  It is well known that the bosonic Hubbard model possesses a Mott insulator
phase. Likewise, it is known that the Dicke model exhibits a self-organized
superradiant phase. By implementing an optical lattice inside of a high finesse
optical cavity both models are merged such that an extended Hubbard model with
cavity-mediated infinite range interactions arises. In addition to a normal
superfluid phase, two superradiant phases are found, one of them coherent and
hence superfluid and one incoherent Mott insulating.
