  We derive the form of the cross-Kerr interaction in a four-level atomic
system in the 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.
