would be entangled if $\rm E_{N}=-Log_{2}V_{s}$ which implies that $\rm Log_{2}V_{s}$ should be negative. That means $$\begin{aligned}
\label{p35}V_{s}<1\end{aligned}$$ represents the condition for detecting entanglement. In order to quantify entanglement using this approach directly, the various correlations in Eq. should be determined applying Eqs. and . To this end, assuming the cavity to be initially in a two-mode vacuum state and the noise force at time $t$ does not statistically related to the cavity mode variables at earlier times, one can readily verify that [@sint] $$\begin{aligned}
\label{p37}\langle\alpha^{*}(t)\alpha(t)\rangle &=
A\left[\frac{L(1-p)^{2}+
Mq_{+}(1-p)}{8B\mu_{+}}\right][1 -
e^{-2\mu_{+}t}] \notag\\&+A\left[\frac{L(1+p)^{2}- Mq_{+}(1+p)}{8B\mu_{-}}\right][1-e^{-2\mu_{-}t}]
\notag\\&+A\left[ \frac{L(1-p^{2}) +Mq_{+}p}{2B(\mu_{+}+\mu_{-})}\right][1
- e^{-(\mu_{+}+\mu_{-})t}],\end{aligned}$$ $$\begin{aligned}
\label{p38}\langle\beta^{*}(t)\beta(t)\rangle &= A\left[\frac{Lq_{-}^{2}
+Mq_{-}(1+p)}{8B\mu_{+}}\right][1 - e^{-2\mu_{+}t}]
\notag\\&+A\left[\frac{Lq_{-}^{2} -Mq_{-}(1-p)}{8B\mu_{-}}\right][1-e^{-2\mu_{-}t}]
\notag\\&-A\left[ \frac{Lq_{-}^{2}+ Mq_{-}p}{2B(\mu_{+}+\mu_{-})}\right][1 -
e^{-(\mu_{+}+\mu_{-})t}],\end{aligned}$$ $$\begin{aligned}
\label{p39}\langle\alpha(t)\beta(t)\rangle &= A\left[\frac{2Lq_{-}(1-p)+M(1-p^{2}+q_{-}q_{+})}{16B\mu_{+}}\right]\notag\\&\times[1
- e^{-2\mu_{+}t}] \notag\\&-A\left[\frac{2Lq_{-}(1+p)- M(1-p^{2}+q_{-}q_{+})}{16B\mu_{-}}\right]\notag\\&\times[1-e^{-2\mu_{-}t}]
\notag\\&+A\left[ \frac{2Lq_{-}p +M(1+p^{2}-q_{-}q_{+})}{4B(\mu_{+}+\mu_{-})}\right]\notag\\&\times[1
- e^{-(\mu_{+}+\mu_{-})t}],\end{aligned}$$ where $$\begin{aligned}
\label{p41}L=2\zeta'^{2}+2\chi-(2\zeta'+\zeta)e^{-\theta},\end{aligned}$$ $$\begin{aligned}
\label{p42}M=\zeta'(1+\zeta'\zeta)+(2-\zeta'\zeta)e^{-\theta}.\end{aligned}$$

In the following, the time evolution of the degree of detectable entanglement is investigated employing the criterion of logarithmic negativity . To achieve the intended goal, $V_{s}$ is plotted against $time$ (a scaled unitless parameter) applying Eqs. , , , , , , , , , and . In order to see the nature of entanglement in depth, various special cases are considered.

When there is no external driving radiation
