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

European Journal of Physics

New approach to deriving the Fokker–Planck equation in the Wigner representation

Ye‐Jun Xu, Chao Li, Yu-hang Ma, Renshi Li

Published Oct 9, 2017• 1 citation
Publisher

Abstract

Abstract We exploit the so-called characteristic operator method to derive the Fokker–Planck equation in the Wigner representation. We introduce the characteristic operator of the Wigner representation and establish a set of fundamental differential equations for the action of the creation and annihilation operators on it. As an example, we show the derivation process of the Fokker–Planck equation for the damped harmonic oscillator in the Wigner representation by using the characteristic operator approach. This method provides us with a new convenient approach to deriving the phase-space Fokker–Planck equation in the Wigner representation.

DOI
10.1088/1361-6404/aa91e9

Related papers

Wigner distribution function for the harmonic oscillator with dissipation within the theory of open quantum systems

Аурелиан Исар

Time evolution of the expectation values of various dynamical operators of the harmonic oscillator with dissipation is analitically obtained within the framework of the Lindblad's theory for open quantum systems. We deduce the density matrix of the damped harmonic oscillator from the solution of the Fokker-Planck equation for the coherent state representation, obtained from the master equation for the density operator. The Fokker-Planck equation for the Wigner distribution function is also solved by using the Wang-Uhlenbeck method of transforming it into a linearized partial differential equation for the Wigner function, subject to either the Gaussian type or the $δ$-function type of initial conditions. The Wigner functions which we obtain are two-dimensional Gaussians with different widths.

arXiv:hep-th/9310065•Oct 12, 1993•2 citations

Damped quantum harmonic oscillator

Аурелиан Исар, A. Sãndulescu

In the framework of the Lindblad theory for open quantum systems the damping of the harmonic oscillator is studied. A generalization of the fundamental constraints on quantum mechanical diffusion coefficients which appear in the master equation for the damped quantum oscillator is presented; the Schrödinger and Heisenberg representations of the Lindblad equation are given explicitly. On the basis of these representations it is shown that various master equations for the damped quantum oscillator used in the literature are particular cases of the Lindblad equation and that the majority of these equations are not satisfying the constraints on quantum mechanical diffusion coefficients. Analytical expressions for the first two moments of coordinate and momentum are also obtained by using the characteristic function of the Lindblad master equation. The master equation 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 is 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 is shown that the quasiprobability distributions are two-dimensional Gaussians with widths determined by the diffusion coefficients. 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. Illustrative examples for specific initial conditions of the density matrix are provided.

arXiv:quant-ph/0602149•Feb 17, 2006•5 citations

Solutions of the damped oscillator Fokker-Planck equation

W. H. Louisell, John H. Marburger

The quantum theory of damping is presented and illustrated by means of a driven damped harmonic oscillator. The theory is formulated in the coherent state representation which illustrates very vividly the nearly classical nature of the problem. In this representation the reduced system density operator equation becomes a Fokker-Planck equation. Green's function solutions are found for the damped oscillator in closed form and as an eigenfunction expansion. In addition, a quantum regression theorem due to Lax is derived in the coherent state representation. The theorem allows two-time averages to be computed from one-time averages.

IEEE Journal of Quantum Electronics•Aug 1, 1967•54 citations

The harmonic oscillator with dissipation within the theory of open quantum systems

Аурелиан Исар

Time evolution of the expectation values of various dynamical operators of the harmonic oscillator with dissipation is analitically obtained within the framework of the Lindblad theory for open quantum systems. We deduce the density matrix of the damped harmonic oscillator from the solution of the Fokker-Planck equation for the coherent state representation, obtained from the master equation for the density operator. The Fokker-Planck equation for the Wigner distribution function, subject to either the Gaussian type or the $δ$-function type of initial conditions, is also solved by using the Wang-Uhlenbeck method. The obtained Wigner functions are two-dimensional Gaussians with different widths.

arXiv:quant-ph/0508140•Aug 18, 2005•0 citations

Fokker-Planck equation of the reduced Wigner function associated to an Ohmic quantum Langevin dynamics

Pedro J. Colmenares

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.

arXiv:1712.02780•Physical review. E•May 18, 2018•16 citations

Fokker–Planck Equations

Bryan J. Dalton, John Jeffers, Stephen Mark Barnett

Abstract This chapter deals with deriving Fokker–Planck equations (FPEs) that govern the behaviour of phase space distribution functions (normalised and unnormalised) for boson and fermion systems due to dynamical or thermal evolution. Specific forms are obtained for master equation dynamics, relating the drift vector and diffusion matrix elements in the FPE to coefficients specifying terms in the Hamiltonian and relaxation operators. The approach used is to derive correspondence rules, which map changes in the density operator when multiplied from left or right by mode annihilation or creation operators (as in evolution equations) onto corresponding changes in distribution functions, these being differentiations and or multiplications by phase space variables (left or right for fermions). Derivations are carried out via changes to characteristic functions or to canonical forms of the density operator. The correct successive application of correspondence rules for fermions is treated, and alternative forms obtained for bosons to correspondence rules and the FPE.

