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Quantum and Semiclassical Optics Journal of the European Optical Society Part B

Connection between approximate Fokker - Planck equations and the Wigner function applied to the micromaser

M. Elk, P. Lambropoulos

Published Feb 1, 1996• 8 citations
Publisher

Abstract

We present a derivation of a Fokker - Planck equation for the micromaser based on the Wigner function and valid in the semiclassical limit. The resulting equation is shown to correspond to the conventionally used approximate Fokker - Planck equations obtained for a continuous approximation to the discrete Fock representation of the density matrix of the light field. We thus establish a connection between the continuous approximation and the Wigner function. Then expressing expectation values as properly symmetrized products, we can extend in an exact way the validity of the Fokker - Planck approach to the micromaser down to the zero pump off-state of few photons and purely thermal statistics.

DOI
10.1088/1355-5111/8/1/004

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We present a derivation of a Fokker - Planck equation for the micromaser based on the Wigner function and valid in the semiclassical limit. The resulting equation is shown to correspond to the conventionally used approximate Fokker - Planck equations obtained for a continuous approximation to the discrete Fock representation of the density matrix of the light field. We thus establish a connection between the continuous approximation and the Wigner function. Then expressing expectation values as properly symmetrized products, we can extend in an exact way the validity of the Fokker - Planck approach to the micromaser down to the zero pump off-state of few photons and purely thermal statistics.
Starting from the master equation for density operator of the atoms and the field mode and applying the operator method of adiabatic elimination of the atomio variables developed by Haake and Lewenstein, we derived the Fokker-Planck Equation for Wigner function of the lightfield and its steady-state solution. With a Gaussian approximation to the solution, analytical and numerical results on the photon statistics for various pumping strengths are obtained. Comparisons with previous works are made.
This article has to do with the derivation and solution of the Fokker-Planck equation associated to the momentum-integrated Wigner function of a particle subjected to a harmonic external field in contact with an ohmic thermal bath of quantum harmonic oscillators. The strategy employed is a simplified version of the phenomenological approach of Schramm, Jung, and Grabert of interpreting the operators as c numbers to derive the quantum master equation arising from a twofold transformation of the Wigner function of the entire phase space. The statistical properties of the random noise comes from the integral functional theory of Grabert, Schramm, and Ingold. By means of a single Wigner transformation, a simpler equation than that mentioned before is found. The Wigner function reproduces the known results of the classical limit. This allowed us to rewrite the underdamped classical Langevin equation as a first-order stochastic differential equation with time-dependent drift and diffusion terms.
We derive simple formulas connecting the generalized Wigner functions for sss-ordering with the density matrix, and vice-versa. These formulas proved very useful for quantum mechanical applications, as, for example, for connecting master equations with Fokker-Planck equations, or for evaluating the quantum state from Monte Carlo simulations of Fokker-Planck equations, and finally for studying positivity of the generalized Wigner functions in the complex plane.
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.
The quantum kinetic equation describing slow atomic motion in laser light is derived by an operational method which provides mathematical expressions with a transparent physical structure. The authors prove in a general way that the coefficients appearing in this equation, which is of a Fokker-Planck type, are simply related to the mean value and to the correlation functions of the Heisenberg radiative force of the semiclassical approach, where the atomic position is treated classically. They derive in particular a new theoretical expression for the damping force responsible for radiative cooling and they interpret it in terms of linear response theory. The authors also obtain a new crossed r-p derivative term, which does not appear in semiclassical treatments, but which they find to be very small in most situations. Finally, all the theoretical expressions derived are valid for any J g to J e transitions and are not restricted to two-level atoms.