Cross-Kerr interaction in a four-level atomic system

We derive the form of the cross-Kerr interaction in a four-level atomic system in an N configuration. We use time-independent perturbation theory to calculate the eigenenergies and eigenstates of the Schrodinger equation for the system. The system is considered as a perturbation of a Raman-resonant three-level lambda scheme for which exact solutions are known. We show that within the strong control field limit the cross-Kerr interaction can arise between two weak probe fields. The strength of this nonlinear coupling is several orders of magnitude larger than that achievable using optical fibres.