solution.

The ground-state energy for the ground state $|-\frac{3}{2}\rangle |0\rangle
$ is $$E_{0}=-\frac{5g^2}{4\omega}-\frac{\Delta }{2}e^{-\frac{g^{2}}{2\omega ^{2}}}-2\chi _{1,0}.$$ The first and second excited energies $\{E_{0}^{k}\}$ ($k=1,2$) can be given by expanding the GRWA Hamiltonian in the basis $|-\frac{3}{2}\rangle
|1\rangle$ and $|-\frac{1}{2}\rangle |0\rangle$ $$H_{\mathtt{GRWA}}=\left(
\begin{array}{cc}
\omega +\mu _{1}(1) & \Delta R_{0,1}^{\prime } \\
\Delta R_{0,1}^{\prime } & \mu _{2}(0)%
\end{array}%
\right) .$$Similarly, $H_{\mathtt{GRWA}}$ is given in terms of $|-\frac{3}{2}\rangle
|2\rangle $, $|-\frac{1}{2}\rangle |1\rangle $, $|\frac{1}{2}\rangle
|0\rangle $ as $$H_{\mathtt{GRWA}}=\left(
\begin{array}{ccc}
2\omega +\mu _{1}(2) & \Delta R_{1,2}^{\prime } & 0 \\
\Delta R_{1,2}^{\prime } & \omega +\mu _{2}(1) & \Delta R_{0,1}^{\prime } \\
0 & \Delta R_{0,1}^{\prime } & \mu _{3}(0)%
\end{array}%
\right) ,$$which provides three analytical excited energies $\{E_{0}^{k}\}$ ($k=3,4,5$).

Energies obtained by the GRWA are presented in dashed lines in Fig. [energy level]{}. Especially, for the resonance case $\Delta =\omega $, the GRWA results are much better than the zeroth-order results (blue dotted lines) in Fig.\[energy level\](b). It ascribes to the effect of the coupling between states with different manifolds. Our approach is basically a perturbative expansion in terms of $\Delta/\omega$. As the increase of the $\Delta /\omega$ , the high order terms in Eq.(5) still cannot be neglected in the intermediate 