  We develop a Ginzburg-Landau theory for the Jaynes-Cummings-Hubbard model
which effectively describes both static and dynamic properties of photons
evolving in a cubic lattice of cavities, each filled with a two-level atom. To
this end we calculate the effective action to first-order in the hopping
parameter. Within a Landau description of a spatially and temporally constant
order parameter we calculate the finite-temperature mean-field quantum phase
boundary between a Mott insulating and a superfluid phase of polaritons.
Furthermore, within the Ginzburg-Landau description of a spatio-temporal
varying order parameter we determine the excitation spectra in both phases and,
in particular, the sound velocity of light in the superfluid phase.
