Spectral form factors and dynamical localization

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.

10 pages, 4 figures

References in corpus (12)

• Semiclassical Foundation of Universality in Quantum Chaos
• Periodic-Orbit Theory of Universality in Quantum Chaos
• Universal spectral form factor for chaotic dynamics
• Statistical properties of the localization measure of chaotic eigenstates and the spectral statistics in a mixed-type billiard
• Effects of stickiness in the classical and quantum ergodic lemon billiard
• Stickiness in generic low-dimensional Hamiltonian systems: A recurrence-time statistics approach
• Phenomenology of quantum eigenstates in mixed-type systems: lemon billiards with complex phase space structure
• Spectral form factor in the double-scaled SYK model
• The level repulsion exponent of localized chaotic eigenstates as a function of the classical transport time scales in the stadium billiard
• Structure, size, and statistical properties of chaotic components in a mixed-type Hamiltonian system
• The distribution of localization measures of chaotic eigenstates in the stadium billiard
• Aspects of diffusion in the stadium billiard