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Physical Review E

Simulation of the time evolution of the Wigner function with a first-principles Monte Carlo method

M. S. Torres, G. Tosi, J. M. A. Figueiredo

Published Sep 4, 2009• 2 citations
Publisher

Abstract

The implementation of Monte Carlo methods acting in the quantum phase space is hindered by the fact that quantum phase-space information is available only through quasiprobability densities. In this work, we present a first-principles Monte Carlo method employing a hidden variables representation. This allows the full quantum time evolution of an arbitrary initial quantum state to be calculated by a classical Monte Carlo algorithm, even for systems subjected to time-dependent potentials. Guidelines for implementing a practical algorithm are presented.

DOI
10.1103/physreve.80.036701

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The implementation of Monte Carlo methods acting in the quantum phase space is hindered by the fact that quantum phase-space information is available only through quasiprobability densities. In this work, we present a first-principles Monte Carlo method employing a hidden variables representation. This allows the full quantum time evolution of an arbitrary initial quantum state to be calculated by a classical Monte Carlo algorithm, even for systems subjected to time-dependent potentials. Guidelines for implementing a practical algorithm are presented.
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The Wigner formulation of quantum mechanics based on the concept of Wigner function (WF), is particularly suitable for the study of quantum transport in mesoscopic systems since it allows to describe quantum mechanical effects in an (r,p) phase space, in analogy to what is done for classical systems. Recently we have introduced the concept of Wigner paths (WPs) in phase space which both provide a pictorial representation of the quantum evolution of the system of interest and constitute a useful tool for providing numerical solutions of the quantum equation describing the time evolution of the system. The theory has been applied to the following cases: (a) The study of the transport properties of the transient regime for homogeneous systems. (b) The comparison between quantum and classical treatment of the scattering mechanism. (c) The study of the reflection of the WF from an infinite potential barrier. Furthermore, the Wigner approach, originally formulated in terms of single-time WF has been extended by introducing the two-time WF. The Fourier transform of this function contains independent information about both momentum and energy of the electron. The Monte Carlo procedure based on the generation of WPs can be used to evaluate the two-time WF of an electron interacting with the phonon gas.
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