Quantum Brownian motion in ratchet potentials: duality relation and its   consequences
  Quantum Brownian motion in ratchet potentials is investigated by means of an approach based on a duality relation. This relation links the long-time dynamics in a tilted ratchet potential in the presence of dissipation with the one in a driven dissipative tight-binding model. The application to quantum ratchet yields a simple expression for the ratchet current in terms of the transition rates in the tight-binding system. 
J. Peguiron and M. Grifoni