The intermediate level statistics in dynamically localized chaotic eigenstates

Abstract

We demonstrate that the energy or quasienergy level spacing distribution in dynamically localized chaotic eigenstates is excellently described by the Brody distribution, displaying the fractional power law level repulsion. This we show in two paradigmatic systems, namely for the fully chaotic eigenstates of the kicked rotator at K = 7, and for the chaotic eigenstates in the mixed-type billiard system (Robnik 1983), after separating the regular and chaotic eigenstates by means of the Poincaré Husimi function, at very high energies with great statistical significance (587654 eigenstates, starting at about 1.000.000 above the ground state). This separation confirms the Berry-Robnik picture, and is performed for the first time at such high energies.Introduction. -One of the main findings in quantum chaos [1-3] of stationary Schrödinger equation is the fact that the statistics of spectral fluctuations of the discrete quantal energy spectra around the smooth mean behaviour of classically chaotic systems obeys the Random Matrix Theory (RMT), in terms of the Gaussian ensembles of random matrices [1,2,4,5], in the sufficiently deep (or far) semiclassical limit. This finding is known as the Bohigas -Giannoni -Schmit (BGS) conjecture first published in [6], although some preliminary ideas were introduced in [7]. By "sufficiently deep (or far) semiclassical limit" we mean that some semiclassical condition is satisfied, namely that all relevant classical transport times, like the typical ergodic time, or diffusion time, are smaller than the so-called Heisenberg time, or break time, given by t H = 2π /∆E, where h = 2π is the Planck constant and ∆E is the mean energy level spacing, such that the mean energy level density is ρ(E) = 1/∆E. If the stated condition is satisfied, the quantum eigenstates as represented in the "quantum phase space" by the Wigner functions, or Husimi functions, are uniformly extended [3], and the spectral statistics is as in RMT, namely like for Gaussian Orthogonal Ensemble (GOE) or Gaussian Unitary Ensemble (GUE), depending on the antiunitary symmetries of the system [1-4, 8, 9]. Here we treat only the

Quantum localization of chaotic eigenstates and the level spacing distribution

“…His result showed that the parameter β, which was obtained using the Izrailev distribution, is functionally related to the localization measure defined as the information entropy of the eigenstates in the angular momentum representation. His results were recently confirmed and extended, with the much greater numerical accuracy and statistical significance [38,39].…”

Survey on the role of accelerator modes for anomalous diffusion: The case of the standard map

“…The main motivation for the present work was our recent research on the quantum kicked rotator [24,25], which is the quantized classical kicked rotator, described by the standard map. For the systematic understanding of the quantum kicked rotator and its semiclassical theory, a complete survey of the standard map is necessary.…”

“…In this paper we study the diffusion properties in area preserving maps, exemplified by the standard map of Chirikov. This work is actually motivated by our extensive study [24,25] of the quantum kicked rotator introduced by Casati et al [26], in which -at the semiclassical level -it is necessary to understand in detail the classical diffusion, in order to set up a theory of (exponential) quantum (or dynamical) localization. Our previous work was stimulated by the series of pioneering and classic papers by Izrailev [27,28,29].…”

Statistical properties of the localization measure in a finite-dimensional model of the quantum kicked rotator

“…Therefore, the problem of calculating the distribution of the localization length (or its inverse) in the semiclassical framework is open for the future work. Also, the derivation of the Brody distribution to explain the level spacing distribution of the energies [29,31,32] in time-independent systems, and of the quasienergies [30,38,39] in time-periodic systems, of chaotic eigenstates, is still open for the future.…”

“…The relevant papers dealing with the mixed type regime after the work [19] are [21][22][23][24][25][26][27][28] and the most recent advance was published in [29][30][31][32]. If the couplings between the regular eigenstates and chaotic eigenstates become important, due to the dynamical tunneling, we can use the ensembles of random matrices that capture these effects [33].…”