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Quantum Optics Journal of the European Optical Society Part B

Time evolution equations for atomic quasiprobability distributions with arbitrary orderings

P. Galatola

Published Jun 1, 1990• 0 citations
Publisher

Abstract

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.

DOI
10.1088/0954-8998/2/3/003

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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.
A study is carried out of the evolution of the classical phase portrait of the density operator of nonlinearly amplified light. A single-mode optical amplifier is considered, with quantumstatistical fluctuations of the two-level active medium and the light being amplified. The Heisenberg-Langevin equations for the field operators are obtained by adiabatic elimination of the atomic variables in the van der Pol approximation. The Fokker-Planck equation for the P-function, corresponding to the Heisenberg-Langevin equations, was solved numerically using the polynomial-expansion method. The initial quasiprobability distribution function corresponded to a pure coherent state. The Wigner quasiprobability distribution was calculated with the help of an integral transform of the P-function. Such quantumstatistical parameters of the amplifier radiation as the mean photon number and its variance, the Mandel parameter, and the mean field and the variance of the phase-amplitude quadratures of the field were calculated. O 1996 American Institute of Physics. [S1063-7761(96)00405-21
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.
The time evolution of the density matrix of the damped harmonic oscillator is studied within the Lindblad theory for open quantum systems. The density matrix is represented via a generating function, which is obtained by solving a time-dependent linear partial differential equation, derived from the master equation of to damped harmonic oscillator. Illustrative examples for specific initial conditions of the density matrix are provided. The Lindblad master equation is also transformed into Fokker-Planck equations for quasiprobability distributions. A comparative study is made for the Glauber P representation, the antinormal-ordering Q representation and Wigner W representation. (Author).
A review of quasiprobability methods for transforming chemical and quantum-optical master equations into Fokker-Planck equations is presented. For cases where conventional representations lead to Fokker-Planck equations with non-positive-definite diffusion coefficients; e.g., sub-Poissonian statistics, a generalization of the representation involving an extension to the complex plane enables analytic results to be obtained for certain nonlinear chemical and optical processes. Alternatively, a different integration measure may be chosen which ensures a positive distribution and Fokker-Planck equation with positive-semidefinite diffusion coefficients. This enables stochastic differential equations to be defined. These methods are applied to two-photon absorption and dispersive bistability in quantum optics where nonclassical photon statistics arise and to two models of nonlinear chemical reactions where sub-Poissonian statistics occur.
Spontaneous emission from a system of NNN identical two-level atoms is considered using a master equation recently derived by the author. The master equation describing the time evolution of the phase-space distribution function associated with the reduced density operator of the atomic system is obtained. This master equation, which is of the type of a Fokker-Planck equation, is used to derive the equation of motion for the mean values of various atomic operators characterizing the physical properties of the system. This leads to a hierarchy of equations, which is decoupled by making a suitable approximation. The intensity of the spontaneously emitted radiation is then calculated. Next, the spontaneous emission from geometrically small systems is considered. For this case, the master equation is solved exactly, and an exact expression for the radiation rate is obtained. The exact solution of the master equation is also used to calculate the normally ordered correlation functions for the electric field. Section V deals with the spontaneous emission from a system of harmonic oscillators, the size of the system being small compared to a wavelength. The master equation for this problem is also solved exactly, and it is shown that this system also leads to superradiant emission in some cases, e.g., if all the oscillstors are excited initially to some coherent state ∣z0〉|{z}_{0}〉∣z0​〉.
The Lindblad master equation for the damped quantum harmonic oscillator is transformed into Fokker–Planck equations for quasiprobability distributions. A comparative study is made for the Glauber P representation, the antinormal-ordering Q representation and the Wigner W representation. It will be proven that the variances for the damped harmonic oscillator found with these representations are the same. By solving the Fokker–Planck equations in the steady state, it will be shown that the quasiprobability distributions are two-dimensional Gaussians with widths determined by the diffusion coefficients.