Abstract

We study the correlations in the quasienergy spectra of systems with dynamical localization, using the quantum kicked rotor (QKR) as a paradigm. Two complementary approaches are taken: We first study the “local spectral density”. For level separations below the mean distance, its two-point correlations are dominated by level attraction rather than level repulsion, a feature known from the electron spectra in disordered solids. We then turn to the unbiased spectra for the QKR on a finite-dimensional Hilbert space (“finite-sample approach”). They are characterized by a novel universal statistics which depends on the ratio γ of the localization length to the basis size. We derive semiclassical expressions for the two-point correlations which interpolate between COE behaviour for γ → ∞ and Poissonian (lack of correlations) for γ → 0. We show how the diffusive nature of the classical dynamics finds its expression in the quantal spectral correlations.

Introduction

One of the most impressive results in “quantum chaology” was that as soon as the corresponding classical dynamics becomes chaotic, the spectral fluctuations, in the limit ħ → 0, obey universal distribution laws, as predicted by random-matrix theory [1–3]. Recently, this quantum–classical correspondence has been extended to the domain of chaotic scattering, where the fluctuations in the S-matrix eigenphases were shown to follow the statistics of Dyson's circular orthogonal ensemble (COE) [4]. The periodic-orbit theory was the main tool for the semiclassical investigation of spectra and their relation to random-matrix theory.