  A cross-Kerr interaction produces a phase shift on two modes of light
proportional to the number of photons in both modes, and is sometimes called
cross-phase modulation. Cross-Kerr nonlinearities have many applications in
classical and quantum nonlinear optics, including the possibility of a
deterministic and all-optical controlled-phase gate. We calculate the one- and
two-photon S-matrix for fields propagating in a medium where the cross-Kerr
interaction is spatially distributed at discrete interaction sites comprised of
atoms. For the interactions considered, we analyze the cases where the photons
co-propagate and counter-propagate through the medium and give a physical
interpretation to the differences between the two cases. Finally, we obtain the
S-matrix in the limit of infinitely long chains, showing that it corresponds to
a perfect controlled-phase operation.
