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

• Published in: Europhysics letters Vol. 77; no. 2; p. 20011
• Main Authors: Coffey, W. T, Kalmykov, Yu. P, Titov, S. V, Mulligan, B. P
• Format: Journal Article
• Language: English
• Published: IOP Publishing 01.01.2007; EDP Sciences
• Subjects: 03.65.Yz; 05.40.-a
• ISSN: 0295-5075; 1286-4854
• DOI: 10.1209/0295-5075/77/20011

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.