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Journal of Physics A Mathematical and Theoretical

A wavefunction description of quantum Fokker–Planck dissipation: derivation and numerical approximation of transient dynamics

Pilar Guerrero, José Luis López, Jesús Montejo–Gámez

Published Dec 20, 2013• 5 citations
Publisher

Abstract

A nonlinear Schrodinger equation describing how a quantum particle interacts with its surrounding reservoir is derived from the Wigner–Fokker–Planck equation (WFPE) via stochastic (Nelsonian) mechanical techniques. This model can be reduced just to a logarithmic Schrodinger equation through a suitable gauge transformation that allows one to explore its steady state dynamics and makes its mathematical and numerical analysis more tractable. The transient behavior of the standard deviation from the mean position associated with its solutions is also studied numerically and compared with that stemming from the WFPE.

DOI
10.1088/1751-8113/47/3/035303

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A nonlinear Schrodinger equation describing how a quantum particle interacts with its surrounding reservoir is derived from the Wigner–Fokker–Planck equation (WFPE) via stochastic (Nelsonian) mechanical techniques. This model can be reduced just to a logarithmic Schrodinger equation through a suitable gauge transformation that allows one to explore its steady state dynamics and makes its mathematical and numerical analysis more tractable. The transient behavior of the standard deviation from the mean position associated with its solutions is also studied numerically and compared with that stemming from the WFPE.
We formally derive two nonlinear Ginzburg-Landau type models starting from the Wigner-Fokker-Planck system, which rules the evolution of a quantum electron gas interacting with a heat bath in thermodynamic equilibrium. These models mainly consist of a quantum, dissipative O(Planck 3) hydrodynamic/O(Planck 4) stochastic correction to the frictional (Caldeira-Leggett-)Schrödinger equation. The main ingredient lies in the use of the hydrodynamic/stochastic fluid model approach associated with the quantum Fokker-Planck equation and the identification of the associated pressure field. Then, Madelung transformations set the problem in the Schrödinger picture of dissipative quantum mechanics. We also describe the stationary dynamics associated with both systems.
A study is provided of dynamics and dissipative tunneling in a symmetric quartic double well potential. The numerical solution for the position autocorrelation function obtained through the Wigner–Fokker–Planck equation is compared with numerically exact results of Stockburger and Mak [J. Chem. Phys. 110, 4983 (1999)]. We find that the Wigner–Fokker–Planck dynamics agree well with the numerically exact computations, they account for both quantum coherences as well as quantum tunneling phenomena. This, in contrast to the mixed quantum classical approximation, which does not perform as well.
This paper is intended to provide the reader with a review of the authors’ latest results dealing with the modeling of quantum dissipation/diffusion effects at the level of Schrodinger systems, in connection with the corresponding phase space and fluid formulations of such kind of phenomena, especially in what concerns the role of the Fokker–Planck mechanism in the description of open quantum systems and the macroscopic dynamics associated with some viscous hydrodynamic models of Euler and Navier–Stokes type.
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.
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The analysis of dissipative transport equations within the framework of open quantum systems with Fokker-Planck-type scattering is carried out from the perspective of a Wigner function approach. In particular, the well-posedness of the self-consistent whole-space problem in 3D is analyzed: existence of solutions, uniqueness and asymptotic behavior in time, where we adopt the viewpoint of mild solutions in this paper. Also, the admissibility of a density matrix formulation in Lindblad form with Fokker-Planck dissipation mechanisms is discussed. We remark that our solution concept allows to carry out the analysis directly on the level of the kinetic equation instead of on the level of the density operator. AMS 1991 Subject classification: 35Q40, 35S10, 81Q99, 81V99 Key words: Open quantum system, Wigner equation, large-time behavior, self-similarity. 1. Introduction This paper is concerned with the analysis of a class of dissipative quantum models that arise when considering the motio...