where $$\begin{aligned}
\label{vkardan}
 \mu_{j}&=&-\frac{1}{3}x_{1}+\frac{2}{3}\sqrt{x_{1}^{2}-3x_{2}}\cos\left[ \theta+\frac{2}{3}(j-1)\pi \right], \;j=1,2,3, \nonumber \\
 \theta &=& \frac{1}{3}\cos^{-1}\left[ \frac{9x_{1}x_{2}-2x_{1}^{3}-27x_{3}}{2(x_{1}^{2}-3x_{2})^{3/2}}\right],
 \end{aligned}$$ with $$\begin{aligned}
\label{x123}
 x_{1}&\dot{=}&V_{A}+V_{B}+V_{C}+\Delta_{3}-2\Delta_{2}, \nonumber \\
 x_{2}&\dot{=}& (V_{A}+V_{B}-\Delta_{2})(V_{C}+\Delta_{3}-\Delta_{2})+V_{B}(V_{A}-\Delta_{2})  \nonumber \\
 &-& \kappa_{1}^{2} - \kappa_{2}^{2}, \nonumber \\
 x_{3}&\dot{=}&V_{B}\left[(V_{A}-\Delta_{2})(V_{C}+\Delta_{3}-\Delta_{2})-\kappa_{2}^{2} \right]  \nonumber \\
 &-& \kappa_{1}^{2}(V_{C}+\Delta_{3}-\Delta_{2}).
 \end{aligned}$$ Also, in the above relations we have defined $$\begin{aligned}
\label{vdefinition1}
 \hspace{-0.5cm}V_{A} &\dot{=}&V(n_{1},n_{2}), \;\; V_{B}\dot{=}V(n_{1}+1,n_{2}), \;\; V_{C}\dot{=}V(n_{1},n_{2}+1), \nonumber \\
 \hspace{-0.5cm}\kappa_{1}&\dot{=}&\lambda_{1}\;\sqrt{n_{1}+1}\;f_{1}(n_{1}+1), \;\; \kappa_{2}\dot{=}\lambda_{2}\;\sqrt{n_{2}+1}\;f_{2}(n_{2}+1),  \nonumber \\
 \hspace{-0.5cm} \Delta_{2} &=& \omega_{2}-\omega_{1}+\Omega_{1},  \;\;\; \Delta_{3}=\omega_{3}-\omega_{1}+\Omega_{2},
 \end{aligned}$$ with $$\begin{aligned}
\label{vdefinition1}
 V(n_{1},n_{2}) \dot{=} \chi\; n_{1} n_{2} g_{1}^{2}(n_{1})g_{2}^{2}(n_{2}).
 \end{aligned}$$ Finally, by preparing the atom initially in the excited state, i.e. $A(0)=1$, $B(0)=C(0)=0$ or equivalently $$\begin{aligned}
\label{sayi}
 |\psi(0)\rangle_{\mathrm{A-F}}=|1\rangle \otimes \sum_{n_{1}=0}^{+\infty}\sum_{n_{2}=0}^{+\infty}q_{n_{1}}q_{n_{2}} |n_{1}, n_{2} \rangle,
 \end{aligned}$$ the following relation for $b_{j}$ may be found $$\begin{aligned}
\label{b123}
 \hspace{-1cm} b_{j}=\frac{\mu_{k}+\mu_{l}+V_{A}+V_{B}-\Delta_{2}}{\mu _{jk} \mu _{jl}}, \; j\neq k\neq l=1,2,3,
 \end{aligned}$$ where $ \mu _{jk}=\mu_{j}-\mu_{k} $. Hence, the wave function $|\psi(t)\rangle$ as given in (\[say\]) is exactly derived. It is valuable to state that the above formalism can be used for any physical system with arbitrary nonlinearity function. In this paper, we use the nonlinearity function $f(n)  = 1/\sqrt{n} = g(n)$ which has been derived by Man’ko [*et al*]{} [@manko2] from the coherent states that have been named by Sudarshan as harmonious states [@harmonious]. This function is a popular nonlinearity 