Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Acta Physica Sinica

THE FORMAL SOLUTIONS FOR FOKKER-PLANCK EQUATION WITH ZERO OR NEGATIVE DIFFUSION COEFFICIENTS AND THEIR APPLICATIONS TO QUANTUM OPTICS

TAN WEI-HAN, Yufang Li, Weiping Zhang, 中国科学院上海光学精密机械研究所

Published Jan 1, 1988• 3 citations
Publisher

Abstract

In this paper, the formal solutions for Fokker-Planck equation with zero or negative diffusion coefficients in quantum optics are given. Their applications to the generation of squeezed states via degenerate parametric amplification or degenerate four wave mixing and other problems are discussed.
DOI
10.7498/aps.37.396

Related papers

Dynamical ordering: Applications to Fokker-Planck equations

Arkadiusz Orłowski, Krzysztof Wódkiewicz

A transformation of the linear second-order Fokker-Planck equations occurring in quantum optics is presented which overcomes difficulties connected with negative definiteness of the diffusion matrices. This transformation is based on a dynamical ordering of quantum quasiprobability distribution functions. The ordering depends on the parameters of the dynamical problem under consideration. An algorithm leading to exact solutions of the dynamically ordered Fokker-Planck equations with arbitrary initial conditions is given, and simple criteria, how to recognize squeezed light in arbitrary ordering, are presented. The problem of Fokker-Planck equations with time-dependent diffusion is also investigated.

Physical Review A•Oct 1, 1994•2 citations

Beyond the Fokker-Planck equation: Stochastic Simulation of Complete Wigner representation for the Optical Parametric Oscillator

Lev I. Plimak, M. K. Olsen, M. Fleischhauer +1 more

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.

•Jan 1, 2001•23 citations

The solution of the Fokker-Planck equation of non-degenerate parametric amplific ation system for generation of squeezed light

Chao Zhao, TAN WEI-HAN, Qizhi Guo +1 more

In this paper, we present the analytical solution of the Fokker-Planck equation of non-degenerate optical parametric amplification(NOPA)for generation of squeezed light. The maximum intra-cavity compression of squeezed light derived from the analytical solution is 1/16(vacuum fluctuations 1/4). To compare it with that of the previous result 1/8 of degenerate optical parametric amplification(DOPA), it seems that the squeezing for NOPA is superior to the squeezing for DOPA.

Acta Physica Sinica•Jan 1, 2003•9 citations

Beyond the Fokker-Planck equation: Stochastic simulation of complete Wigner representation for the optical parametric oscillator

L. I Plimak, M. K. Olsen, M Fleischhauer +1 more

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.

arXiv:quant-ph/0108128•Europhysics Letters (EPL)•Nov 1, 2001•36 citations

Quantum Statistics of Single-mode Processes Preserving Generalized Superposition of Coherent and Chaotic Fields

Vlasta Peřinová, A. Lukš, Przemyslaw Szlachetka

A unified approach to the quantum statistics of single-mode linear processes described by the general Fokker-Planck equation and the generalized initial superposition of coherent and chaotic fields is developed to reveal the common traits in the mathematical models of several quantum optical systems. Processes such as the degenerate parametric amplification and the harmonic oscillator eventually coupled to the thermal or rigged reservoir are involved as special cases. A sufficient condition for the validity of the master equation is formulated as its being formally independent of possible reservoirs. The dependence of non-classical phenomena such as the sub-Poisson photon statistics and squeezing on the drift and diffusion coefficients is discussed.

Journal of Modern Optics•Nov 1, 1989•15 citations

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

Giacomo Mauro D’Ariano, Chiara Macchiavello, Saverio Moroni

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.

Modern Physics Letters B•Feb 20, 1994•7 citations
A transformation of the linear second-order Fokker-Planck equations occurring in quantum optics is presented which overcomes difficulties connected with negative definiteness of the diffusion matrices. This transformation is based on a dynamical ordering of quantum quasiprobability distribution functions. The ordering depends on the parameters of the dynamical problem under consideration. An algorithm leading to exact solutions of the dynamically ordered Fokker-Planck equations with arbitrary initial conditions is given, and simple criteria, how to recognize squeezed light in arbitrary ordering, are presented. The problem of Fokker-Planck equations with time-dependent diffusion is also investigated.
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 unified approach to the quantum statistics of single-mode linear processes described by the general Fokker-Planck equation and the generalized initial superposition of coherent and chaotic fields is developed to reveal the common traits in the mathematical models of several quantum optical systems. Processes such as the degenerate parametric amplification and the harmonic oscillator eventually coupled to the thermal or rigged reservoir are involved as special cases. A sufficient condition for the validity of the master equation is formulated as its being formally independent of possible reservoirs. The dependence of non-classical phenomena such as the sub-Poisson photon statistics and squeezing on the drift and diffusion coefficients is discussed.
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.