Dynamical localization in kicked rotator as a paradigm of other systems: spectral statistics and the localization measure

TL;DR

This paper investigates the spectral statistics and eigenfunction localization in the quantum kicked rotator, demonstrating the Brody distribution's effectiveness and revealing a scaling law between spectral repulsion and localization.

Contribution

It provides new insights into the spectral statistics of the kicked rotator, showing Brody distribution's superiority and establishing a first-order scaling law relating spectral and eigenfunction properties.

Findings

Spectral statistics are well described by the Brody distribution.

A scaling law exists between spectral repulsion and localization length.

Eigenfunction localization exhibits a first-order scaling relationship.

Abstract

We study the intermediate statistics of the spectrum of quasi-energies and of the eigenfunctions in the kicked rotator, in the case when the corresponding system is fully chaotic while quantally localized. As for the eigenphases, we find clear evidence that the spectral statistics is well described by the Brody distribution, notably better than by the Izrailev's one, which has been proposed and used broadly to describe such cases. We also studied the eigenfunctions of the Floquet operator and their localization. We show the existence of a scaling law between the repulsion parameter with relative localization length, but only as a first order approximation, since another parameter plays a role. We believe and have evidence that a similar analysis applies in time-independent Hamilton systems.