Spectral form factors and dynamical localization

TL;DR

This paper introduces a spectral analysis method to determine quantum transport times and dynamical localization in stadium billiards, linking spectral form factors with localization measures.

Contribution

It presents a novel spectral fluctuation approach to estimate transport times, connecting spectral form factors with dynamical localization in quantum billiards.

Findings

Transport times correlate linearly with level repulsion exponents.

Spectral form factors reveal transition from non-universal to universal regimes.

Method applied to 22 stadium billiards across localization transition.

Abstract

Quantum dynamical localization occurs when quantum interference stops the diffusion of wave packets in momentum space. The expectation is that dynamical localization will occur when the typical transport time of the momentum diffusion is greater than the Heisenberg time. The transport time is typically computed from the corresponding classical dynamics. In this paper, we present an alternative approach based purely on the study of spectral fluctuations of the quantum system. The information about the transport times is encoded in the spectral form factor, which is the Fourier transform of the two-point spectral autocorrelation function. We compute large samples of the energy spectra (of the order of $10^6$ levels) and spectral form factors of 22 stadium billiards with parameter values across the transition between the localized and extended eigenstate regimes. The transport time is obtained from the point when the spectral form factor transitions from the non-universal to the universal regime predicted by random matrix theory. We study the dependence of the transport time on the parameter value and show the level repulsion exponents, that are known to be a good measure of dynamical localization, depend linearly on the transport times obtained in this way.