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The paper studies the dynamics of transitions between the levels of a Wannier-Stark ladder induced by a resonant periodic driving. The analysis of the problem is done in terms of resonance quasienergy states, which take into account the metastable character of the Wannier-Stark states. It is shown that the periodic driving creates from a localized Wannier-Stark state an extended Bloch-like state with a spatial length varying in time as ~ t^1/2. Such a state can find applications in the field of atomic optics because it generates a coherent pulsed atomic beam.

A simple method of calculating the Wannier-Stark resonances in 2D lattices is suggested. Using this method we calculate the complex Wannier-Stark spectrum for a non-separable 2D potential realized in optical lattices and analyze its general structure. The dependence of the lifetime of Wannier-Stark states on the direction of the static field (relative to the crystallographic axis of the lattice) is briefly discussed.

The paper discusses the metastable states of a quantum particle in a periodic potential under a constant force (the model of a crystal electron in a homogeneous electric ,eld), which are known as the Wannier-Stark ladder of resonances. An ecient procedure to ,nd the positions and widths of resonances is suggested and illustrated by numerical calculation for a cosine potential.