Spectral properties of systems with dynamical localization. I. The local spectrum

Abstract

The authors study the correlations in the quasi-energy (QE) spectra of systems with dynamical localization, using the quantum kicked rotor (QKR) as a paradigm. The specific spatial structure of the QE eigenstates is taken into account by investigating the local spectrum, which gives each eigenstate an individual weight according to its overlap with some reference state. Two-point correlations in the local spectrum are related by Fourier transform to the time evolution of the probability to stay at the initial state. They devise a scaling theory for this dynamical quantity in the case of the QKR, containing the participation ratio as a single parameter. It implies that the local spectrum is characterized by positive correlations, in contrast to the unbiased spectra in classically chaotic systems with a bounded phase space. This is consistent with recent results on spectral properties of systems with Anderson localization. A scheme for experimental measurements of spectral two-point correlation functions is proposed.