  We discuss the simulation of non-perturbative cavity-QED effects using
systems of trapped ions. Specifically, we address the implementation of
extended Dicke models with both collective dipole-field and direct
dipole-dipole interactions, which represent a minimal set of models for
describing light-matter interactions in the ultrastrong and deep-strong
coupling regime. We show that this approach can be used in state-of-the-art
trapped ion setups to investigate excitation spectra or the transition between
sub- and superradiant ground states, which are currently not accessible in any
other physical system. Our analysis also reveals the intrinsic difficulty of
accessing this non-perturbative regime with larger numbers of dipoles, which
makes the simulation of many-dipole cavity QED a particularly challenging test
case for future quantum simulation platforms.
