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

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.