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Europhysics Letters (EPL)

Particle-Spin Coupling in a Chaotic System: Localization-Delocalization in the Husimi Distributions

Marcus A. M. de Aguiar, K. Furuya, Caio Lewenkopf, Maria Carolina Nemes

Published May 15, 1991• 39 citations
Publisher

Abstract

The wave functions of the Dicke Hamiltonian, describing a spin coupled to a bosonic mode, are studied via Husimi distributions. A classical analogue of this system is also obtained. For several energy ranges studied, the Husimi distribution of the wave functions show the scar of simple periodic orbits when projected into the boson phase space. Surprisingly, these same distributions, when projected into the spin phase space, are spread through large regions. An explanation of this fact is given in terms of semiclassical theory and border effects associated with nonsemiclassical behaviour.

DOI
10.1209/0295-5075/15/2/003

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There is no unique way to quantify the degree of delocalization of quantum states in unbounded continuous spaces. In this work, we explore a recently introduced localization measure that quantifies the portion of the classical phase space occupied by a quantum state. The measure is based on the α -moments of the Husimi function and is known as the Rényi occupation of order α . With this quantity and random pure states, we find a general expression to identify states that are maximally delocalized in phase space. Using this expression and the Dicke model, which is an interacting spin-boson model with an unbounded four-dimensional phase space, we show that the Rényi occupations with α>1 are highly effective at revealing quantum scars. Furthermore, by analyzing the high moments ( α>1 ) of the Husimi function, we are able to identify qualitatively and quantitatively the unstable periodic orbits that scar some of the eigenstates of the model.
Measuring the degree of localization of quantum states in phase space is essential for the description of the dynamics and equilibration of quantum systems, but this topic is far from being understood. There is no unique way to measure localization, and individual measures can reflect different aspects of the same quantum state. Here we present a general scheme to define localization in measure spaces, which is based on what we call Rényi occupations, from which any measure of localization can be derived. We apply this scheme to the four-dimensional unbounded phase space of the interacting spin-boson Dicke model. In particular, we make a detailed comparison of two localization measures based on the Husimi function in the regime where the model is chaotic, namely, one that projects the Husimi function over the finite phase space of the spin and another that uses the Husimi function defined over classical energy shells. We elucidate the origin of their differences, showing that in unbounded spaces the definition of maximal delocalization requires a bounded reference subspace, with different selections leading to contextual answers.
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We study the distribution of zeros of the Husimi functions for the spin-boson model, following an approach introduced by Leboeuf and Voros. The interest lies in the model's double feature of possessing both a classical integrable to chaotic transition and an unbounded four-dimensional phase space. The latter gives rise to several new questions regarding the Husimi zeros which are discussed and partially answered. Some significant results occur in spite of the fact that we treat the case of spin 1/2.
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