Two photons co- and counter-propagating through $N$ cross-Kerr sites

TL;DR

This paper analyzes the quantum scattering of one and two photons through a chain of cross-Kerr interaction sites, revealing how propagation direction affects the interaction and demonstrating a perfect controlled-phase operation in the infinite chain limit.

Contribution

It provides a detailed calculation of the two-photon S-matrix for spatially distributed cross-Kerr sites and compares co- and counter-propagating photon scenarios.

Findings

The S-matrix differs for co- and counter-propagating photons.

In the infinite chain limit, the system implements a perfect controlled-phase gate.

The physical interpretation of propagation direction effects enhances understanding of photon interactions.

Abstract

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.