Complete evolution equation for the joint amplitude in photon-pair generation through spontaneous four-wave mixing

As four-wave-mixing-based photon-pair sources mature, accurate modeling of the photon-pair properties becomes important. Unlike spontaneous parametric down-conversion, four-wave mixing is accompanied by a number of parasitic effects such as nonlinear self- and cross-phase modulation. Currently, most modelings of photon-pair states are analytic in nature, which limits the number and type of effects that can be taken into account. In this work, we derive a complete, dual-pump evolution equation for the joint amplitude of photon pairs, wherein any desired effects can be included. We describe how to efficiently obtain numerical solution to this equation using a split-step approach. Lastly, we cover a few analytical solutions and compare two schemes for pure-photon generation under three different parasitic effects. We show how one scheme is highly sensitive to parasitic effects, while the other is very robust.

Figure 1

The initializing, repeating, and finalization parts of the split-step algorithm. Each part is bounded by the dashed line and the order of steps in each part is indicated with numbers. The spontaneous scattering effects ($\mathcal{S}$) are indicated in blue, the linear effects ($\mathcal{L}$) are in red, while the nonlinear effects ($\mathcal{N}$) are in black. Small arrows represent a half-step of $\mathrm{\Delta }z/2$ while long arrows represent a full step of $\mathrm{\Delta }z$.

Figure 2

Normalized absolute value of the JSA showing the effect of dispersion fluctuations, nonlinear phase modulation and group-velocity dispersion in the asymmetric scheme compared to no effects. The heralded photon purities are indicated in each case.

Figure 3

Normalized absolute value of the JSA showing the effect of dispersion fluctuations, nonlinear phase modulation and group-velocity dispersion in the collision scheme compared to no effects. The heralded photon purities are indicated in each case.

Figure 4

Heralded purity versus propagation length for the asymmetric scheme under no effects, dispersion fluctuations, nonlinear phase modulation, and higher-order dispersion.

Figure 5

Heralded purity versus propagation length for the collision scheme under no effects, dispersion fluctuations, nonlinear phase modulation, and higher-order dispersion. The inset shows a magnified view of the indicated region.

Figure 6

Normalized absolute value of the JSA showing the combined effect of group-velocity dispersion and dispersion fluctuations on the asymmetric scheme (left) and collision scheme (right).