\hat a_{ j}^\dagger \hat a_{ j} + \Delta \sum_{ j} 
\hat\sigma^\dagger_{ j} \hat\sigma^-_{ j} + g\sum_{ j} \left(\hat\sigma ^\dagger_{ j} \hat a_{ j} + \hat a_{ j}^\dagger \hat\sigma_{ j}^-\right)\\
   +\sum_{d} t_{ d} \sum_{ j} \left(\hat a^\dagger_{ j} \hat a_{ j+ d} + \hat a^\dagger_{ j+ d} \hat 
a_{ j}\right)\label{eq:JCHreal}\end{gathered}$$ and discuss the main quantities needed in order to calculate its phase diagram. The system (\[eq:JCHreal\]) comprises bosonic and spin degrees of freedom, the specific interpretation of which depends on the actual physical system. Depending on the implementation, $\hat a_j^\dagger$ and $\hat a_j$ describe the creation and annihilation of a photon (phonon) at the $j$th cavity (ion), $\hat \sigma_j^\pm$ are the spin flip operators between the internal states of the atom (ion) and $\Delta$ is the transition energy of the atom (the detuning of the external laser field from the red motional sideband). $g$ describes the cavity-mediated atom-photon coupling (the phonon-ion coupling in the Lamb Dicke limit) and $\omega$ is the cavity resonance (the local oscillation) frequency. Between separated cavities (ions), there is a photon (phonon) transfer described in (\[eq:JCHreal\]) by the distance dependent hopping amplitude $t_d$.\
In the limit of vanishing hopping $t_d\equiv0$, the resulting Jaynes-Cummings model can easily be 