reduction in the negativity $S_{X_{1}}$ and fast oscillatory behaviour are the results of considering the detuning effect.

### Sum squeezing

Here, we are going to investigate sum squeezing phenomenon. In order to study this nonclassicality indicator, two Hermitian operators $\hat{Y}_{1}$ and $\hat{Y}_{2}$ are given by [@hillery] $$\begin{aligned}
\label{ss1}
 \hat{Y}_{1} = \frac{1}{2}(\hat{a}_{1}\hat{a}_{2}+\hat{a}^{\dag}_{1}\hat{a}^{\dag}_{2}), \;
 \hat{Y}_{2} = \frac{1}{2i}(\hat{a}_{1}\hat{a}_{2}-\hat{a}^{\dag}_{1}\hat{a}^{\dag}_{2}),
 \end{aligned}$$ where $[\hat{Y}_{1},\hat{Y}_{2}]=\frac{i}{2}(\hat{a}^{\dag}_{1}\hat{a}_{1}+\hat{a}^{\dag}_{2}\hat{a}_{2}+1)$ yields the uncertainly relation $\langle (\Delta \hat{Y}_{1})^{2} \rangle \langle (\Delta \hat{Y}_{2})^{2} \rangle \geq\frac{1}{16}|\langle \hat{a}^{\dag}_{1}\hat{a}_{1}+\hat{a}^{\dag}_{2}\hat{a}_{2}+1\rangle)|^{2}$. Similar to the previous section, we can define the normalized sum squeezing parameters correspond to operators $Y_{1}$ and $Y_{2}$ respectively as $$\begin{aligned}
\label{ssqy2}
 \hspace{-1cm}S_{Y_{1}}&=&\frac{2 \mathrm{Re} \langle \hat{a}_{1}^{2} \hat{a}_{2}^{2} \rangle+2 \langle \hat{a}_{1}^{\dag}\hat{a}_{1}\hat{a}_{2}^{\dag}\hat{a}_{2}  \rangle -4 \left( \mathrm{Re} \langle \hat{a}_{1} \hat{a}_{2} \rangle \right)^{2} }{\langle \hat{a}_{1}^{\dag}\hat{a}_{1} \rangle+ \langle \hat{a}_{2}^{\dag}\hat{a}_{2} \rangle +1}, \nonumber\\
 \hspace{-1cm}S_{Y_{2}}&=&\frac{ 2 \langle \hat{a}_{1}^{\dag}\hat{a}_{1}\hat{a}_{2}^{\dag}\hat{a}_{2}  \rangle - 2 \mathrm{Re} \langle \hat{a}_{1}^{2} \hat{a}_{2}^{2} \rangle -4 \left( \mathrm{Im} \langle \hat{a}_{1} \hat{a}_{2} \rangle \right)^{2} }{\langle \hat{a}_{1}^{\dag}\hat{a}_{1} \rangle+ \langle \hat{a}_{2}^{\dag}\hat{a}_{2} \rangle +1}.
 \end{aligned}$$ Figure \[sumsq\] describes the sum squeezing for different chosen parameters assumed in figure \[quant.ph-sq\] with constant (intensity-dependent) coupling relating to the upside (downside) plots. It is seen from the upside plot ($f_{i}(n_{i}) = 1$) of figure \[sumsq\](a) that in the absence of the Kerr medium and in the resonance condition, the state 