the transition $|a\rangle\leftrightarrow|c\rangle$ is electric dipole forbidden. Moreover, as a result of the emerging various quantum effects, it is assumed that the atoms can only be initially prepared in a partial maximum coherent superposition of the upper and lower energy levels (a more general case has been presented in [@sinta]).

On account of this, the initial state of the three-level atom can be written as $$\begin{aligned}
\label{p02}|\Psi_{A}(0)\rangle\; ={1\over\sqrt{2}}\big[|a\rangle +
e^{i\varphi}|c\rangle\big],\end{aligned}$$ where $\varphi$ is an arbitrary phase randomly distributed about a fixed mean value $\varphi_{0}$. Based on this assumption, the phase can be defined as $\varphi=\varphi_{0}+\delta\varphi$ in which $\delta\varphi$ is taken to be small random fluctuations around $\varphi_{0}$ that can be adjusted at will by proper choice of the phase of the cavity radiation [@oc283781]. In line with Eq. , the corresponding initial reduced atomic density operator appears to be $$\begin{aligned}
\label{p03}\hat{\rho}_{A}(0)& =
{1\over2}\big[|a\rangle\langle a| + e^{-i\varphi}|a\rangle\langle
c| + e^{i\varphi}|c\rangle\langle a| +
|c\rangle\langle c|\big],\end{aligned}$$ where $e^{\pm i\varphi}/2$ stand for the initial atomic coherence.

Addressing the contribution of every phase change does not seem practically realistic. Consequently, assuming the phase fluctuations as a Gaussian random process [@pra77062308; @pra444688; @oc283781] and using the deviation of the phase fluctuation instead of the actual phase 