In this thesis we describe a novel regime of cavity quantum electrodynamics, where a single atom is coupled to a multi-mode Fabry-Perot cavity with a strength much larger than its free spectral range. In this regime, the atom acting as a quantum impurity mediates interactions between many-body states of radiation in the multi-mode cavity. This novel regime of cavity QED is experimentally realized by coupling superconducting artificial atoms to a high impedance 1-D superconducting transmission line cavity. We study the problem of single photon decay in these strongly non-linear cavities with discrete energy levels. By engineering the properties of the artificial atoms, we alter interaction and connectivity between many-body states of radiation, and we observe two distinct effects. For the case of a multi-mode Fabry-Perot coupled to a fluxonium artificial atom, the interactions mediated by the atom attempts to down convert a single photon into many low frequency photons but fails because of limited connectivity in the many-body Fock space. This phenomenon of many-body localization of radiation gives rise to striking spectral features where a single standing wave resonance of the cavity is replaced by a fine structure of satellite peaks. On the other hand, for the case of a transmon coupled galvanically to the cavity, the interaction splits a single photon at high energy into a shower of odd number of lower energy photons. In this case the single standing wave resonance of the cavity acquires a shorter lifetime which can be calculated using Fermi's golden rule and matches our theoretical model without any adjustable parameters.