Directional emission of single photons from small atomic samples

We provide a formalism to describe deterministic emission of single photons with tailored spatial and temporal profiles from a regular array of multi-level atoms. We assume that a single collective excitation is initially shared by all the atoms in a metastable atomic state, and that this state is coupled by a classical laser field to an optically excited state which rapidly decays to the ground atomic state. Our model accounts for the different field polarization components via re-absorption and emission of light by the Zeeman manifold of optically excited states.