  We study the quantum version of a tilting and flashing Hamiltonian ratchets,
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 which 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.
