  Controlling light-matter based quantum systems in the strong coupling regime
allows for exploring quantum simulation of many-body physics in nowadays
architectures. For instance, the atom-field interaction in a cavity QED network
provides control and scalability for quantum information processing. Here, we
propose the control of single- and two-body Jaynes-Cummings systems in a
non-equilibrium scenario, which allows us to establish conditions for the
coherent and incoherent interchange of polariton branches. Our findings provide
a systematic approach to manipulate polaritons interchange, that we apply to
reveal new insights in the transition between Mott Insulator- and
Superfluid-like states. Furthermore, we study the asymmetry in the absorption
spectrum by triggering the cavity and atomic losses as a function of the
atom-cavity detuning and photon's hopping.
