medium and the constants $\lambda_{1},\lambda_{2}$ determine the strength of the atom-field couplings. In above relation, $\hat{{\mathbb R}}_{j}=\hat{a}_{j}\hat{g}_{j}(\hat{n}_{j})$ and $\hat{{\mathcal{A}}}_{j}=\hat{a}_{j}\hat{f}_{j}(\hat{n}_{j})$ with $\hat{{\mathbb R}}_{j}^{\dag}$ and $\hat{{\mathcal{A}}}_{j}^{\dag}$ as their respective conjugate hermitians, where $\hat{n}_{j}=\hat{a}_{j}^{\dag}\hat{a}_{j}$. Note that, $g_{j}(\hat{n}_{j})$ and $f_{j}(\hat{n}_{j})$ are two generally different operator-valued functions which depend on the light intensity and specify the deformation of the Kerr medium and intensity-dependent atom-field coupling, respectively.\
The wave function $|\psi(t)\rangle$ corresponding to the whole system may be proposed in the form $$\begin{aligned}
\label{say}
 |\psi(t)\rangle&=&\sum_{n_{1}=0}^{+\infty}\sum_{n_{2}=0}^{+\infty}q_{n_{1}}q_{n_{2}}\Big[ A(n_{1},n_{2},t) e^{-i\gamma_{1} t}|1,n_{1},n_{2}\rangle \nonumber \\
 \hspace{-2cm}&+& B(n_{1}+1,n_{2},t)e^{-i\gamma_{2} t}|2,n_{1}+1,n_{2}\rangle  \nonumber \\
 \hspace{-2cm}&+& C(n_{1},n_{2}+1,t)e^{-i\gamma_{3} t}|3,n_{1},n_{2}+1\rangle  \Big],
 \end{aligned}$$ where $q_{n_{1}}$ and $q_{n_{2}}$ describe the amplitudes of the initial states of the field associated to each mode and $A$, $B$ and $C$ are the time-dependent atomic probability amplitudes which have to be evaluated. Also we have set $$\begin{aligned}
 \gamma_{1}&\doteq&\omega_{1}+n_{1}\Omega_{1}+n_{2}\Omega_{2}, \nonumber \\
 \gamma_{2}&\doteq&\omega_{2}+(n_{1}+1)\Omega_{1}+n_{2}\Omega_{2}, \nonumber \\
 \gamma_{3}&\doteq&\omega_{3}+n_{1}\Omega_{1}+(n_{2}+1)\Omega_{2}.
 \end{aligned}$$ By applying the probability amplitudes method, and after some lengthy but straightforward manipulations we obtained the probability amplitudes $A$, $B$ and $C$ (which determine the explicit form of the wave function of whole system) in the following form [@usJOSA]: $$\begin{aligned}
\label{abc}
 A(n_{1},n_{2},t)&=&-e^{-i\Delta_{2}t}\sum_{j=1}^{3}(\mu_{j}+V_{B})b_{j}e^{i\mu_{j}t}, \nonumber \\
 B(n_{1}+1,n_{2},t)&=&\sum_{j=1}^{3}\kappa_{1}\;b_{j} e^{i\mu_{j} t}, \nonumber \\
 C(n_{1},n_{2}+1,t)&=&\frac{e^{i(\Delta_{3}-\Delta_{2})t}}{\kappa_{2}}\sum_{j=1}^{3}\Big[(\mu_{j}+V_{B}) \nonumber \\
 &\times& (\mu_{j}+V_{A} - \Delta_{2}) - \kappa_{1}^{2}\Big]b_{j}e^{i\mu_{j}t},
 \end{aligned}$$ 