||The simplest form of dipole interaction between an atom and a single photon field mode, is described by the vacuum Rabi model. Strong atom-photon coupling, which is described by the simpler Jaynes-Cummings model, has been achieved in many platforms, such as cavity and circuit quantum electrodynamics (QED) and has brought a lot of success towards experimental quantum information processing over the past years. The full Rabi model dynamics can only be obtained in the so-called ultra-strong and deep-strong coupling regimes where the interaction coupling strength is comparable or higher than the natural system frequencies. However, due to our inability to achieve such high coupling strengths, these regimes remain largely unexplored in the lab. In this thesis, we investigate the possibility of reaching these regimes in a circuit QED setup, by means of a recently proposed analog-digital quantum simulation. Following a detailed numerical model of the proposed scheme, where we include the most important experimental limitations, we demonstrate the feasibility of the proposal using a transmon coupled to a 2D superconducting resonator, for a certain range of design parameters. Moreover, we show that the Wigner function representation of the resonator state in phase space is instrumental in order to probe the signature of deep-strong coupling in the system. Following these results, we design a device that will enable us to carry out the experiment with high fidelity measurements and perform direct Wigner tomography inside the resonator.