Bose-Einstein condensates in an optical cavity with sub-recoil bandwidth

This article provides a brief synopsis of our recent work on the interaction of Bose-Einstein condensates with the light field inside an optical cavity exhibiting a bandwidth on the order of the recoil frequency. Three different coupling scenarios are discussed giving rise to different physical phenomena at the borderline between the fields of quantum optics and many-body physics. This includes sub-recoil opto-mechanical cooling, cavity-controlled matter wave superradiance and the emergence of a superradiant superfluid or a superradiant Mott insulating many-body phase in a self-organized intra-cavity optical lattice with retarded infinite range interactions.