led to more tractable situation [@sint]. With similar conviction, employing the property of the Gaussian variables [@method], one can express $$\begin{aligned}
\label{p21}\langle\exp\pm i\delta\varphi\rangle=\exp-\langle\delta\varphi^{2}/2\rangle.\end{aligned}$$ Moreover, for a Gaussian random process $\langle\delta\phi\rangle$ is zero where $\langle\delta^{2}\phi/2\rangle=\theta$ represents the deviation which is generally designated as phase fluctuation.

Furthermore, following the detailed proceduce of obtaining the master equation and equations of evolution presented elsewhere [@sinta; @sint], it is possible to verify that $$\begin{aligned}
\label{p04}\alpha(t) = C_{+}(t)\alpha(0) + D_{+}(t)\beta^{*}(0) + E_{+}(t)
 + F_{+}(t),\end{aligned}$$ $$\begin{aligned}
\label{p05}\beta(t) =  C_{-}(t)\beta(0)+D_{-}(t)\alpha^{*}(0)
+ E_{-}(t) + F_{-}(t),\end{aligned}$$ where $$\begin{aligned}
\label{p06}C_{\pm}(t) = \frac{1}{2}\left[(1\pm p)e^{-\mu_{-}t} +
(1\mp p)e^{-\mu_{+}t}\right],\end{aligned}$$ $$\begin{aligned}
\label{p07}D_{\pm}(t) = \frac{q_{\pm}}{2}[e^{-\mu_{+}t} - e^{-\mu_{-}t}],\end{aligned}$$ $$\begin{aligned}
\label{p08}E_{+}(t)& =  \frac{1}{2}\int_{0}^{t}[(1+p)e^{-\mu_{-}(t-t')} \notag\\&+
(1-p)e^{-\mu_{+}(t-t')}]f_{a}(t')dt',\end{aligned}$$ $$\begin{aligned}
\label{p09}E_{-}(t) &= \frac{1}{2}\int_{0}^{t}[(1-p)e^{-\mu_{-}(t-t')}
\notag\\&+ (1+p)e^{-\mu_{+}(t-t')}]f_{b}(t')dt',\end{aligned}$$ $$\begin{aligned}
\label{p10}F_{+}(t) = \frac{q_{+}}{2}\int_{0}^{t}[e^{-\mu_{+}(t-t')} -
e^{-\mu_{-}(t-t')}]f^{*}_{b}(t')dt',\end{aligned}$$ $$\begin{aligned}
\label{p11}F_{-}(t) = \frac{q_{-}}{2}\int_{0}^{t}[e^{-\mu_{+}(t-t')} -
e^{-\mu_{-}(t-t')}]f_{a}^{*}(t')dt',\end{aligned}$$ with $$\begin{aligned}
\label{p12}\mu_{\pm} &= {\kappa\over2}+{A\over2B}\left[(2\zeta'+\zeta)e^{-\theta}\right.\notag\\&\left.\pm\sqrt{\zeta'^{2}(1+\zeta\zeta')^{2}+4(\zeta'^{2}+\chi)^{2}-
[(2-\zeta'\zeta)e^{-\theta}]^{2}}\right],\end{aligned}$$ $$\begin{aligned}
\label{p13}p={2\big[\zeta'^{2}+\chi\big]\over
\sqrt{\zeta'^{2}(1+\zeta\zeta')^{2}+4(\zeta'^{2}+\chi)^{2}-
[(2-\zeta'\zeta)e^{-\theta}]^{2}}},\end{aligned}$$ $$\begin{aligned}
\label{p14}q_{\pm}={-\zeta'(1+\zeta'\zeta)\pm\big[(2-\zeta'\zeta)e^{-\theta}\big]
\over
\sqrt{\zeta'^{2}(1+\zeta\zeta')^{2}+4(\zeta'^{2}+\chi)^{2}-
[(2-\zeta'\zeta)e^{-\theta}]^{2}}},\end{aligned}$$ in which $$\begin{aligned}
\label{p15}A = \frac{2r_{a}g^{2}}{\gamma^{2}},\end{aligned}$$ $$\begin{aligned}
\label{p16}B=(4+\zeta^{2})(1+\zeta'\zeta),\end{aligned}$$ $\zeta={\Omega/\gamma}$, $\zeta'={\Omega/\Gamma}$, and $\chi=\gamma/\Gamma.$ It is worth noting that $\kappa$ is the cavity damping constant, $\Gamma$ is the spontaneous atomic damping rate, $\gamma$ is the rate at which the coherent superposition decays and $r_{a}$ is the rate at which the atoms are injected into the cavity.

In addition, the corresponding stochastic noise 