Two-body problem in periodic potentials

Abstract

We investigate the problem of two atoms interacting via a short range s -wave potential in the presence of a deep optical lattice of arbitrary dimension D. Using a tight-binding approach, we derive analytical results for the properties of the bound state and the scattering amplitude. We show that the tunneling through the barriers induces a dimensional crossover from a confined regime at high energy to an anisotropic three-dimensional regime at low energy. The critical value of the scattering length needed to form a two-body bound state shows a logaritmic dependence on the tunneling rate for D=1 and a power law for D>1. For the special case D=1, we also compare our analytical predictions with exact numerics, finding remarkably good agreement. © 2006 The American Physical Society.