  We consider variants of the Jaynes-Cummings-Hubbard model of lattice
polaritons, taking into account next-nearest-neighbor, diagonal and long-range
photon hopping in one and two dimensions. These models are relevant for
potential experimental realizations of polariton Mott insulators based on
trapped ions or microwave stripline resonators. We obtain the Mott-superfluid
phase boundary and calculate excitation spectra in the Mott phase using
numerical and analytical methods. Including the additional hopping terms leads
to a larger Mott phase in the case of trapped ions, and to a smaller Mott phase
in the case of stripline resonators, compared to the original model with
nearest-neighbor hopping only. The critical hopping for the transition changes
by up to about 50 percent in one dimension, and by up to about 20 percent in
two dimensions. In contrast, the excitation spectra remain largely unaffected.
