Matter--wave emission in optical lattices: Single particle and collective effects

We introduce a simple set--up corresponding to the matter-wave analogue of impurity atoms embedded in an infinite photonic crystal and interacting with the radiation field. Atoms in a given internal level are trapped in an optical lattice, and play the role of the impurities. Atoms in an untrapped level play the role of the radiation field. The interaction is mediated by means of lasers that couple those levels. By tuning the lasers parameters, it is possible to drive the system through different regimes, and observe phenomena like matter wave superradiance, non-Markovian atom emission, and the appearance of bound atomic states.