providing profound results for the phase diagram and other ground state quantities [@Koch2009; @Pippan2009; @Makin2008; @Aichhorn2008; @Rossini2007; @Lei2008; @Schmidt2009; @Greentree2006]. In the present paper we show that in the strong interaction limit and near commensurate filling simple approximate analytic solutions of the JCHM can be found if there is translational invariance, i.e. for an infinite homogeneous system or periodic boundary conditions realizable e.g. with ions in a race-track Paul trap design. These solutions provide a good analytic approximation to the full ground state phase diagram.

This paper is structured as followed. In section \[sec:intro\] we will briefly summarize the main properties of the Jaynes-Cummings model, together with other important quantities needed later on. In section \[sec:Approximations\] we introduce two different approximation schemes, both giving analytic results for the critical hopping amplitude for the Mott insulator (MI) to superfluid (SF) transition. In section \[sec:Application\] we apply both approximation to the simple cubic nearest-neighbor JCH model describing an array of coupled cavities and to the special case of an linear ion chain.

the JCH model {#sec:intro}
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In this section, we will shortly review the main features of the JCH model defined by the Hamiltonian $$\begin{gathered}
  \hat H = \omega \sum_{ j} 