Periodically driven quantum ratchets:: Symmetries and resonances

We study the quantum version of a tilting and flashing Hamiltonian ratchet, consisting of a periodic potential and a time-periodic driving field. The system dynamics is governed by a Floquet evolution matrix bearing the symmetry of the corresponding Hamiltonian. The dc-current appears due to the desymmetrization of Floquet eigenstates, which become transporting when all the relevant symmetries are violated. Those eigenstates that mostly contribute to a directed transport reside in phase space regions corresponding to classical resonances. Quantum dynamics leads to the dependence of the average velocity on the initial phase of the ac-field. A resonant enhancement (or suppression) of the dc-current, due to avoided crossings between different Floquet states, takes place upon tuning some control parameters. Our studies are predominantly aimed at experimental realizations of ac-driven quantum ratchets with cold atoms.