Master equation in phase space applied to the quantum Brownian motion in a tilted periodic potential

Quantum effects in the Brownian motion in a tilted cosine potential are treated in the high temperature and weak bath-particle coupling limit via the semiclassical master equation for the time evolution of the reduced Wigner function in phase space. The differential recurrence relation generated from the master equation by expanding the periodic Wigner function expressed as a function of wave number k in Fourier series via Floquet's theorem is solved for bounded periodic initial conditions using matrix-continued fractions yielding both the time-independent and the time-dependent bounded periodic solutions. Just as the classical problem, the non-periodic solution is determined from the recurrence relation by integrating w.r.t. k over the first Brillouin zone with the initial conditions generated by the non-periodic Wigner distribution corresponding to zero probability current. The time-independent periodic (locked) solution is used to estimate quantum effects in the dc current-voltage characteristic of a Josephson junction including the capacitance, while the time-dependent non-periodic (running) solution is used to determine the dynamic structure factor and ultimately via integration over the first Brillouin zone the lifetime of the zero-voltage state. The current-voltage characteristics displayed as average velocity versus tilt reproduce in the high damping limit those from the semiclassical Smoluchowski equation, while the decrease in the lifetime of the zero-voltage state due to (the barrier lowering) quantum effects appears to be very sensitive to the tilt.