quantum simulation. The most prominent example are certainly ultra-cold atoms in optical lattices [@Bloch2008; @Greiner2002], which are almost ideal representations of various types of Hubbard models [@Essler2008; @Jaksch1998; @Albus2003; @Schneider2008; @Joerdens2008]. However due to their finite mass, atomic systems represent, with few exceptions, only models with explicit particle number conservation. On the other hand different quantum optical systems employing photons or quasi-particles such as phonons have been suggested recently, which are not limited by this constraint. For example an array of coupled high-$Q$ micro-cavities containing a two-level atom and a photon is described by the Jaynes-Cummings-Hubbard model (JCHM) [@Hartmann2006; @Hartmann2008; @Hartmann2008a; @Angelakis2007]. It is a combination of two well-known systems, the Jaynes-Cummings model [@Jaynes1963; @Shore1993] describing the coupling of a single two-level system to a bosonic mode and the hard-core Bose-Hubbard model [@Fisher1989] which describes the interaction and tunneling of bosons on a lattice. Recently we have shown that a modification of the JCHM can also be implemented in a linear ion trap, which has the advantage of an easier experimental realization since the required parameter regime is already realizable with current technology [@Ivanov2009]. A large variety of analytic and numeric methods was applied to the JCHM and related models, 