In cavity quantum electrodynamics (QED), the interaction between an atomic transition and the cavity field is measured by the vacuum Rabi frequency . The analogous term circuit QED has been introduced for Josephson junctions, because superconducting circuits behave as arti ficial atoms coupled to the bosonic field of a resonator. In the regime with the vacuum Rabi frequency comparable to the two-level transition frequency, superradiant quantum phase transitions for the cavity vacuum have been predicted, e.g. within the Dicke model. In my Ph.D thesis, possible implementations of the Dicke model in circuit QED systems are theoretically investigated for di fferent kind of Josephson atoms, both in the case of inductive and capacitive coupling. In the thermodynamic limit of a large number of arti ficial atoms, predictions and constraints are explored for the occurrence of the quantum phase transition, with a doubly degenerate vacuum (ground state) and a spontaneous photonic coherence above a quantum critical coupling. In the finite-size case, the robustness and protection of the vacuum degeneracy in the ultrastrong coupling regime are studied, leading to possible applications for Quantum Computation using multiple resonators. Finally, a generalized Dicke model is proposed for which a doubly superradiant phase with four degenerate vacua is predicted.
