Effective Hamiltonians for fastly driven tight-binding chains

We consider a single particle tunnelling in a tight-binding model with nearest-neighbour couplings, in the presence of a periodic high-frequency force. An effective Hamiltonian for the particle is derived using an averaging method resembling classical canonical perturbation theory. Three cases are considered: uniform lattice with periodic and open boundary conditions, and lattice with a parabolic potential. We find that in the latter case, interplay of the potential and driving leads to appearance of the effective next-nearest neighbour couplings. In the uniform case with periodic boundary conditions the second- and third-order corrections to the averaged Hamiltonian are completely absent, while in the case with open boundary conditions they have a very simple form, found before in some particular cases by S. Longhi (2008) [10]. These general results may found applications in designing effective Hamiltonian models in experiments with ultracold atoms in optical lattices, e.g. for simulating solid-state phenomena.