the number-phase entropic uncertainty relation by considering the two-mode Pegg-Barnett formalism, from which the entropy squeezing can be investigated, some of the nonclassical criteria of the obtained states such as Mandel $Q$ parameter, CSI and two different types of squeezing parameter are studied. Finally, section \[conclusion\] contains a summary and concluding remarks.

A brief review on the model
===========================

Let us give a brief review on the model which has been extensively discussed by us in [@usJOSA], in which a three-level atom (in a $\Lambda$ configuration) interacts with a two-mode quantized electromagnetic field which oscillates with frequencies $\Omega_{1}$ and $\Omega_{2}$ in an optical cavity surrounded by a Kerr medium with intensity-dependent coupling. According to the generalized JCM, the Hamiltonian for this system in the RWA can be written as $\hat{H} =\hat{H}_{0}+\hat{H}_{1}$ where ($\hbar=c=1$) $$\begin{aligned}
\label{hamiltonih0}
 \hat{H}_{0}&=& \sum_{j=1}^{3} \omega_{j}\hat{\sigma}_{jj}+\sum_{j=1}^{2} \Omega_{j} \hat{a}^{\dag}_{j} \hat{a}_{j},
 \end{aligned}$$ and $$\begin{aligned}
\label{hamiltonih1}
 \hat{H}_{1} &=& \chi \hat{{\mathbb R}}_{1}^{\dag} \hat{{\mathbb R}}_{1} \hat{{\mathbb R}}_{2}^{\dag} \hat{{\mathbb R}}_{2}+\lambda_{1}(\hat{{\mathcal{A}}}_{1} \hat{\sigma}_{12}+\hat{\sigma}_{21}\hat{{\mathcal{A}}}_{1}^{\dag}) \nonumber \\
 &+& \lambda_{2}(\hat{{\mathcal{A}}}_{2}\hat{\sigma}_{13}+\hat{\sigma}_{31}\hat{{\mathcal{A}}}_{2}^{\dag}),
 \end{aligned}$$ where $\hat{\sigma}_{ij}$ is the atomic lowering and raising operator between $|i\rangle$ and $|j\rangle$ defined by $\hat{\sigma}_{ij}=|i\rangle \langle j|,(i,j=1,2,3),\hat{a}_{j}$ ($\hat{a}_{j}^{\dag}$) is the bosonic annihilation (creation) operator of the field mode $j$, $\chi $ denotes the third-order susceptibility of cross-Kerr 