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Physical Review A

Zeros of the Husimi functions of the spin-boson model

M. B. Cibils, Y. Cuche, P. Lebœuf, Walter Felipe Wreszinski

Published Oct 1, 1992• 34 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.46.4560

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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.
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.
The Husimi distribution is proposed for a phase-space analysis of quantum phase transitions in the Dicke model of spin-boson interactions. We show that the inverse participation ratio and Wehrl entropy of the Husimi distribution give sharp signatures of the quantum phase transition. The analysis is done using two frameworks: a numerical treatment and an analytical variational approximation. Additionally, we propose a characterization of the Dicke model quantum phase transition by means of the zeros of the Husimi distribution in the variational approach.
The quantisation of the two-dimensional toric and spherical phase spaces is considered in analytic coherent state representations. Every pure quantum state admits therein a finite multiplicative parametrisation by the zeros of its Husimi function. For eigenstates of quantised systems, this description explicitly reflects the nature of the underlying classical dynamics: in the semiclassical regime, the distribution of the zeros in the phase space becomes one-dimensional for integrable systems, and highly spread out (conceivably uniform) for chaotic systems. This multiplicative representation thereby acquires a special relevance for semiclassical analysis in chaotic systems.
We identify a transition to chaos in the semiclassical spin-boson model that occurs for relatively large boson fields, as one of two periodic orbits becomes unstable. We have studied the quantum dynamics in the vicinity of this transition, as characterized by (i) phase-space trajectories followed by quantum expectation values, (ii) spectra of such trajectories, (iii) subsystem entropies for the spin and boson systems, and (iv) growth of operator variances for the boson system. We find that the transition to chaos in the classical system has no apparent effect on any of these variables, in the spin-1/2 case. This is in disagreement with some claims made in earlier studies of this system. \textcopyright{} 1996 The American Physical Society.