  Recent experiments on strongly coupled cavity quantum electrodynamics present
new directions in "matter-light" systems. Following on from our previous work
[Phys. Rev. Lett. 102, 135301 (2009)] we investigate Bose-Hubbard models
coupled to a cavity light field. We discuss the emergence of photoexcitations
or "polaritons" within the Mott phase, and obtain the complete variational
phase diagram. Exploiting connections to the super-radiance transition in the
Dicke model we discuss the nature of polariton condensation within this novel
state. Incorporating the effects of carrier superfluidity, we identify a
first-order transition between the superradiant Mott phase and the single
component atomic superfluid. The overall predictions of mean field theory are
in excellent agreement with exact diagonalization and we provide details of
superfluid fractions, density fluctuations, and finite size effects. We
highlight connections to recent work on coupled cavity arrays.
