Ultrastrong-coupling QPT phenomena in few-qubit circuit QED system

We consider the system of a few qubits interacting with a single mode resonator in superconducting circuit QED system. We get an effective low-energy Hamiltonian in the limit where the ratio of the qubit frequency to the resonator frequency tends to infinity. In the one-qubit case, three secondorder superradiant phases occur and Goldstone mode emerges under the condition of ultrastrong coupling strength. Moreover, a first-order phase transition occurs between two different superradiant phases. In the two-qubit case, a two-qubit Hamiltonian with qubit-qubit interactions is analyzed fully quantum mechanically. We show that quantum phase transition is inhibited even in the ultrastrong coupling regime in this model. In addition, in the three qubits model, the superradiant quantum phase transition is retrieved in the ultrastrong coupling regime. Furthermore, the N-qubit model with U(1) symmetry is studied. We find that the superradiant phase transition is inhibited or restored with the qubit-number parity.