  We rigorously investigate the quantum dissipative dynamics of a ratchet
system described by a periodic potential model based on the Caldeira-Leggett
Hamiltonian with a biharmonic force. In this model, we use the reduced
hierarchy equations of motion in the Wigner space representation. These
equations represent a generalization of the Gaussian-Markovian quantum
Fokker-Planck equation introduced by Tanimura and Wolynes (1991), which was
formulated to study non-Markovian and non-perturbative thermal effects at
finite temperature. This formalism allows us to treat both the classical limit
and the tunneling regimes, and it is helpful for identifying purely quantum
mechanical effects through the time evolution of the Wigner distribution. We
carried out extensive calculations of the classical and quantum currents for
various temperatures, coupling strengths, and barrier heights. Our results
reveal that at low temperature, while the quantum current is larger than the
classical current in the case of a high barrier, the opposite is true in the
case of a low barrier. We find that this behavior results from the fact that
the tunneling enhances the current in the case of a high barrier, while it
suppresses the current in the case of a low barrier. This is because the effect
of the ratchet potential is weak in the case of a low barrier, due to the large
dispersion of the distribution introduced by tunneling. This causes the
spatio-temporal asymmetry, which is necessary for ratchet current, to be weak,
and as a result, the net current is suppressed.
