Analytical results for a conditional phase shift between single-photon pulses in a nonlocal nonlinear medium

TL;DR

This paper provides an analytical solution demonstrating that a conditional phase shift of π with high fidelity between two single-photon pulses in a nonlocal nonlinear medium is theoretically possible, supporting prior numerical findings.

Contribution

It offers a general analytical framework for understanding conditional phase shifts in nonlocal nonlinear media, addressing previous theoretical objections.

Findings

Analytical proof of π phase shift feasibility

Supports previous numerical results

Addresses theoretical objections to single-photon phase shifts

Abstract

It has been suggested that second-order nonlinearities could be used for quantum logic at the single-photon level. Specifically, successive two-photon processes in principle could accomplish the phase shift (conditioned on the presence of two photons in the low frequency modes) $ |011 \rangle \longrightarrow i|100 \rangle \longrightarrow -|011 \rangle $. We have analyzed a recent scheme proposed by Xia et al. to induce such a conditional phase shift between two single-photon pulses propagating at different speeds through a nonlinear medium with a nonlocal response. We present here an analytical solution for the most general case, i.e. for an arbitrary response function, initial state, and pulse velocity, which supports their numerical observation that a $\pi$ phase shift with unit fidelity is possible, in principle, in an appropriate limit. We also discuss why this is possible in this system, despite the theoretical objections to the possibility of conditional phase shifts on single photons that were raised some time ago by Shapiro and by one of us.