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 analytic description of dc current rectification and current reversals. Transport characteristics for various limiting cases -- such as the classical limit, limit of high or low frequencies, and/or high temperatures -- 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 a 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 non-trivial exponent, $jpropto T^{-17/6}$.
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