Spectral Form Factor and Dynamical Localization

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, which are known to be a good measure of dynamical localization, depend linearly on the transport times obtained in this way.

Automated summary

Quantum dynamical localization refers to the cessation of wave packet diffusion in momentum space due to quantum interference. It is expected to occur when the typical transport time for momentum diffusion exceeds the Heisenberg time. This paper presents an alternative approach that investigates the transport time through the study of spectral fluctuations of the quantum system, specifically the spectral form factor. The results show a linear relationship between the level repulsion exponents, a measure of dynamical localization, and the transport times obtained from the spectral form factors.