Ultrastrong-coupling quantum-phase-transition phenomena in a few-qubit   circuit QED system
  We study ultrastrong-coupling quantum-phase-transition phenomena in a few-qubit system. In the one-qubit case, three second-order transitions occur and the 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 the quantum phase transition is inhibited even in the ultrastrong-coupling regime in this model. In addition, in the three-qubit model, the superradiant quantum phase transition is retrieved in the ultrastrong-coupling regime. Furthermore, the N-qubit model with U(1) symmetry is studied and we find that the superradiant phase transition is inhibited or restored with the qubit-number parity. 
Wen-Juan Yang, Xiang-Bin Wang
quant-ph