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Physical Review A

Causality and propagation in the Wigner, Husimi, Glauber, and Kirkwood phase-space representations

Bilha Segev

Published Apr 19, 2001• 8 citations
Publisher

Abstract

Time evolution is considered in phase space in terms of evolution kernels for various phase-space quasi-distributions. The propagators for the Wigner function, the standard-ordered function, the Kirkwood (antistandard-ordered) function, the Glauber P and Q functions, and the Husimi function are explicitly written as bilinear transforms of the evolution operator. Free propagation, propagation in dispersive media, and scattering, are studied, and manifestations of causality and interference are analyzed. It is shown that free propagation and scattering in the Husimi, Glauber, and Kirkwood representations with the underlying dynamics of the Schr\"odinger equation involve divergent evolution kernels connecting distant phase-space points at all times. The time evolution is much simpler in the Wigner representation where (i) free propagation is a simple classical translation involving no interference, and (ii) analytical properties of the scattering matrix restrict the velocities of propagation so that no information can travel due to scattering faster than free motion. As an example, a correlation is found between the coordinate and momentum of particles detected after they are released from a box. Propagators with relativistic dispersion relations of free photons or Klein-Gordon particles are briefly discussed in an Appendix.

DOI
10.1103/physreva.63.052114

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Time evolution is considered in phase space in terms of evolution kernels for various phase-space quasi-distributions. The propagators for the Wigner function, the standard-ordered function, the Kirkwood (antistandard-ordered) function, the Glauber P and Q functions, and the Husimi function are explicitly written as bilinear transforms of the evolution operator. Free propagation, propagation in dispersive media, and scattering, are studied, and manifestations of causality and interference are analyzed. It is shown that free propagation and scattering in the Husimi, Glauber, and Kirkwood representations with the underlying dynamics of the Schr"odinger equation involve divergent evolution kernels connecting distant phase-space points at all times. The time evolution is much simpler in the Wigner representation where (i) free propagation is a simple classical translation involving no interference, and (ii) analytical properties of the scattering matrix restrict the velocities of propagation so that no information can travel due to scattering faster than free motion. As an example, a correlation is found between the coordinate and momentum of particles detected after they are released from a box. Propagators with relativistic dispersion relations of free photons or Klein-Gordon particles are briefly discussed in an Appendix.
We present a time evolution equation that provides a novel basis for the treatment of quantum systems in phase space and for the investigation of the quantum-classical correspondence. Through the use of a generalized Husimi transform, we obtain a phase space representation of the time-dependent Schrödinger equation directly from the coordinate representation. Such an equation governs the time evolution of densities such as the Husimi density entirely in phase space, without recourse to a coordinate or momentum representation. As an application of the phase-space Schrödinger equation, we compute the eigenfunctions of the harmonic oscillator in phase space, relate these to the Husimi transform of coordinate representation eigenstates, and investigate the coherent state, its time evolution, and classical limit (ℏ→0) for the probability density generated by this state. Finally, we discuss our results as they relate to the quantum-classical correspondence, and quasiclassical trajectory simulations of quantum dynamics.
This thesis describes quantum mechanics in the phase space formulation. We introduce quantization and in particular the Weyl quantization. We study a general class of phase space distribution functions on phase space. The Wigner distribution function is one such distribution function. The Wigner distribution function in general attains negative values and thus can not be interpreted as a real probability density, as opposed to for example the Husimi distribution function. The Husimi distribution however does not yield the correct marginal distribution functions known from quantum mechanics. Properties of the Wigner and Husimi distribution function are studied to more extent. We calculate the Wigner and Husimi distribution function for the energy eigenstates of a particle trapped in a box. We then look at the semi classical limit for this example. The time evolution of Wigner functions are studied by making use of the Moyal bracket. The Moyal bracket and the Poisson bracket are compared in the classical limit. The phase space formulation of quantum mechanics has as advantage that classical concepts can be studied and compared to quantum mechanics. For certain quantum mechanical systems the time evolution of Wigner distribution functions becomes equivalent to the classical time evolution stated in the exact Egerov theorem. Another advantage of using Wigner functions is when one is interested in systems involving mixed states. A disadvantage of the phase space formulation is that for most problems it quickly loses its simplicity and becomes hard to calculate.
This tutorial gives an overview of the use of the Wigner function as a tool for modeling optical field propagation. Particular emphasis is placed on the spatial propagation of stationary fields, as well as on the propagation of pulses through dispersive media. In the first case, the Wigner function gives a representation of the field that is similar to a radiance or weight distribution for all the rays in the system, since its arguments are both position and direction. In cases in which the field is paraxial and where the system is described by a simple linear relation in the ray regime, the Wigner function is constant under propagation along rays. An equivalent property holds for optical pulse propagation in dispersive media under analogous assumptions. Several properties and applications of the Wigner function in these contexts are discussed, as is its connection with other common phase-space distributions like the ambiguity function, the spectrogram, and the Husimi, P, Q, and Kirkwood–Rihaczek functions. Also discussed are modifications to the definition of the Wigner function that allow extending the property of conservation along paths to a wider range of problems, including nonparaxial field propagation and pulse propagation within general transparent dispersive media.
We study the Wigner kernel and the Gabor matrix associated with the propagators of a broad class of linear evolution equations, including the complex heat, wave, and Hermite equations. Within the framework of time-frequency analysis, we derive explicit expressions for the Wigner kernels of Fourier multipliers and establish quantitative decay estimates for the corresponding Gabor matrices. These results are obtained under symbol regularity conditions formulated in the Gelfand-Shilov scale and ensure exponential off-diagonal decay or quasi-diagonality of the matrix representation. We believe this approach can be extended to more general symbols in the pseudodifferential setting, improving the existing results in terms of their Gabor matrix decay. For the complex heat equation, we obtain closed-form formulas exhibiting both dissipative and oscillatory behavior governed respectively by the real and imaginary parts of the diffusion parameter. The modulus of the Gabor matrix is shown to display Gaussian decay and temporal spreading consistent with diffusion phenomena. In contrast, the complex Hermite equation is analyzed via Hörmander's metaplectic semigroup, where the propagator decomposes as the product of a real Hermite semigroup and a fractional Fourier transform. In this setting, the Gabor matrix retains its Gaussian shape while undergoing a pure rotation on the time-frequency plane, reflecting the symplectic structure of the underlying flow. The analysis provides a unified operator-theoretic and phase-space perspective on parabolic and hyperbolic evolution equations, linking the geometry of their symbols with the sparsity and localization properties of their Gabor representations. Explicit formulas are given in a form suitable for numerical computation and visualization of phase-space dynamics.
We apply quasi-distribution methods developed for quantum mechanics to the propagation of pulses in dispersive media with attenuation. We show that a Schrödinger type equation follows for propagation of the pulse for each mode. One then transforms the equation to obtain an equation of evolution in the phase space of position and wavenumber. In this paper we emphasize windowed wave functions and their corresponding phase space quasi-distributions. We obtain the time evolution equation, discuss possible approximations, and compare to the Wigner distribution approximation previously derived by Loughlin and Cohen by different methods.