A generalized atomic system of N-level atom within the framework of ($$N-1$$) JCM interacting with one-mode field solved via supersymmetric approach

We study a generalized atomic system of N-level atom within the framework of ([Formula: see text]) JCM interacting with one-mode field in the presence of Kerr medium and multi-photon transitions. We solve this system via supersymmetric approach. We show that the system possesses supersymmetric structure and construct its supersymmetric generators and their properties and diagonalize its Hamiltonian using supersymmetric unitary transformation. The atom-field eigenfunctions and eigenvalues of the system are obtained in two different cases of the initial conditions of the atom and the field mode. The evolution of the quasi-probability distribution Q-function and the atomic inversion is investigated for one case of the initial conditions. Their graphical results are presented and discussed for three special cases of the level number N, namely, [Formula: see text] which represents the three-level Λ-type system, [Formula: see text] which represents the four-level m-configuration system, and [Formula: see text] which represents the five-level double Λ-type system. The influence of the detuning and Kerr medium parameters on the behavior of these nonclassical quantities is presented graphically and discussed and found that they have important effects. We end by discussion and conclusions and some features and comments.