  We discuss the Jaynes-Cummings-Hubbard model (JCHM) describing the
superfluid-Mott insulator transition of polaritons (i.e., dressed photon-qubit
states) in coupled qubit-cavity arrays in the crossover from strong to weak
correlations. In the strongly correlated regime the phase diagram and the
elementary excitations of lattice polaritons near the Mott lobes are calculated
analytically using a slave boson theory (SBT). The opposite regime of weakly
interacting polariton superfluids is described by a weak-coupling mean-field
theory (MFT) for a generalised multi-mode Dicke model. We show that a
remarkable relation between the two theories exists in the limit of large
photon bandwidth and large negative detuning, i.e., when the nature of
polariton quasiparticles becomes qubit-like. In this regime, the weak coupling
theory predicts the existence of a single Mott lobe with a change of the
universality class of the phase transition at the tip of the lobe, in perfect
agreement with the slave-boson theory. Moreover, the spectra of low energy
excitations, i.e., the sound velocity of the Goldstone mode and the gap of the
amplitude mode match exactly as calculated from both theories.
