Quantum scaling laws in the onset of dynamical delocalization

Abstract

We study the destruction of dynamical localization, experimentally observed in an atomic realization of the kicked rotor, by a deterministic Hamiltonian perturbation, with a temporal periodicity incommensurate with the principal driving. We show that the destruction is gradual, with well defined scaling laws for the various classical and quantum parameters, in sharp contrast with predictions based on the analogy with Anderson localization.PACS numbers: 05.45. Mt, 32.80.Lg, 03.65.Yz, 05.70.Fh Quantum chaos is defined as the dynamical behavior of a quantum system whose classical limit is chaotic. This has triggered a large number of studies trying to relate classical properties to quantum properties, e.g. Lyapunov exponents to quantum fidelity [1, 2], or to detect quantum stability in a quantum-chaotic system [3].Quantum-chaotic dynamics manifest themselves by characteristic behaviors in which quantum interference plays an important role, making the dynamics distinct from classical dynamics. An example, that shall concern us particularly here, is dynamical localization (DL) [4,5], observed in time-periodic systems. DL is the suppression of the classical chaotic diffusion by quantum interference due to long-range coherence in momentum space; it manifests itself after a typical "localization time" as an exponential localization of the average momentum distribution. Because the system is time-periodic, one can use the Floquet theorem to build a basis of quasi-eigenstates (states that are left unchanged, except for a phase factor, under the temporal evolution over one period). This makes it possible to map the quasieigenstates of the time-periodic system on the true eigenstates of a quasi-random static one-dimensional system, which presents the non-trivial Anderson localization. Anderson (or strong) localization has been a major subject in physics in the last decades, with implications in several areas, beyond the primary field of solid state physics [6,7,8,9]. In this paper, we show that studying the breakdown of dynamical localization may also bring some new insight on the physics of Anderson localization. The latter is known to be strongly dependent on the number of freedoms, with marginal localization in dimension 2 and the coexistence of localized and delocalized statesdepending on the parameters -in dimension 3. By playing with the temporal dependance of the Hamiltonian -for example by adding incommensurate frequencies to make a quasi-periodic Hamitonian -it is possible to study temporal equivalents of the Anderson model in various dimensions. What happens if we introduce progressively a second (incommensurate) frequency in the system, by increasing its strength from zero? As the system is quasiperiodic with two basic frequencies, it is reasonnable to assume that it can be mapped onto a two-dimensional Anderson model [10,11], which, for a small perturbation, is a quasi-1D model, and one could then expect localization to be preserved, at the cost of an increased localization length. Theoretical studies based o...

Classical diffusive dynamics for the quasiperiodic kicked rotor

“…This phenomenon has been shown to be equivalent to the Anderson localisation in 1D (5)(6)(7). An impressive surge of theoretical and experimental work on the subject followed (8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20).…”

Decoherence effects in the dynamics of interacting ultracold bosons in disordered lattices

“…The 3D localisation was also studied with both fermions [4] and bosons [5]. The quasiperiodic kicked rotor can also realize an equivalent of both 1D [6,7,8] and 3D systems [9,10]; the latter allowing a detailed experimental study of the Anderson transition, including measurements of the critical wavefunction [11] and of its critical exponent [10,12,13]. Ultracold atom systems allow a large control of decoherence, which is essentially due to spontaneous emission, and of atom-atom interactions, for which an experimental "knob" is provided by the Feshbach resonances [14], making them an ideal tool for studies of the interplay of quantum effects, disorder, interactions, and decoherence.…”