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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Signatures of quantum chaos in Wigner and Husimi representations

S.B. Lee, Michael D. Feit

Published Jun 1, 1993• 9 citations
Publisher

Abstract

In this paper, we study the quantum manifestations of classical chaos in phase space using Wigner and Husimi distribution functions. We test the claim that Husimi represents the correspondence better than Wigner does. The results show the claim is valid. We also use a quantum dissipation scheme empirically for classically damped motions often characterized by strange attractors. We believe quantum resemblance to classical distributions can be regarded as signatures of quantum chaos in phase space.

DOI
10.1103/physreve.47.4552

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In this paper, we study the quantum manifestations of classical chaos in phase space using Wigner and Husimi distribution functions. We test the claim that Husimi represents the correspondence better than Wigner does. The results show the claim is valid. We also use a quantum dissipation scheme empirically for classically damped motions often characterized by strange attractors. We believe quantum resemblance to classical distributions can be regarded as signatures of quantum chaos in phase space.
We study the correspondence between quantum and classical dynamics with Wigner and Husimi distribution functions, respectively. The nature of quantum-mechanical regular and irregular behaviors is discussed with the use of Wigner and Husimi distribution functions. By numerical studies of the chaotic state, the Husimi distribution function is found to be a better representation than the Wigner distribution function, because coarse graining is usually involved implicitly in an observational process using the Wigner distribution function.
The correspondence between classical and quantum mechanics is an important subject for the better understandings of “quantum chaos”. In particular, it is very important to investigate the correspondence between distribution functions in classical mechanics and in phase space representation of quantum mechanics. This is the review of our recent progresses in the study of distribution functions in classical and quantum mechanics, namely distribution functions in classical mechanics and in coarse-grained classical mechanics as well as the Wigner function and the Husimi function. Topics dealt with include formulations of the Wigner representation, the Husimi representation and coarse-grained classical mechanics, and their applications to the analyses of the eigenstates and time developments of the distribution functions.
In classical mechanics, driven systems with dissipation often exhibit complex, fractal dynamics known as strange attractors. This paper addresses the fundamental question of how such structures manifest in the quantum realm. We investigate the quantum Duffing oscillator, a paradigmatic chaotic system, using the Caldirola-Kanai framework, where dissipation is integrated directly into a time-dependent Hamiltonian. By employing the Husimi distribution to represent the quantum state in phase space, we present the first visualization of a quantum strange attractor within this model. Our simulations demonstrate how an initially simple Gaussian wave packet is stretched, folded, and sculpted by the interplay of chaotic dynamics and energy loss, causing it to localize onto a structure that beautifully mirrors the classical attractor. This quantum "photograph" is inherently smoothed, blurring the infinitely fine fractal details of its classical counterpart as a direct consequence of the uncertainty principle. We supplement this analysis by examining the out-of-time-ordered correlator, which shows that stronger dissipation clarifies the exponential growth associated with the classical Lyapunov exponent, thereby confirming the model's semiclassical behavior. This work offers a compelling geometric perspective on open chaotic quantum systems and sheds new light on the quantum-classical transition.
The quantum dynamics in Husimi representation is studied for the commutation relation, time development of distribution function and eigen distribution function. In the integrable system, the Husimi eigen distribution function is confined almost on the classical torus and can be approximated by Gaussian distribution with the variance (Δξl)2∼ℏ/2(\Delta\xi_{l})^{2}{\sim}\hbar/2(Δξl​)2∼ℏ/2 around the torus. Further, considering the correspondence between quantum and classical mechanics, the Husimi distribution function is a better representation than the Wigner distribution function, because coarse graining is usually involved in observational process.
Abstract This paper considers a Husimi representation of quantum mechanics in which the (stationary) state of a system or ensemble is described by a Husimi function and an observable is described by a phase space function or distribution such that the expectation value of the observable is given by an integral over phase space of the product of that function or distribution and the Husimi function. The density matrix, Wigner function, and Husimi function are considered to be alternative ways of describing the state of a system or ensemble, and methods of recovering the Wigner function or density matrix from the Husimi function are discussed. The classical limits of the Wigner and Husimi functions and of the relationship between them are considered. © 1993 John Wiley & Sons, Inc.
We investigate the quantum-classical correspondence in open quantum many-body systems using the SU(3) Bose-Hubbard trimer as a minimal model. Combining exact diagonalization with semiclassical Langevin dynamics, we establish a direct connection between classical trajectories characterized by fixed-point attractors, limit cycles, or chaos and the spectral and structural properties of the quantum steady state. We show that classical dynamical behavior, as quantified by the sign of the Lyapunov exponent, governs the level statistics of the steady-state density matrix: non-positive exponents associated with regular dynamics yield Poissonian statistics, while positive exponents arising from chaotic dynamics lead to Wigner-Dyson statistics. Strong symmetries constrain the system to lower-dimensional manifolds, suppressing chaos and enforcing localization, while weak symmetries preserve the global structure of the phase space and allow chaotic behavior to persist. To characterize phase-space localization, we introduce the phase-space inverse participation ratio IPR, which defines an effective dimension D of the Husimi distribution's support. We find that the entropy scales as S∝ln⁡NDS \propto \ln N^DS∝lnND, consistently capturing the classical nature of the underlying dynamics. This semiclassical framework, based on stochastic mixtures of coherent states, successfully reproduces not only observable averages but also finer features such as spectral correlations and localization properties. Our results demonstrate that dissipative quantum chaos is imprinted in the steady-state density matrix, much like in closed systems, and that the interplay between dynamical regimes and symmetry constraints can be systematically probed using spectral and phase-space diagnostics. These tools offer a robust foundation for studying ergodicity, localization, and non-equilibrium phases of open quantum systems.