Classical and quantum transport in deterministic Hamiltonian ratchets

We study directed transport in classical and quantum area-preserving maps, periodic in space and momentum. On the classical level, we show that a sum rule excludes directed transport of the entire phase space, leaving only th e possibility of transport in (dynamically defined) subsets, such as regula r islands or chaotic areas. As a working example, we construct a mapping wi th a mixed phase space where both the regular and the chaotic components su pport directed currents, but with opposite sign. The corresponding quantum system shows transport of similar strength, associated to the same subsets of phase space as in the classical map.