level]{} with dash-dotted lines. In large detuning regime $\Delta/\omega =0.1$, the zeroth-order results agree well with the numerical ones from weak to strong coupling regimes in Fig. \[energy level\](a). But the RWA fails to give correct energies as the coupling strength $g/\omega$ increases. Because of the coupling of the qubit and the original oscillator, the latter should be displaced. Thus, the displaced oscillator state in the zeroth-order approximation, $|n\rangle _{j}=\exp [\frac{jg}{\omega }%
(a^{\dagger}-a)]|n\rangle (j=\pm \frac{3}{2},\pm \frac{1}{2})$, plays a more important role than the original oscillator state $|n\rangle$ in the RWA, resulting in more accurate eigen-energies in Eq. ( \[zeroenergy\]) and eigenfunctions in Eq. ( \[eigenstates\]). However, there is a noticeable deviation of the zeroth-order approximated results for the resonance case $%
\Delta /\omega =1$, indicating that the higher-order terms in Eq. (\[H\_11\]) should be taken into account. Physically, qubit states with different $n$ manifolds should be coupled by the interactions.

![image](elevel.eps)

*First-order approximation:* Keeping the linear terms in $a$ and $%
a^{\dag }$ and neglecting all higher order terms in the interaction Hamiltonian $H_{1}$(\[H\_11\]) gives $$H_{1}=\Delta \{J_{x}G_{0}\left( a^{\dagger }a\right) +iJ_{y}[F_{1}\left(
a^{\dagger }a\right) a^{\dagger }-aF_{1}\left( a^{\dagger }a\right) ]\}.
\label{H_1}$$The term $F_{1}\left( a^{\dagger }a\right) a^{\dagger }$ describes the photon hopping from state 