Rocked quantum periodic systems in the presence of coordinate-dependent friction

We study rocked periodic systems with quantum fluctuations and coordinate-dependent friction. The steady current along the direction of the external driving force is evaluated numerically in terms of the path-integral Monte Carlo method and discussed in terms of the steepest descent approximation. Above the crossover temperature T-c a classical-like distribution function is given in order to calculate the least nonvanishing eigenvalue of the Fokker-Planck equation, thus one can finally obtain an expression of the current for moderate-to-strong damping. The influence of nonlinear quantum dissipation on the magnitude and the direction of the average current is observed. The classical current is also calculated using the Langevin simulation. The results show that when the friction is a periodic function of the coordinate, a net current averaging over the two directions arises even in the absence of both spatial and time asymmetry.