Quantum localization of chaotic eigenstates and the level spacing distribution

Abstract

The phenomenon of quantum localization in classically chaotic eigenstates is one of the main issues in quantum chaos (or wave chaos), and thus plays an important role in general quantum mechanics or even in general wave mechanics. In this work we propose two different localization measures characterizing the degree of quantum localization, and study their relation to another fundamental aspect of quantum chaos, namely the (energy) spectral statistics. Our approach and method is quite general, and we apply it to billiard systems. One of the signatures of the localization of chaotic eigenstates is a fractional power-law repulsion between the nearest energy levels in the sense that the probability density to find successive levels on a distance $S$ goes like $\propto S^{\beta }$ for small $S$, where $0\le \beta \le 1$, and $\beta =1$ corresponds to completely extended states. We show that there is a clear functional relation between the exponent β and the two different localization measures. One is based on the information entropy and the other one on the correlation properties of the Husimi functions. We show that the two definitions are surprisingly linearly equivalent. The approach is applied in the case of a mixed-type billiard system (Robnik 1983), in which the separation of regular and chaotic eigenstates is performed.

7 pages, 5 figures

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• Statistics of phase space localization measures and quantum chaos in the kicked top model
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• Exponential Fermi acceleration in general time-dependent billiards
• Multifractality in quasienergy space of coherent states as a signature of quantum chaos
• Quantum localization measures in phase space
• Phenomenology of quantum eigenstates in mixed-type systems: lemon billiards with complex phase space structure
• Structure, size, and statistical properties of chaotic components in a mixed-type Hamiltonian system
• Identification of quantum scars via phase-space localization measures
• Aspects of diffusion in the stadium billiard
• Statistical properties of the localization measure in a finite-dimensional model of the quantum kicked rotator
• Classical and quantum mixed-type lemon billiards without stickiness
• Power-law decay of the fraction of the mixed eigenstates in kicked top model with mixed-type classical phase space
• Further results on the power-law decay of the fraction of the mixed eigenstates in kicked-top model with mixed-type classical phase space
• Spectral form factors and dynamical localization
• Classical and Quantum Elliptical Billiards: Mixed Phase Space and Short Correlations in Singlets and Doublets
• Entropic comparison of Landau-Zener and Demkov interactions in the phase space of a quadrupole billiard
• Phase-space localization at the lowest Landau level
• Loss of integrability in a system with two-photon interactions