Phase-noise limitations on single-photon cross-phase modulation with differing group velocities

A framework is established for evaluating cphase gates that use single-photon cross-phase modulation (XPM) originating from the Kerr nonlinearity. Prior work [J. H. Shapiro, Phys. Rev. A 73, 062305 (2006)], which assumed that the control and target pulses propagated at the same group velocity, showed that the causality-induced phase noise required by a noninstantaneous XPM response function precluded the possibility of high-fidelity $\pi$-radian conditional phase shifts. The framework presented herein incorporates the more realistic case of group-velocity disparity between the control and target pulses, as employed in existing XPM-based fiber-optical switches. Nevertheless, the causality-induced phase noise identified by Shapiro [J. H. Shapiro, Phys. Rev. A 73, 062305 (2006)] still rules out high-fidelity $\pi$-radian conditional phase shifts. This is shown to be so for both a reasonable theoretical model for the XPM response function and for the experimentally measured XPM response function of silica-core fiber.

Figure 1

Fidelity upper-bound $F_{\max }$ for the single-resonance, two-pole response function plotted vs the normalized damping parameter $\Gamma$.

Figure 2

The Raman response of silica-core fiber, as measured by Stolen et al. [21]. (a) Frequency response. (b) Temporal response.

Figure 3

Heat map of $F_1^{\max }$ vs $\eta u$ and $L/u$ for Dirac-$\delta$ pulses with $t_d=L/2u$.

Figure 4

Heat map of $F_1^{\max }$ vs $\eta u$ and $L/u$ for Gaussian pulses with $t_{\psi }=3\mathrm{ps}$ and $t_d=L/2u$.