Landau-Stark states and cyclotron-Bloch oscillations of a quantum particle

Recent experimental progress in the creation of synthetic electric and magnetic fields, acting on cold atoms in a two-dimensional lattice, has attracted renewed interest to the problem of a quantum particle in the Hall configuration. The present work contains a detailed analysis of the eigenstates of this system, called Landau-Stark states, and of the associated dynamical phenomenon of cyclotron-Bloch oscillations. It is shown that Landau-Stark states and cyclotron-Bloch oscillations crucially depend on two factors. The first is the orientation of the electric field relative to the primary axes of the lattice. The second is ratio between the frequencies of Bloch and cyclotron oscillations, that is also the ratio between the magnitudes of electric and magnetic fields. The analysis is first carried out in the tight-binding approximation, where the magnetic field is characterized by the Peierls phase entering the hopping matrix elements. Agreement of this analysis with the full quantum theory is also studied.