Localization of eigenstates and mean Wehrl entropy
Introduction
Analysis of quantum chaotic systems is often based on the statistical properties of the spectrum of the Hamiltonian H (in the case of autonomous systems) or the Floquet operator F (in the case of periodically perturbed systems). In general, quantized analogues of classically chaotic systems display spectral fluctuations conforming the predictions of random matrices. Depending on the geometrical properties of the system one uses orthogonal, unitary or symplectic ensemble [1], [2].
Another line of research deals with eigenstates of the analyzed quantum system. One is interested in their localization properties, which can be characterized by the eigenvector distribution [3], [4], [5], [6] the entropic localization length [7] or the inverse participation ratio [8]. All this quantities, however, are based on the expansion of an eigenstate on a given basis , which may be chosen arbitrarily. If one chooses (with a bad will), as the eigenbasis of F, all these quantities carry no information whatsoever. One may ask, therefore, to what extend the quantitative analysis based on the eigenvector statistics is reliable.
Let G denote a unitary operator, such that is its eigenbasis. We showed [9], [10], [11] that the eigenvector statistics of a quantum map F conforms the prediction of random matrices, if operators F and G are relatively random, i.e., their commutators are sufficiently large.
In this paper, we advocate an alternative method of solving the problems with arbitrariness of the choice of the expansion basis. Instead of working in a finite discrete basis, we shall use the coherent states expansion of the eigenstates of F. For several examples of compact classical phase spaces one may construct a canonical family of the generalized coherent states [12]. Localization properties of any pure quantum state may be characterized by the Wehrl entropy, equal to the average log of its overlap with a coherent state [13], [14]. We propose to describe the structure of a given Floquet operator F by the mean Wehrl entropy of its eigenstates. This quantity, explicitly defined without any arbitrariness, is shown to be a useful indicator of quantum chaos.
This paper is organized as follows. In Section 2, we review the definition of the Husimi distribution, stellar representation, and the Wehrl entropy. For concreteness, we work with SU(2) vector coherent states, linked to the algebra of the angular momentum operator and corresponding to the classical phase-space isomorphic with the sphere. In Section 3, we define the mean Wehrl entropy of eigenstates and present analytical results obtained for low-dimensional Hilbert spaces. Exemplary application of this quantity to the analysis of the quantum map describing the model of the periodically kicked top is provided in Section 4.
Section snippets
Husimi distribution and stellar representation
Consider a compact classical phase-space , a classical area preserving map and a corresponding quantum map F acting in an N-dimensional Hilbert space . A link between classical and quantum mechanics can be established via a family of generalized coherent states |α〉. For several examples of the classical phase spaces, there exist a canonical family of coherent states. It forms an overcomplete basis and allows for an identity resolution . Any mixed quantum state, described by
Mean Wehrl entropy of eigenstates of quantum map
Consider a quantum pure state in the N-dimensional Hilbert space. Its Wehrl entropy computed in the vector coherent states representation may vary from 1−1/N, for a coherent state, to the number of order of , for the typical delocalized state. This difference suggests a simple measure of localization of eigenstates of a quantum map F. Denoting its eigenstates by we define the mean Wehrl entropy of eigenstatesThis quantity may be straightforwardly computed
Mean Wehrl entropy for the kicked top
In order to demonstrate the usefulness of the mean Wehrl entropy in the analysis of quantum chaotic systems we present numerical results obtained for the periodically kicked top. This model is most suitable for the investigation of quantum chaos [34], [1]. Classical dynamics takes place on the sphere, while the quantum map is defined in terms of the components of the angular momentum operator J. The size of the Hilbert space is determined by the quantum number j and equals N=2j+1. One-step
Concluding remarks
The Wehrl entropy of a given state characterizes its localization in the classical phase space. We have shown that the mean Wehrl entropy of eigenstates of a given evolution operator F may serve as a useful indicator of quantum chaos. Let us emphasize that this quantity, linked to the classical phase space by a family of coherent states, does not depend on the choice of basis. This contrasts the others quantities, like eigenvector statistics, localization entropy, inverse participation
Acknowledgements
I am indebted to W. Słomczyński for fruitful discussions and a constant interest in the progress of this research. I am also thankful to M. Kuś and P. Pakoński for helpful remarks. It is a pleasure to thank Bernhard Mehlig for the invitation to Dresden and the Center for Complex Systems for a support during the workshop. Financial support from Polski Komitet Badań Naukowych in Warsaw under the grant no 2P-03B/00915 is gratefully acknowledged.
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