Effective cross-Kerr Hamiltonian for a nonresonant four-level atom

We derive an effective cross-Kerr Hamiltonian for the four-level atom interacting with three electromagnetic fields in the N configuration. When the atom has relaxed into the ground state, a cross-Kerr nonlinearity arises between two weak probe fields. As a development on earlier work, we derive the form of the atom-field interaction for all detunings and include the spontaneous decay of the upper atomic levels. In general, the atom will also display a linear and self-Kerr response, but if certain resonance conditions are satisfied then only the cross-Kerr interaction will remain. We consider the application of our theory to cold four-level rubidium atoms. The electrical susceptibilities of the probe transitions are explored, and it is shown that a large, pure cross-Kerr nonlinearity can be generated with vanishing absorption of both probe fields.