\left[ \frac{g}{\omega }\left( a^{\dagger
}-a\right) \right] +iJ_{y}\sinh \left[ \frac{g}{\omega }\left( a^{\dagger
}-a\right) \right] \right\}.\label{H_11}  \notag \\\end{aligned}$$Then We can expand the even and odd functions $\cosh (y)$ and $\sinh (y)$, respectively, as $\cosh \left[ \frac{g}{\omega }\left(
a^{\dagger }-a\right) \right] =G_{0}\left( a^{\dagger }a\right) +G_{1}\left(
a^{\dagger }a\right) \left( a^{\dagger }\right) ^{2}+a^{2}G_{1}\left(
a^{\dagger }a\right) +...$ and $\sinh \left[ \frac{g}{\omega }\left(
a^{\dagger }-a\right) \right] =F_{1}\left( a^{\dagger }a\right) a^{\dagger
}-aF_{1}\left( a^{\dagger }a\right) +F_{2}\left( a^{\dagger }a\right) \left(
a^{\dagger }\right) ^{3}-a^{3}F_{2}\left( a^{\dagger }a\right) +...$, where $%
G_{i}(a^{\dagger }a)(i=0,1,...)$ and $F_{j}(a^{\dagger }a)(j=1,2,...)$ are coefficients that depend on the cavity number operator $\widehat{n}=a^{\dagger}a$ and the dimensionless parameter $g/\omega$. A different order of approximations can then be performed by neglecting some terms in the expansions.

*zeroth-order approximation:* In the zeroth-order approximation, we only keep the first term $G_{0}\left( a^{\dagger }a\right) $ in $\cosh %
\left[ \frac{g}{\omega }\left( a^{\dagger }-a\right) \right] $, and the Hamiltonian is approximated as $$H^{^{0th}}=\omega a^{\dagger }a-\frac{g^{2}}{\omega }J_{z}^{2}+\Delta
J_{x}G_{0}\left( a^{\dagger }a\right) .$$ In the basis of the oscillator state $|n\rangle$, the term $G_{0}\left( a^{\dagger }a\right) $ only has non-vanishing diagonal element $$G_{0}(n)=\langle n|\cosh \left[ \frac{g}{\omega }\left( a^{\dagger
}-a\right) \right] |n\rangle =e^{-\frac{g^{2}}{2\omega ^{2}}}L_{n}(\frac{%
g^{2}}{\omega ^{2}}),$$ where Laguerre polynomials $L_{n}^{m-n}(x)=\sum_{i=0}^{\min
\{m,n\}}(-1)^{n-i}\frac{m!x^{n-i}}{(m-i)!(n-i)!i!}$. Note that only the oscillator number operator $\widehat{n}$ appears, 