Presentation Quantum chaos with cold atoms in a standing wave Laboratoire de Physique des Lasers Atomes et Molécules (PhLAM) Villeneuve dAscq, France Pascal Szriftgiser Jean Ringot Hans Lignier J. C. G. In collaboration with Dominique Delande LKB - Paris

System The system Cold atoms in a standing wave d L /2 V0V0 dB H = P 2 /2 + K cos ( ) n (t-n) Theory: Graham, Schlautman, Zoller PRA 45, 19 (1992) Exp: Moore et al., PRL 73, 2974 (1994); Klappauf et al. PRL 81, 1203 (1998)

Quantum chaos Classical phase portrait K = 0.9 K = 2 K = 5 K = 10 Regular dynamics: Periodic orbits Mixt dynamics: Stability islands Mixt: Small islands Chaotic: Ergodic diffusion

Quantum chaos Momentum evolution P(p) ~ e -|p|/p L Log P p

Quantum chaos Dynamical localization t Quantum Classical P(p)P(p) P(p)P(p) P(p)P(p) Theory: Casati, Chirikov, Ford, Izrailev (1979) Experience : Moore, Robinson, Bharucha, Williams, Raizen (1994) t L ~ t H = h/ Heisenberg time

System Momentum distribution measurement Stimulated Raman transitions R 1 2 hf Resonant probe F = 3 F = 4 v 0 = R /2k L v r /10

Quantum chaos Experiment

Dynamical localization Momentum distributions f (kHz) Final distribution (50 kicks) p/hk Initial distribution Exponencial fit Gaussian fit

Dynamical localization Quasi-periodic kicks f1f1 f2f2 f 1 /19 0

Dynamical localization Dynamical localization with two colors For « irrational » values of the frequency ratio, the classical diffusive behavior is preserved Initial distribution Freq. ratio = Freq. ratio = Initial distribution Frequency ratio = 1,0833… Frequency ratio = 1 Number of atoms (log) Momentum (hk)

Dynamical localization Localization measurement Initial Localized Delocalized Consevation of the atom number The population P(0) of the 0 velocity class is a mesurement of the degree of localization

Dynamical localization 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 Breaking the periodicity destroys localization J. Ringot, P. Szriftgiser, J.C.G., D. Delande, PRL 85, 2341 (2000)

Sub-Fourier Sub-Fourier lines Atomic signal Frequency ratio r F 12 Experimental F 12 The lines ARE NOT FTs of a temporal signal! « Sub-Fourier » resolution: f 1 f 2 for T mes < 1/|f 1 -f 2 | P. Szriftgiser, J. Ringot, D. Delande, J.C.G., submited to PRL Resolution ~ 1/T exp Exp) F

Sub-Fourier = /14 = /37 Quantum interference sensitivity ot frequency and phase Interpretation

Sub-Fourier Interpretation Kicks Width number of kicks K [1+ A cos(2 rt)] n (t-n) r

Sub-Fourier Physical mechanism = P2P2 N r -1 r e -i N/rN/rN N ~ 0 periodicity N ~ /2 quasi-periodicity Résolution is due to the dynamical spectrum, not to the excitation spectrum

Sub-Fourier Evolution of the width N r N Fourier limit K = 14 K = 28 K = 42 1µs1µs 2µs2µs 3µs3µs 1/N 2 = D N N r -1 r ~ N 2 Before localization = PL2PL2 N r -1 r ~ N After localization

Conclusion Complex dynamics – unexpected results Simplicity and versatility Detailed experimental study of the linewidth Interpretation – physical mechanisms New conditions: anomalous diffusion r ~1/N 3 Quatum-chaotic signal processing ultrafast frequency locking (?)

Funding CNRS Ministère de la Recherche Région Nord-Pas de Calais Comunidade Européia

Sub-Fourier Resolution (quantitative) f f1f1 f2f2 F 12 r = 0.87 F 12 r F 12 /2 F 12 measures the Fourier resolution

Sub-Fourier Excitation spectrum TF T 1/T TF 1/ 1/T

DC Bias4.6 GHz Generation of the Raman beams Master S +1 S -1 FP Raman setup Experimental setup

Sub-Fourier Sub-Fourier resonances Atomic signal Frequency ratio r F 12 Experimental F 12 Experimental) F P. Szriftgiser, J. Ringot, D. Delande, J.C.G., submetido a Nature

Sub-Fourier Sub-Fourier resonances Atomic signal Frequency ratio r F 12 Experimental F 12 Experimental) F P. Szriftgiser, J. Ringot, D. Delande, J.C.G., submetido a Nature

Dynamical localization Kinetic energy Kicks tLtL

System Velocity distribution for the sisyphus molasses 72 kHz T = 3.3 µK Raman detuning

Active magnetic field compensation ON+ offset OFF+offset 1 kHz ~ 300 µG ~ v R /8 ON+offset (kHz) BW ~ 500 Hz Magnetic field compensation Experimental setup Bx By Bz