Quantization of some generalized Jaynes–Cummings models in a Kerr-like medium

Based on fundamental principles of quantum mechanics, we present a method for a rigorous analytic construction of the spectra of the one- and two-photon Jaynes–Cummings models in a Kerr medium. To obtain an idea of the method, we consider a first generalized Jaynes–Cummings model with a real, linear superpotential using techniques of supersymmetric quantum mechanics. The Hamiltonian of this model is written as a combination of operators generating the underlying superalgebra whose elements are defined as differential matrix operators. Based on the formalism of supersymmetric quantum mechanics and the properties of sets of common observables, we derive solutions of the one- and two-photon models expressed in terms of confluent hypergeometric functions. Finally, using numerical analysis, we study the influence of the Kerr effect on the energy spectra of the physical system.