  We study a generalized cold atom Bose Hubbard model, where the periodic
optical potential is formed by a cavity field with quantum properties. On the
one hand the common coupling of all atoms to the same mode introduces cavity
mediated long range atom-atom interactions and on the other hand atomic
backaction on the field introduces atom-field entanglement. This modifies the
properties of the associated quantum phase transitions and allows for new
correlated atom-field states including superposition of different atomic
quantum phases. After deriving an approximative Hamiltonian including the new
long range interaction terms we exhibit central physical phenomena at generic
configurations of few atoms in few wells. We find strong modifications of
population fluctuations and next-nearest neighbor correlations near the phase
transition point.
