The interaction between a ♦-type four-level atom and a single-mode field in the presence of Kerr medium with intensity-dependent coupling involving multi-photon processes has been studied. Using the generalized (nonlinear) Jaynes-Cummings model, the exact analytical solution of the wave function for the considered system under particular condition, has been obtained when the atom is initially excited to the topmost level and the field is in a coherent state. Some physical properties of the atom-field entangled state such as linear entropy showing the entanglement degree, Mandel parameter, mean photon number and normal squeezing of the resultant state have been calculated. The effects of Kerr medium, detuning and the intensity-dependent coupling on the temporal behavior of the latter mentioned nonclassical properties have been investigated. It is shown that by appropriately choosing the evolved parameters in the interaction process, each of the above nonclassicality features, which are of special interest in quantum optics as well as quantum information processing, can be revealed.[10], entanglement [11], atomic dipole squeezing [12,13] and so on. This model has been extended in different directions such as multi-mode fields [14,15,16], multi-atoms interaction [17,18], multi-level atoms [19] and Kerr nonlinearity [20]. The interaction between two-level and different types of three-level atoms with quantized cavity field have been reviewed by Yoo and Eberly in [2]. Also, in recent years, researchers have strongly focused on the nonlinear interaction between a two-or multi-level atom with cavity field which leads to the deformed JCM. The latter nonlinear JCM, which firstly suggested in [21,22] describes the dependence of atom-field coupling on the light intensity. The revival-collapse phenomenon in the quadrature squeezing has been observed in the interaction between a two-level atom with single-mode cavity field of the multi-photon intensity-dependent JCM [23]. A nonlinear Jaynes-Cummings model which constructed from the standard JCM by deforming the single-mode field operators using f -deformed oscillator introduced by Man , ko et al [24] has been studied in [25]. The nonlinear interaction between a three-level atom in Λ-and V -configuration with a twomode field under a multi-photon process has been studied respectively in [26] and [27]. The interaction between a Λ-and V -type three-level atom with intensity-dependent coupling in a Kerr medium respectively studied in [28] and [29]. Recently, one of us has studied the entanglement dynamics of the nonlinear interaction between a Λ-type three-level atom with a two-mode cavity field in the presence of a cross-Kerr medium and its deformed counterpart [30], intensitydependent atom-field coupling and the detuning parameters [31,32]. Also, the authors have investigated a three-level atom in motion which interacts with a single-mode field in an optical cavity in an intensity-dependent coupling regime [33]. Sub-Poissonian statistics, population inversion and entropy squeezi...