one can deduce that intensity-dependent coupling, as well as the deformed Kerr medium possesses a direct role in disappearing the classicality feature and so improving this nonclassicality indicator. Also, it is found that the depth of negativity of Mandel parameter in the presence of the detuning parameters may be decreased when intensity-dependent coupling is considered. At last, in the constant coupling regime, the typical collapse and revival exhibition as a nonclassical phenomenon can be observed.

The Cauchy-Schwartz inequality {#CSInequality}
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In this section we pay attention to CSI for checking another aspect of the nonclassicality of the state vector of the system. In the light of the Agarwal exploration [@agarwal] on the link between $P$-function [@GSPfunction1; @GSPfunction2; @GSPfunction3] and CSI, he showed that if the quantity $I_{0}$, as defined by $$\begin{aligned}
\label{CSinequality}
 I_{0}=\frac{\left( \langle a_{1}^{\dag 2}a_{1}^{2}\rangle \langle a_{2}^{\dag 2} a_{2}^{2} \rangle \right)^{1/2}}{|\langle a_{1}^{\dag}a_{1} a_{2}^{\dag} a_{2} \rangle|}-1.
 \end{aligned}$$ gets negative (positive) values, the corresponding $P$-function would be negative (positive), i.e. the nonclassical (classical) behavior is appeared. Notice that, commonly, one may expect that Glauber-Sudarshan $P$-distribution function, in the sense that it is a distribution function, have to be positive, while there exists quantum states for which $P$-function is negative or highly 