atomic population inversion, photon counting statistics, quantum phase properties of the cavity field etc have been discussed. Quantum properties of a $\Lambda$-type three-level atom interacting with a single-mode field in a Kerr medium with intensity-dependent coupling and in the presence of the detuning parameters have been studied by us [@us]. The ability of the nonlinear JCM in generating a class of $SU(1,1)$ coherent states of the Gilmore-Perelomov type and also $SU(2)$ group was shown by one of us [@miry1]. In addition, as a results of a system in which a two-level atom interacts alternatively with a dispersive quantized cavity field and a resonant classical field, it is recently proposed a theoretical scheme from which the nonlinear elliptical states can be generated [@miry2]. In particular and in direct relation to the present work, more recently nonlinear interaction between a three-level atom (in a $\Lambda$ configuration) and a two-mode cavity field in the presence of a cross-Kerr medium and its deformed counterpart [@newhonarasa], intensity-dependent atom-field coupling and the detuning parameters has been discussed by us [@usJOSA]. Briefly, we studied the effects of these latter parameters on a few physical properties of the obtained state vector of the entire quantum system. Specifically, using 