Quantum master equation in phase space: Application to the Brownian motion in a periodic potential

W. T. Coffey; Yu. P. Kalmykov; S. V. Titov; B. P. Mulligan

doi:10.1209/0295-5075/77/20011

All issues

Volume 77 / No 2 (January 2007)

EPL, 77 2 (2007) 20011

Abstract

Issue		EPL Volume 77, Number 2, January 2007


Article Number		20011
Number of page(s)		6
Section		General
DOI		https://doi.org/10.1209/0295-5075/77/20011
Published online		15 January 2007

EPL, 77 (2007) 20011

Quantum master equation in phase space: Application to the Brownian motion in a periodic potential

W. T. Coffey¹, Yu. P. Kalmykov², S. V. Titov¹^,3 and B. P. Mulligan¹

¹ Department of Electronic and Electrical Engineering, Trinity College - Dublin 2, Ireland
² Laboratoire MEPS, Université de Perpignan - 52 Avenue de Paul Alduy, 66860 Perpignan Cedex, France
³ Institute of Radio Engineering and Electronics, Russian Academy of Sciences - Vvedenskii Square 1, Fryazino, 141190, Russia

Received: 14 November 2006
Accepted: 23 November 2006

Abstract

The quantum Brownian motion of a particle in a periodic potential $V(x)=-V_{0}{\rm cos}(x/x_{0})$ is treated using the master equation for the Wigner distribution function $W(x, p, t)$ in phase space $(x, p)$ . Explicit equations for the diffusion coefficients of the master equation for this dissipative quantum system are derived. The dynamic structure factor $S(k, t)$ and escape rate $\Gamma$ are evaluated using matrix continued fractions. The matrix continued fraction calculation for $\Gamma$ agrees well with the analytical solution of the corresponding quantum Kramers turnover problem.

PACS: 03.65.Yz – Decoherence; open systems; quantum statistical methods / 05.40.-a – Fluctuation phenomena, random processes, noise, and Brownian motion

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