Entanglement of a multi-photon three-level atom interacting with a single-mode field in the presence of nonlinearities

General formalisms of a three-level atom in different configurations (namely, V, ??, and ?? configurations) interacting with multi-photon process of a single-mode field are investigated. Analytical expressions of the unitary operator within the framework of the dressed states and density operators of the consideration systems are derived. The temporal evolution of atomic quantum entropy when the field is initially in the binomial state and the atom is initially in its upper most state is considered. The effect of multiplicity, Kerr-like medium, detuning and different forms of intensity-dependent coupling functional on the degree of entanglement (DEM) are discussed. Details of the phenomenon differ from one configuration to the other.