Effects of self- and cross-phase modulation on photon purity for four-wave-mixing photon pair sources

We consider the effect of self-phase modulation and cross-phase modulation on the joint spectral amplitude of photon pairs generated by spontaneous four-wave mixing. In particular, the purity of a heralded photon from a pair is considered in the context of schemes that aim to maximize the purity and minimize correlation in the joint spectral amplitude using birefringent phase matching and short pump pulses. We find that nonlinear phase-modulation effects will be detrimental and will limit the quantum interference visibility that can be achieved at a given generation rate. An approximate expression for the joint spectral amplitude with phase modulation is found by considering the group velocity walk-off between each photon and the pump but neglecting the group-velocity dispersion at each wavelength. The group-velocity dispersion can also be included with a numerical calculation, and it is shown that it has only a small effect on the purity for the realistic parameters considered.

Figure 1

Joint spectral amplitudes calculated simply from the pump, signal, and idler group velocities. (a) General example with ${\beta }_{1p}>{\beta }_{1i}>{\beta }_{1s}$ results in a highly correlated signal and idler, indicated by the diagonal nature of the main peak, (b) ${\beta }_{1p}={\beta }_{1i}$ results in the central peak becoming vertical and uncorrelated, and (c) when ${\beta }_{1p}-{\beta }_{1s}\approx {\beta }_{1i}-{\beta }_{1p}$, the phase-matching condition lies at $+{45}^{\circ }$ while the energy matching lies at $-{45}^{\circ }$. Tuning the pump bandwidth can make the central peak circular and uncorrelated. All JSAs are plotted as absolute values, ignoring the complex phase, and are shown normalized to have a maximum of 1.

Figure 2

Joint temporal amplitudes when the idler and pump are group velocity matched. For increasing fiber length compared to the length of the pump pulse $\tau$, the group velocity walk-off of the signal smears out the JTA. (a) ${\beta }_{1s}L/\tau =2$, (b) ${\beta }_{1s}L/\tau =5$, and (c) ${\beta }_{1s}L/\tau =10$.

Figure 3

JSA for increasing generation rates $R$, showing the correspondingly increasing effects of SPM and XPM. The initial effect is to broaden the idler, eventually leading to a splitting and distortion of the JSA which reduces the purity $P$. (a) $R$ approaching zero, $P=89\%$. (b) $R=0.1,P=83\%$. (c) $R=0.2,P=78\%$.

Figure 4

Purity plotted against generation rate for four choices of fiber parameters. As $R$ increases, SPM and XPM become significant and reduce $P$. Red: ${\beta }_{1s}L/\tau =40$. Green: ${\beta }_{1s}L/\tau =20$. Blue: ${\beta }_{1s}L/\tau =10$. Purple: ${\beta }_{1s}L/\tau =5$. The black dashed line shows the purity at a given generation rate after numerical optimization of ${\beta }_{1s}L/\tau$. It can be seen that a large value of ${\beta }_{1s}L/\tau$ is desirable at low rates, but for higher rates it will cause a more rapid falloff in purity.

Figure 5

(a) Joint temporal amplitude when the signal and idler are equally spaced in ${\beta }_1$ about the pump. The pump duration is optimized to avoid correlation, but the hard edges to the JTA caused by the sudden beginning and end of the nonlinearity mean that some correlation is inevitable. (b) Corresponding JSA with a low generation rate. (c) Corresponding JSA at a higher generation rate, $R=0.2$, showing the effects of SPM and XPM. The JSA is broadened in one direction.

Figure 6

Purity against generation rate for the symmetric scheme. Blue (dark gray): $\tau$ is kept constant. Green (light gray): $\tau$ is increased with $R$ to reoptimize. Again, the phase modulation reduces $P$ as $R$ is increased, although here $P$ is lower to start with than in the asymmetric scheme because there is more correlation in the sinc ripples of the JSA.

Figure 7

Purity as a function of the effective generation rate $RT$ when filtering is applied to one of the photons (the herald) resulting in transmission $T$. Unfiltered generation rates $R$ are shown from 0.02 to 0.1.

Figure 8

JSA from the numerical model including dispersion for realistic fiber parameters with a length of 50 cm and a pump bandwidth 2 nm. The dispersion causes some curvature of the JSA, which may introduce correlation.

Figure 9

Purity against generation rate $R$ for different strengths of dispersion, with an initial pump bandwidth of 1 nm. $R=0$ corresponds to the case with no non-linear phase modulation. (a) $L=0.5$ m and (b) $L=1$ m. In each case, black circles indicate no dispersion, blue squares show realistic dispersion, and red triangles illustrate double-strength dispersion. Surprisingly, at higher rates the dispersive case sometimes does better than the case with no dispersion.

Figure 10

Purity against generation rate with anomalous dispersion for an initial pump bandwidth of 2 nm. (a) $L=0.5$ m and (b) $L=1$ m. In each case, black circles indicate no dispersion, and blue squares show realistic dispersion. Here, a very slight benefit can be seen at the low rate for the dispersive case compared to the case without dispersion.