  We derive a cross-Kerr type effective 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 show in general that
the atom will also display a linear and self-Kerr response. However, if certain
resonance conditions are satisfied then the linear and self-Kerr interactions
will vanish. The electrical susceptibilities of the probe transitions are also
explored and it is shown that a large, pure cross-Kerr nonlinearity can be
generated with vanishing absorption of both probe fields.
