Controlling Discrete and Continuous Symmetries in “Superradiant” Phase Transitions with Circuit QED Systems

We explore theoretically the physics of a collection of two-level systems coupled to a single-mode bosonic field in the nonstandard configuration where each (artificial) atom is coupled to both field quadratures of the boson mode. We show that such an unusual coupling scheme can be implemented in circuit QED systems, where artificial Josephson atoms are coupled both capacitively and inductively to a superconducting resonator. We demonstrate that it is possible to pass from a discrete, paritylike ${\mathbb{Z}}_{2}$ symmetry to a continuous $U(1)$ with the appearance of photonic Goldstone and amplitude modes above a critical point even in the ultrastrong coupling regime (where the rotating wave approximation for the interaction between field and two-level systems is no longer applicable). We determine the rich phase diagram showing ``superradiant'' phases with different symmetries and phase boundaries of both first and second order.