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Physical Review A

Wigner distribution function of a simple optical system: An extended-phase-space approach

Sumiyoshi Abe, Norikazu Suzuki

Published Jan 1, 1992• 13 citations
Publisher

Abstract

We construct the Wigner distribution function for a coupled boson-fermion system. This function is regarded as a certain ``superfield'' defined on the extended phase space, the points of which are labeled by the anticommuting canonical c variables together with the ordinary ones. In this approach, the variables of a fermion and a boson are treated on a completely equal footing, and the phase-space representation is fully realized. We apply the formalism to the kinetic theory of the optical Dicke model with a two-level atom, and show how systematically a set of the generalized Fokker-Planck equations, which describes the effective dynamics of the radiation field, is derived from a single superfieldlike equation.

DOI
10.1103/physreva.45.520

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We construct the Wigner distribution function for a coupled boson-fermion system. This function is regarded as a certain ``superfield'' defined on the extended phase space, the points of which are labeled by the anticommuting canonical c variables together with the ordinary ones. In this approach, the variables of a fermion and a boson are treated on a completely equal footing, and the phase-space representation is fully realized. We apply the formalism to the kinetic theory of the optical Dicke model with a two-level atom, and show how systematically a set of the generalized Fokker-Planck equations, which describes the effective dynamics of the radiation field, is derived from a single superfieldlike equation.
The Wigner function, which provides a phase-space description of quantum systems, has various applications in quantum mechanics, quantum kinetic theory, quantum optics, radiation transport and others. The concept of the Wigner function has been extended to quantum fields, scalar and electromagnetic. Then, one deals with the Wigner functional which gives a distribution of field and its conjugate momentum. We introduce here the Wigner functional of fermionic fields of the values in a Grassmann algebra. Properties of the functional are discussed and its equation of motion, which is of the Liouville form, is derived.
In this paper the equations for quantum optical pulse propagation in nonlinear and dispersive single-mode fibers are presented in terms of two phase-space formulations based on the positive-P and the Wigner distributions. Included are the effects due to the coupling of the electromagnetic modes to the vibrational states of a vitreous silica fiber. By making use of the well-known equivalence of Fokker-Planck and Ito stochastic equations, we demonstrate two alternative methods for formulating the equations of motion as coupled stochastic c-number equations for the propagating field. The first method involves a representation of the density operator in terms of the positive P distribution function. This leads to exact stochastic equations of motion. The second method makes use of the Wigner distribution function. This method, which requires truncation of third-order derivative terms in the corresponding Fokker-Planck equation, is necessarily approximate. However, we discuss certain advantages to the Wigner approach that have made it the preferable method for exploratory work.
We show that the most complete and accurate description of fluctuations in quantum-optical systems is that obtained using the generalized Wigner distribution for the macroscopic observables of the system. The smallness of the inverse of the saturation photon number entitles us to neglect the terms with derivatives of order higher than second order in the time-evolution equation for the quasiprobability distribution. This treatment allows us to also describe correctly nonclassical effects such as photon antibunching or "squeezing," which are maltreated or even destroyed if one neglects the atom-atom correlations. We derive and analyze the Fokker-Planck equation for the Wigner function in the case of the usual laser and compare the results with those following from other approaches. In the region very high above threshold we find a small antibunching effect, which was not discovered in previous treatments due to the neglect of atom-atom correlations.
Th author introduces an exact and simple technique to convert the atomic part of the master equation for the density operator of a quantum optical system into a partial differential equation for a quasiprobability distribution in an arbitrary representation. He focusses on the special cases of two-level atoms and normal and symmetrical orderings, corresponding respectively to the P-function and Wigner representations. As an example of his results he derives the equation for the Wigner function of single-mode optical bistability with two-level atoms, and shows how he recovers the well known Fokker-Planck approximation.
Abstract This chapter provides an overview of the phase space method for treating the physics of systems of bosons and fermions in regimes where quantum degeneracy occurs. Key concepts such as modes, creation and annihilation operators, distribution functions in phase space depending on c-number (bosons) or Grassmann (fermions) phase space variables, quantum correlation functions, and phase space integrals are introduced, together with how the dynamics of the system density operator as described via Liouville–von Neumann or master equations is translated into Fokker–Planck equations for the distribution function, and then how phase space integrals for quantum correlation functions can be determined via Langevin stochastic equations that are equivalent to the Fokker–Planck equations, and which involve stochastic phase space variables. The extension of the phase space approach to systems with large mode numbers, involving field annihilation and creation operators, distribution functionals, and functional Fokker–Planck and Langevin stochastic field equations is also reviewed.
This is the first in a series of papers dealing with quantum or Wigner probability distributions and the dynamics of early-Universe phase transitions. In this paper we argue that the inflaton field (a real gauge-singlet scalar field) when spatially averaged over a causal horizon behaves as if it were a dissipative quantum-mechanical system in one dimension. Realizing that the mathematical phase-space or stochastic description of Wigner is the only consistent way to describe these systems and still preserve the canonical commutation relations we devote this paper to investigating various properties of the Wigner formalism. This framework rests on the quasiprobability distribution and its evolution equation: a generalized Fokker-Planck equation. We first show that this generalized Fokker-Planck equation is equivalent to Schr\"odinger's equation for nondissipative pure states. We then show how one computes ground-state energies within this formalism (for nondissipative systems) by calculating ground-state energy levels for a variety of anharmonic potentials. Finally we compute the quantum-mechanical effective potential in one dimension to order g\ensuremath{\Elzxh}.