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Modern Physics Letters B

ON THE MONTE CARLO SIMULATION APPROACH TO FOKKER-PLANCK EQUATIONS IN QUANTUM OPTICS

Giacomo Mauro D’Ariano, Chiara Macchiavello, Saverio Moroni

Published Feb 20, 1994• 7 citations
Publisher

Abstract

The suitability of the Monte Carlo simulation approach to Fokker-Planck equations in quantum optics is investigated. The method is especially useful for multimode analysis, and hence for studying realistic models of nonlinear optical systems. Here it is illustrated on the basis of two simple examples of application: i) a one-dimensional Fokker-Planck equation in the number representation, which describes a simple model of optical amplifier, and ii) the Van der Pol two-dimensional equation in the P-function representation.

DOI
10.1142/s0217984994000248

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The suitability of the Monte Carlo simulation approach to Fokker-Planck equations in quantum optics is investigated. The method is especially useful for multimode analysis, and hence for studying realistic models of nonlinear optical systems. Here it is illustrated on the basis of two simple examples of application: i) a one-dimensional Fokker-Planck equation in the number representation, which describes a simple model of optical amplifier, and ii) the Van der Pol two-dimensional equation in the P-function representation.
A novel numerical method to solve Fokker-Planck equations in quantum optics is presented, based on a Monte Carlo simulation of the probability diffusion process. The method is especially useful for multimode analysis and hence for studying realistic models of nonlinear optical systems. Two simple examples are given: the first is a 1D Fokker-Planck equation in the number representation, which describes a simple model of optical amplifier; the second is a 2D equation in the P-function representation-the quasi-probability for normal-ordered averages-which corresponds to the customary Van der Pol model of the laser threshold. In this case also the field correlation function and spectrum are numerically simulated.
We demonstrate a method which allows the stochastic modelling of quantum systems for which the generalised Fokker-Planck equation in the phase space contains derivatives of higher than second order. This generalises quantum stochastics far beyond the quantum-optical paradigm of three and four-wave mixing problems to which these techniques have so far only been applicable. To verify our method, we model a full Wigner representation for the optical parametric oscillator, a system where the correct results are well known and can be obtained by other methods. 1 I.
We demonstrate a method which allows the stochastic modelling of quantum systems for which the generalised Fokker-Planck equation in the phase space contains derivatives of higher than second order. This generalises quantum stochastics far beyond the quantum-optical paradigm of three- and four-wave mixing problems to which these techniques have so far only been applicable. To verify our method, we model a full Wigner representation for the optical parametric oscillator, a system where the correct results are well known and can be obtained by other methods.
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.
The Fokker-Planck approach is used to study the statistics of an optical pulse propagating in semiconductor optical amplifiers in which the amplified spontaneous emission noise dominates. The probability density functions (pdfs) of the pulse peak sample are calculated for different saturation levels. Analytical results are possible for an ideal linear amplifier and when the simplest first-order approximation for nonlinear amplification is used. In other cases, we solve numerically the Fokker-Planck equation and the appropriate Langevin equation. Multicanonical Monte Carlo simulations ensure efficient calculations of the pdfs whose high- and low-power tails deviate from noncentral chi-square statistics for moderate and deep saturation levels, thus implying that the electric field process is not Gaussian.
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