  Classical Hamiltonian ratchets have been recently successfully realized using
cold atoms in driven optical lattices. Here we study the current rectification
of the motion of a quantum particle in a periodic potential exposed to an
external ac field. The dc current appears due to the desymmetrization of
Floquet eigenstates, which become transporting. Quantum dynamics enhances the
dependence of the current on the initial phase of the driving field. By
changing the laser field parameters which control the degree of space-time
asymmetry, Floquet eigenstates are tuned through avoided crossings. These
quantum resonances induce resonant changes of the resulting current. The width,
strength and position of these quantum resonances are tunable using control
parameters of the experimental realization with cold atoms.
