Quantum Brownian motion in ratchet potentials

We investigate the dynamics of quantum particles in a ratchet potential subject to an ac force field. We develop a perturbative approach for weak ratchet potentials and force fields. Within this approach, we obtain an analytical description of dc current rectification and current reversals. Transport characteristics for various limiting cases\char22{}such as the classical limit, the limit of high or low frequencies, and the limit of high temperatures \char21{} are derived explicitly. To gain insight into the intricate dependence of the rectified current on the relevant parameters, we identify characteristic scales and obtain the response of the ratchet system in terms of scaling functions. We pay special attention to inertial effects and show that they are often relevant, for example, at high temperatures. We find that the high-temperature decay of the rectified current follows an algebraic law with a nontrivial exponent, $j\ensuremath{\propto}{T}^{\ensuremath{-}17/6}.$