Experimental Evidence of Dynamical Localization and Delocalization in a Quasiperiodic Driven System

Abstract

This paper presents the first experimental evidence of the transition from dynamical localization to delocalization under the influence of a quasi-periodic driving on a quantum system. A quantum kicked rotator is realized by placing cold atoms in a pulsed, far-detuned, standing wave. If the standing wave is periodically pulsed, one observes the suppression of the classical chaotic diffusion, i.e. dynamical localization. If the standing wave is pulsed quasi-periodically with two different frequencies, dynamical localization is observed or not, depending on the two frequencies being commensurable or incommensurable. One can thus study the transition from the localized to the delocalized case as a function of the effective dimensionality of the system. Dynamical localization (DL) is a specifically quantum phenomenon taking place in time-periodic systems whose corresponding classical dynamics displays chaotic diffusion. While, in the classical limit, because of the diffusion process, the system spreads indefinitely in the phase space, the quantum system follows the classical diffusive dynamics for short time only, but after some localization time freezes its evolution with no further increase of the average energy.This behavior, attributed to quantum interferences among the diffusive paths which for long times are in the average completely destructive, was numerically observed at the end of the 70's on the one-dimensional kicked rotator exposed to periodic kicks [1], a paradigmatic simple system whose classical dynamics can be reduced to iterations of the Chirikov's standard map.The possibility of observing DL with a system constituted of cold atoms placed in a far-detuned standing wave has been theoretically suggested in 1992 [2] and experimentally observed in 1994 [3]. A crucial question is whether DL is robust versus perturbations of the system. Indeed, as it strongly relies on quantum interferences, it is expected to be rather fragile. As a matter of fact, it has been experimentally shown that DL can be partly or totally destroyed by decoherence (i.e. coupling of the system to external degrees of freedom; in the present context spontaneous emission plays such a role) and noise, that is deviation from strict periodicity [4,5].Moreover, there is a relevant connection of DL with the Anderson localization taking place in disordered systems. Indeed, the periodically kicked rotator problem can be mapped on a one-dimensional Anderson model, that is a model of a particle moving along a one-dimensional chain of sites [6], with coupling between neighbors and diagonal disorder, i.e. pseudo-random values of the potential energies on each site. In 1-d, although the classical motion is diffusive, the quantum eigenstates are all localized. Because of the similarity of the Anderson and kicked top models, Anderson localization and DL are similar quantum phenomena, the first one taking place along space coordinates, the second one along the time coordinate with localization in the momentum domain. It is well known that Anderson l...

Non-dispersive wave packets in periodically driven quantum systems

“…For simplicity, we choose a single cosine function to define the periodic driving. For a more complicated dependence on time [86], it is enough to expand it in a Fourier series, see section 3.5.…”

Kicked-rotor quantum resonances in position space

“…The experimental realization of the QKR has triggered in the last decade an impressive number of studies involving dynamical localization [5,6,7,8,9,10,11], quantum transport [12,13,14,15], ratchets [16,17,18], chaosassisted tunneling [19,20], classical and quantum resonances [21,22,23,24]. Quantum resonances have been used in studies of fundamental aspects of quantum chaos such as quantum stabilization [25] or measurements of the gravitation [26].…”

Maximum group velocity in a one-dimensional model with a sinusoidally varying staggered potential

“…The existence of an upper bound on the speed has been explained using the notion of the counter-propagation of quasiparticles ("holon" and "doublon") generated due to the quench. Moreover, a dynamical localization-to-delocalization transition has been observed in a quantum kicked rotator, realized by placing cold atoms in a pulsed, far-detuned, standing wave, by measuring the number of zero velocity atoms under the influence of a quasiperiodic driving 55 . In connection to our work, the dynamical localization we predict can be experimentally observed by realizing the hard core boson model in an optical lattice with a sinusoidally varying alternating on-site potential and measuring the current stroboscopically starting from an initial current carrying ground state (obtained by applying a synthetic gauge potential 56 to the one-dimensional optical lattice).…”