  We discuss analytic approximations to the ground state phase diagram of the
homogeneous Jaynes-Cummings-Hubbard (JCH) Hamiltonian with general short-range
hopping. The JCH model describes e.g. radial phonon excitations of a linear
chain of ions coupled to an external laser field tuned to the red motional
sideband with Coulomb mediated hopping or an array of high-$Q$ coupled cavities
containing a two-level atom and photons. Specifically we consider the cases of
a linear array of coupled cavities and a linear ion chain. We derive
approximate analytic expressions for the boundaries between Mott-insulating and
superfluid phases and give explicit expressions for the critical value of the
hopping amplitude within the different approximation schemes. In the case of an
array of cavities, which is represented by the standard JCH model we compare
both approximations to numerical data from density-matrix renormalization group
(DMRG) calculations.

Other Condensed Matter;Quantum Gases