quantum optics [@Scully], to quantum information [Wallraff]{} to condensed-matter physics [@Leggett]. In early work on cavity quantum electrodynamics (QED), the qubit-cavity coupling strength was much smaller than the cavity transition frequency, the rotating-wave approximation (RWA) can be applied, and an analytical exact solution can be derived straightforwardly [@jaynes]. With recent advances in the circuit QED using superconducting qubits, it is possible to engineer systems for which the qubits are so far detuned from the cavity, or are coupled to the cavity in a ultra-strong coupling regime where the coupling strength is comparable to the cavity transition frequency, that the RWA is demonstrated to fail to describe the system correctly [Niemczyk,pfd,fedorov,devoret,you,li]{}. The counter-rotating-wave (CRW) interactions in the qubit-cavity systems are therefore expected to play a crucial role. Under the RWA, the ground state is simply a direct product of the low state of the qubit and the vacuum cavity. The CRW interactions lead to a squeezed vacuum state containing virtual photons [@Ashhab; @ciuti]. The analytical exact study in the full model is highly nontrivial. There have been numerous theoretical studies on one- and two-qubit and cavity coupling systems, including the adiabatic approximation [Ashhab1,agarwal]{}, a Bargmann space technique [@Braak; @Zhong], an extended 