Quantum phases of incommensurate optical lattices due to cavity backaction

Ultracold bosonic atoms are confined by an optical lattice inside an optical resonator and interact with a cavity mode, whose wave length is incommensurate with the spatial periodicity of the confining potential. We predict that the intracavity photon number can be significantly different from zero when the atoms are driven by a transverse laser whose intensity exceeds a threshold value and whose frequency is suitably detuned from the cavity and the atomic transition frequency. In this parameter regime the atoms form clusters in which they emit in phase into the cavity. The clusters are phase locked, thereby maximizing the intracavity photon number. These predictions are based on a Bose-Hubbard model, whose derivation is here reported in detail. The Bose-Hubbard Hamiltonian has coefficients which are due to the cavity field and depend on the atomic density at all lattice sites. The corresponding phase diagram is evaluated using Quantum Monte Carlo simulations in one-dimension and mean-field calculations in two dimensions. Where the intracavity photon number is large, the ground state of the atomic gas lacks superfluidity and possesses finite compressibility, typical of a Bose-glass.