Jaynes-Cummings model in a finite Kerr medium

We introduce a spin model which exhibits the main properties of a Kerr medium to describe an intensity dependent coupling between a two-level atom and the radiation field. We select a unitary irreducible representation of the su(2) Lie algebra such that the number of excitations of the field is bounded from above. We analyze the behavior of both the atomic and the field quantum properties and its dependence on the maximal number of excitations.