Classical diffusive dynamics for the quasiperiodic kicked rotor

Abstract

We study the classical dynamics of a quasiperiodic kicked rotor, whose quantum counterpart is known to be an equivalent of the 3D Anderson model. Using this correspondence allowed for a recent experimental observation of the Anderson transition with atomic matter waves. In such a context, it is particularly important to assert the chaotic character of the classical dynamics of this system. We show here that it is a 3D anisotropic diffusion. Our simple analytical predictions for the associated diffusion tensor are found in good agreement with the results of numerical simulations.

Phase diagram of the anisotropic Anderson transition with the atomic kicked rotor: theory and experiment

“…Note that the linear dispersion along directions 2 and 3 implies that V (p) is indeed quasi-periodic along the transverse directions. However, due to the pseudo-randomness of the phases in the longitudinal direction[37], the classical dynamics is still fully diffusive in all directions (for all the parameters considered in this study) and indeed hardly distinguishable from that of the corresponding 3D kicked rotor associated with pseudo-random phases along the three directions[12,26,38].New Journal of Physics 15 (2013) 065013 (http://www.njp.org/)…”

Experimental Observation of Two-Dimensional Anderson Localization with the Atomic Kicked Rotor

“…where y n = y(t = n + ), y = x i , p i (i = 1, 2). If K is sufficiently large, the classical dynamics is almost fully chaotic [29]. For the 1D problem (ε = 0), this takes place for K 6.…”

“…by assuming that positions of consecutive kicks are uniform uncorrelated variables (see [29] for the essentially identical calculation in 3D). It is diagonal in the (1,2) directions with:…”

“…ii) The diffusion process for the 2D quasiperiodic kicked rotor is not isotropic. As shown in [29] and discussed in the Supplemental Material [14], the quasiperiodic kicked rotor can be mapped on a 2D Anderson-like model, whose dynamics at short time is indeed diffusive, but anisotropic. Along the "physical" direction (which coincides with the atom momentum component along the standing wave), the diffusion constant is -for small ε -almost equal to the one of the periodic kicked rotor, D 11 ≈ K 2 /4; along the other (virtual) direction, the diffusion constant is D 22 ≈ K 2 ε 2 /8, so that it vanishes in the limit ε → 0, where one must recover the usual 1D periodic kicked rotor.…”

“…A second, more fundamental, phenomenon is that Eq. ( 3) assumes that the classical diffusion constant is simply K 2 /4, which is valid only for K 1, whereas oscillatory corrections at moderate K are known to exist for the 1D kicked rotor [34] and to persist even for the 3D QPKR [29]. This dependence is thus not eliminated by the normalization to E kin (ε = 0).…”