  Jaynes-Cummings-Hubbard lattices provide unique properties for the study of
correlated phases as they exhibit convenient state preparation and measurement,
as well as "in situ" tuning of parameters. We show how to realize charge
density and supersolid phases in Jaynes-Cummings-Hubbard lattices in the
presence of long-range interactions. The long-range interactions are realized
by the consideration of Rydberg states in coupled atom-cavity systems and the
introduction of additional capacitive couplings in quantum-electrodynamics
circuits. We demonstrate the emergence of supersolid and checkerboard solid
phases, for calculations which take into account nearest neighbour couplings,
through a mean-field decoupling.
