intensity T < t < T T: kick s period Graham, Schlautman, Zoller (99) Moore, Robinson, Bharucha, Sundaram, Raizen, PRL 75, 4598 (995) Standing wave intensity v.s. time

4 The kicked rotor classical dynamic K = 0 K = 0.0 K ~ K = 5 The standard map: B. V. Chirikov, Phys. Rep. 5, 63 (979) + = + = t t t t t t P K P P θ θ θ sin ( ) ( ) ( ) p M T k P x k T t t n t K P t H L L n,, / ' ' cos / ', = = = + = θ δ θ θ The whole classical dynamic is given by only one parameter: / 8 0 τ V ω T K r = τ: pulse duration ( << T ) p time p Dt = Gaussian distribution K>>

5 Quantized standard map Same Hamiltonian: H Schrödinger equation: ( t, θ ) = P / + K cos( θ ) δ ( t n) Ψ iκ t = HΨ Two parameters: κ and K n κ = 8ω r T scaled Planck constant + i i P + Quantization of the map: Ψ ( n + ) = exp K cos( θ ) exp ( n ) κ κ Ψ

6 Kicked Rotor Quantum Dynamics p p loc P(p) Classical evolution p = Dt Quantum evolution P(p) P(p) 0 T H : localisation time time * Periodic system: Floquet theorem * Suppression of classical diffusion * Exponential localization in the p-space Casati, Chirikov, Ford, Izrailev (979)

7 Dynamical Localization Localisation time: T H K κ Typical experimental values: 0 K < κ < kicks 0 kicks 0 kicks 50 kicks 00 kicks 00 kicks < T H < 5 Kicks -600 p/ k Experiment => atomic velocity measurement

8 A Raman experiment on caesium atoms 00 GHz Optical transition F=4 9. GHz F=3 δ, detuning ~ khz Ground state Resonant transition (with a null magnetic field) for: δ = kv atome + Cte M. Kasevich and S. Chu, Phys. Rev. Lett., 69, 74 (99)

9 Raman beam generation Beat power (dbm) FWHM ~ Hz Hz Beat frequency: Hz DC Bias 4.6 GHz FP S + Master S -

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

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12 Experimental observation of (one color) dynamical localization 0. Initial gaussian distribution Distribution after 50 kicks f (khz) p/hk Gaussian fit Exponential fit Kick s period: T = 7 µs (36 khz), 50 pulses of τ = 0.5 µs duration. K~0, κ~.4 B. G. Klappauf, W. H. Oskay, D. A. Steck and M. G. Raizen, Phys. Rev. Lett., 8, 03 (998)

13 One colour modulation : H Two colours modulation : r = f /f, frequency ratio of two pulse series: f f Two colours modulation ( t, θ ) = P / + K cos( θ ) δ ( t n) H n ( t, θ ) P / + K cos( θ ) δ ( t n) + δ ( t n / r + φ) = n n -Periodicity breaking and Floquet s states. -Relationship between frequency modulation and effective dimensionality. -Dynamical localisation and Anderson localisation. time G. Casati, I. Guarneri and D. L. Shepelyansky, Phys. Rev. Lett., 6, 345 (989)

14 Two-colours dynamical localization breaking φ = 80 The population P(0) of the 0 velocity class is a measurement of the degree of localization Initial distribution Localized Delocalized Standing wave intensity v.s. time Freq. ratio = Freq. ratio = 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, 74 (000).

15 «Localization spectrum» Localization P(0) / /4/3 /3 5/3 4/3 3/4 3/ 5/4 Φ = Frequency ratio

16 Sub-Fourier lines Atomic signal Experimental FT FT (Exp) FT r = 0.87 f f Frequency ratio r.0.5 FT f Pascal Szriftgiser, Jean Ringot, Dominique Delande, Jean Claude Garreau, PRL, 89, 40 (00)

17 First Interpretation The higher harmonics in the excitation spectrum are responsible of the higher resolution: () The resonance s width is independent of the kick s strength K () If the pulse width is increased => the resonance s width should increase as well (3) The resonance s width decay as /T excitation sequence Experimental points at N =0, for τ =,,3 µs Assuming: K τ Resonance width N µs µs 3 µs Fourier limit K = 4 K = 8 K = Pulse number N 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

18 Let s come back to the periodic case: the Floquet s States For a mono-color experiment: F: Floquet operator i i P Ψ ( n + ) = F Ψ ( n), F = exp K cos( θ ) exp κ κ An infinity of eigenstates φ k : F φ k > = e iε(k) φ k > 0 0 In the Floquet s states basis: < φ k φ k > K = 0, κ = Ψ n ( n) = F Ψ( 0) = ck exp( inε k ) k k φ Ψ( 0) c k = k Only the significant states are taken into account: c k > φ Momentum

19 The non periodic case: Dynamic of the Floquet s States K κ K+δK κ+δκ Only the significant states are plotted ( c k > 0.000): time K = 0, κ = Avoided crossings Momentum H. Lignier, J. C. Garreau, P. Szriftgiser, D. Delande, Europhys. Lett., 69, 37 (005) C

20 Partial Reversibility in DL Destruction Kicks number Momentum distribution 0.9 P = 0 µw P = 50 µw Kicks number (first series)

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

22 Ψ n ( n) = F Ψ( 0 ) = ck exp( inε k ) φ k c φ Ψ( 0) p k k = k * ( nt ) = ckck ' exp[ in( ε k ε k ' )] φk ' p k k, k ' At long time (i.e. after localization time), the interference terms will on the average cancel out: p c k φk p φk k φ Adiabatic case: Different state + random phase Intermediate case: Diabatic case: Same state + random phase