  We use an analytic slave-boson approach to calculate the elementary
excitations of the Jaynes-Cummings-Hubbard Model (JCHM) describing strongly
correlated polaritons on a lattice in various quantum optical systems. In the
superfluid phase near the Mott transition we find a gapless, linear Goldstone
mode and a gapped Amplitude mode corresponding to phase and density
fluctuations, respectively. The sound velocity of the Goldstone mode develops a
peculiar anomaly as a function of detuning at low densities, which persists
into the weakly interacting regime of a polariton BEC.
