Conditional phase shift between single photon pulses with different velocities in a Kerr medium

Spectral entanglement is a potentially significant obstacle to the eventual realization of quantum logical gates at the single-photon level using optical nonlinearities \footnote{J Gea-Banacloche, Phys. Rev. A \textbf{81}, 043823 (2010)}. It has recently been pointed out \footnote{D. J. Brod, J. Combes, and J. Gea-Banacloche, Phys. Rev. A \textbf{94}, 023833 (2016)} that this unwanted effect can be virtually eliminated by setting up a situation where conservation of momentum and energy lead to non-equivalent algebraic conditions on the wavevectors and frequencies of the interacting photons. We verify that this may be the case, in principle, for two photons traveling through a nonlocal Kerr medium with different velocities (co- or counterpropagating). The role of the nonlocality is merely to make the theory well behaved, and is essentially equivalent to a truncation of the medium's bandwidth, as we also verify here.