Statistical properties of the localization measure in a finite-dimensional model of the quantum kicked rotator

Abstract

We study the quantum kicked rotator in the classically fully chaotic regime K = 10 and for various values of the quantum parameter k using Izrailev's N -dimensional model for various N ≤ 3000, which in the limit N → ∞ tends to the exact quantized kicked rotator. By numerically calculating the eigenfunctions in the basis of the angular momentum we find that the localization length L for fixed parameter values has a certain distribution, in fact its inverse is Gaussian distributed, in analogy and in connection with the distribution of finite time Lyapunov exponents of Hamilton systems. However, unlike the case of the finite time Lyapunov exponents, this distribution is found to be independent of N , and thus survives the limit N = ∞. This is different from the tight-binding model of Anderson localization. The reason is that the finite bandwidth approximation of the underlying Hamilton dynamical system in the Shepelyansky picture (D.L. Shepelyansky, Phys. Rev. Lett. 56, 677 (1986)) does not apply rigorously. This observation explains the strong fluctuations in the scaling laws of the kicked rotator, such as e.g. the entropy localization measure as a function of the scaling parameter Λ = L/N , where L is the theoretical value of the localization length in the semiclassical approximation. These results call for a more refined theory of the localization length in the quantum kicked rotator and in similar Floquet systems, where we must predict not only the mean value of the inverse of the localization length L but also its (Gaussian) distribution, in particular the variance. In order to complete our studies we numerically analyze the related behavior of finite time Lyapunov exponents in the standard map and of the 2×2 transfer matrix formalism. This paper is extending our recent work (T. Manos and M. Robnik, Phys. Rev. E 87, 062905 (2013)).

A Brief Introduction to Stationary Quantum Chaos in Generic Systems

“…For details see reference [52]. Nevertheless, one should observe some scattering of points around the mean value, noted already by Izrailev [35] in the case of the quantum kicked rotator, which probably is due to the fact that the localization measure has a certain distribution rather than a sharp value, as has been observed recently in the kicked rotator by Manos and Robnik [71].…”

The distribution of localization measures of chaotic eigenstates in the stadium billiard

“…In both cases the scattering of points around the mean linear behaviour is significant, and it is related to the fact that the local- ization measure A of eigenstates has some distribution P (A), as observed and discussed in Ref. [39] for the quantum kicked rotator, and discussed for the stadium billiard in the previous section IV.…”

“…[12] it has been demonstrated that β is a unique function of A in the billiard with the mixed phase space [13,14], but is not linear. Finally, Manos and Robnik [39], have observed that the localization measure in the quantum kicked rotator has a nearly Gaussian distribution, and this was a motivation for the work in the present paper, where we study the distributions of A in the stadium billiard, systematically in almost all regions of interest (determined by the shape parameter ε and the energy), and also the dependence of its standard deviation on the control parameter α, while the dependence of < A > on α as an empirical rational function is already known from our previous paper [16].…”

Aspects of diffusion in the stadium billiard

“…It was mainly Izrailev who has studied the relation between the spectral fluctuation properties of the quasienergies (eigenphases) of the quantum kicked rotator and the localization properties [14][15][16]. This picture has been recently extended in [17][18][19][20] and is typical for chaotic time-periodic (Floquet) systems. Similar analysis in the case of the stadium as a time independent system is in progress [21].…”