Phase-matched matter wave collisions in periodic potentials

Quantum scattering of particles in a periodic potential differs from scattering of free particles because of the non-trivial relation between energy and momentum, and because the potential can extract or provide momentum in the form of integer multiples of the lattice momentum vectors. In this paper, we study solutions for the single-particle dynamics in periodic potentials in one and more dimensions, and we identify the possible final states for various collision processes. This analysis leads to an interpretation of recently performed experiments on four-wave mixing and collisions of Bose - Einstein condensates and to predictions for further collision experiments.