•Nov 13, 2014•10 citations
Abstract We exploit the so-called characteristic operator method to derive the Fokker–Planck equation in the Wigner representation. We introduce the characteristic operator of the Wigner representation and establish a set of fundamental differential equations for the action of the creation and annihilation operators on it. As an example, we show the derivation process of the Fokker–Planck equation for the damped harmonic oscillator in the Wigner representation by using the characteristic operator approach. This method provides us with a new convenient approach to deriving the phase-space Fokker–Planck equation in the Wigner representation.
Time evolution of the expectation values of various dynamical operators of the harmonic oscillator with dissipation is analitically obtained within the framework of the Lindblad's theory for open quantum systems. We deduce the density matrix of the damped harmonic oscillator from the solution of the Fokker-Planck equation for the coherent state representation, obtained from the master equation for the density operator. The Fokker-Planck equation for the Wigner distribution function is also solved by using the Wang-Uhlenbeck method of transforming it into a linearized partial differential equation for the Wigner function, subject to either the Gaussian type or the δδδ-function type of initial conditions. The Wigner functions which we obtain are two-dimensional Gaussians with different widths.
In the framework of the Lindblad theory for open quantum systems the damping of the harmonic oscillator is studied. A generalization of the fundamental constraints on quantum mechanical diffusion coefficients which appear in the master equation for the damped quantum oscillator is presented; the Schrödinger and Heisenberg representations of the Lindblad equation are given explicitly. On the basis of these representations it is shown that various master equations for the damped quantum oscillator used in the literature are particular cases of the Lindblad equation and that the majority of these equations are not satisfying the constraints on quantum mechanical diffusion coefficients. Analytical expressions for the first two moments of coordinate and momentum are also obtained by using the characteristic function of the Lindblad master equation. The master equation is transformed into Fokker-Planck equations for quasiprobability distributions. A comparative study is made for the Glauber PPP representation, the antinormal ordering QQQ representation and the Wigner WWW representation. It is 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 is shown that the quasiprobability distributions are two-dimensional Gaussians with widths determined by the diffusion coefficients. 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. Illustrative examples for specific initial conditions of the density matrix are provided.
The quantum theory of damping is presented and illustrated by means of a driven damped harmonic oscillator. The theory is formulated in the coherent state representation which illustrates very vividly the nearly classical nature of the problem. In this representation the reduced system density operator equation becomes a Fokker-Planck equation. Green's function solutions are found for the damped oscillator in closed form and as an eigenfunction expansion. In addition, a quantum regression theorem due to Lax is derived in the coherent state representation. The theorem allows two-time averages to be computed from one-time averages.
Time evolution of the expectation values of various dynamical operators of the harmonic oscillator with dissipation is analitically obtained within the framework of the Lindblad theory for open quantum systems. We deduce the density matrix of the damped harmonic oscillator from the solution of the Fokker-Planck equation for the coherent state representation, obtained from the master equation for the density operator. The Fokker-Planck equation for the Wigner distribution function, subject to either the Gaussian type or the δδδ-function type of initial conditions, is also solved by using the Wang-Uhlenbeck method. The obtained Wigner functions are two-dimensional Gaussians with different widths.
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.
Abstract This chapter deals with deriving Fokker–Planck equations (FPEs) that govern the behaviour of phase space distribution functions (normalised and unnormalised) for boson and fermion systems due to dynamical or thermal evolution. Specific forms are obtained for master equation dynamics, relating the drift vector and diffusion matrix elements in the FPE to coefficients specifying terms in the Hamiltonian and relaxation operators. The approach used is to derive correspondence rules, which map changes in the density operator when multiplied from left or right by mode annihilation or creation operators (as in evolution equations) onto corresponding changes in distribution functions, these being differentiations and or multiplications by phase space variables (left or right for fermions). Derivations are carried out via changes to characteristic functions or to canonical forms of the density operator. The correct successive application of correspondence rules for fermions is treated, and alternative forms obtained for bosons to correspondence rules and the FPE.