  Cold atoms, driven by a laser and simultaneously coupled to the quantum field
of an optical resonator, can self-organize in periodic structures. These
structures are supported by the optical lattice, which emerges from the laser
light they scatter into the cavity mode, and form when the laser intensity
exceeds a threshold value. We study theoretically the quantum ground state of
these structures above the pump threshold of self-organization, by mapping the
atomic dynamics of the self-organized crystal to a Bose-Hubbard model. We find
that the quantum ground state of the self-organized structure can be the one of
a Mott-insulator or a superfluid, depending on the pump strength of the driving
laser. For very large pump strengths, where the intracavity intensity is
maximum and one would expect a Mott-insulator state, we find intervals of
parameters where the system is superfluid. These states could be realized in
existing experimental setups.
