Rigorous two-body problem in a 1D lattice with long-range interactions

International audienceWe study the scattering and the bound-state properties of two ultracold particles interacting via a short-range and a dipolar (long-range) interaction in a strictly 1D optical lattice. Our rigorous model take naturally into account the effects of the shape of the fundamental energy band and also of the role of the excited energy bands of the lattice. We first compare our exact results for a short-range interaction with the usual Bose-Hubbard (BH) model. By doing this, we highlight the effect of the excited bands, the non on-site interaction and the tunneling between distant neighbor that are not included into the usual BH model. A key result of our model is that the effective interaction deeply depend on the particle quasi-momenta and can then present a resonant behavior even for deep lattices. As a second phase of this study, we focus on the scattering of two polar particles in a lattice. The long-range interaction and the energy band structure of the lattice induce peculiar Wigner threshold laws. Our exact two-body calculations allow us to assess the validity of an extended Bose-Hubbard model for dipolar gases. This work was supported by the Agence Nationale de la Recherche (contract no. ANR-12-BS04-0020-01