& -\frac{%
K_{4}}{C_{4}} \\
\frac{1}{C_{1}} & \frac{1}{C_{2}} & \frac{1}{C_{3}} & \frac{1}{C_{4}}%
\end{array}%
\right) ,$$where $K_{i}$ has been defined in Eq.( \[ki\]) for $n=0$, and the normalized parameter is $C_{i}=\sqrt{2+2K_{i}^{2}}$. The corresponding eigenvalues are $\varepsilon_{i,0}$ in Eq.( \[zeroenergy\]).

In terms of the transformation $S^{\dagger}H_{1}^{^{\prime }}S$, the Hamiltonian $H^{1st}$ of the three-qubit Dicke model can be approximated as

$$\begin{aligned}
H_{\texttt{GRWA}} &=&\omega a^{\dagger }a+\mu _{1}(a^{\dagger }a)|-\frac{3}{2}\rangle \langle -\frac{%
3}{2}|+\mu _{2}(a^{\dagger }a)|-\frac{1}{2}\rangle \langle -\frac{1}{2}|+\mu _{3}(a^{\dagger }a)|\frac{1}{2%
}\rangle \langle \frac{1}{2}|+\mu _{4}(a^{\dagger }a)|\frac{3}{2}\rangle \langle \frac{3}{2}%
|  \nonumber  \label{GRWAham} \\
&&+\Delta F_{1}\left( a^{\dagger }a\right) [\frac{-\sqrt{3}K_{2}+K_{1}(\sqrt{%
3}+2K_{2})}{C_{1}C_{2}}(a|-\frac{1}{2}\rangle \langle -\frac{3}{2}|+h.c)  \nonumber \\
&&+\frac{-\sqrt{3}K_{3}+K_{2}(\sqrt{3}-2K_{3})}{C_{2}C_{3}}(a|\frac{1}{2}%
\rangle \langle -\frac{1}{2}|+h.c)
\nonumber \\
&&+\frac{-\sqrt{3}K_{4}+K_{3}(\sqrt{3}+2K_{4})}{C_{3}C_{4}}(a|\frac{3}{2}%
\rangle \langle \frac{1}{2}|+h.c)],\end{aligned}$$

where $\mu _{i}(a^{\dagger }a)=\varepsilon _{i,0}-\Delta [G_{0}\left( a^{\dagger }a\right) -\beta ]\frac{2K_{i}[\sqrt{3%
}-(-1)^{i}K_{i}]}{_{C_{i}^{2}}}$. There are only the energy-conserving terms $(a|-\frac{1}{2}\rangle \langle -%
\frac{3}{2}|+h.c)$, $(a|\frac{1}{2}\rangle \langle -\frac{1}{2}|+h.c)$, and $%
(a|\frac{3}{2}\rangle \langle \frac{1}{2}|+h.c)$ with renormalized coefficients, originating from the CRW terms $iJ_{y}[F_{1}\left(
a^{\dagger }a\right) a^{\dagger }-aF_{1}\left( a^{\dagger }a\right)
]$. The dominated effect of the original CRW terms is considered here. Because it is the three-qubit Dicke model Hamiltonian in the same RWA form with renormalized coefficients, the present approach essentially borrows the basic idea of the GRWA proposed by Irish for the one-qubit model [@irish].

Note that the individual bosonic creation 