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arXiv:1504.03776 · Physical Review A

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

Bryn A. Bell, Alex R. McMillan, Will McCutcheon, John Rarity

Published Nov 20, 2015• 27 citations
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Abstract

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.

DOI
10.1103/physreva.92.053849

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•Jan 1, 2017•16 citations
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.
Summary form only given. Sources of indistinguishable single photons are a key component of many optical quantum systems such as linear optical quantum computers [1] or quantum communication [2]. Spontaneous four-wave mixing is a promising method for in-fiber photon-pair generation with tailored spectral properties. However, a number of parasitic effects can negatively influence the quality of photons heralded from the pair. In this work, we investigate the impact on the quantum purity of heralded photons of arguably the three most important effects: nonlinear phase modulation (NPM), higher-order dispersion (HOD) and dispersion fluctuations (DF) along the fiber leading to variations in phase matching. For the first time, we directly compare two schemes under each of these effects. The first is the asymmetric scheme in which one of the quantum fields is group-velocity matched to the single pump field while the other walks off [3]. The second is the symmetric collision scheme, which uses two pump fields with different group velocities that pass through each other with each of the two quantum fields copropagating with one of the pumps [4]. In all calculations presented here, we use identical Gaussian pumps ∝ exp[-t2/(2T20)] with T0= 1 ps and a group-slowness difference of Δβ1= 1 x 10-11s/m between all non-copropagating fields. For both schemes we include the effect of NPM using analytical time-domain models [4,5] and choose a high generation probability of R= 0.2 to illustrate the impact. The effect of HOD is included with a novel analytical model for the asymmetric scheme and a numerical split-step method [5] for the collision scheme with all fields experiencing group-velocity dispersion of β2= 50 x 10-2652/m. Lastly, the effect of DFs is investigated using a Brownian-motion model for the phase-matching frequency [6] with a standard deviation σω= 0.5/T0and a correlation length in fluctuations of 10 cm.
A time-domain approach using the momentum operator is used to model spontaneous four-wave mixing in a lossless nonlinear waveguide. The effects of self- and cross-phase modulation on the photon-pair production rate and heralded photon purity are investigated. We show that in the special case where only one half of the photon-pair state is filtered the generation rate and purity of the heralded photons are unmodified by the presence of self- and cross-phase modulation. The significance of this special case arises when we consider heralded single-photon sources, where future schemes are likely to only filter the herald photon to ensure a high heralding efficiency is maintained.
We present a general model, based on a Hamiltonian approach, for the joint quantum state of photon pairs generated through pulsed spontaneous four-wave mixing, including nonlinear phase modulation and a finite material response time. For the case of a silica fiber, it is found that the pair-production rate depends weakly on the waveguide temperature, due to higher-order Raman scattering events, and more strongly on pump-pair frequency detuning. From the analytical model, a numerical scheme is derived, based on the well-known split-step method. This scheme allows computation of joint states where nontrivial effects are included, such as group-velocity dispersion and Raman scattering. In this work, the numerical model is used to study the impact of the noninstantaneous response on the prefiltering purity of heralded single photons. We find that for pump pulses shorter than 1 ps, a significant detuning-dependent change in quantum-mechanical purity may be observed in silica. This shows that Raman scattering not only introduces noise, but can also drastically change the spectral correlations in photon pairs when pumped with short pulses.
We model the spectral quantum-mechanical purity of heralded single photons from a photon-pair source based on nondegenerate spontaneous four-wave mixing taking the impact of distributed dispersion fluctuations into account. The considered photon-pair-generation scheme utilizes pump-pulse walk-off to produce pure heralded photons and phase matching is achieved through the dispersion properties of distinct spatial modes in a few-mode silica step-index fiber. We show that fiber-core-radius fluctuations in general severely impact the single-photon purity. Furthermore, by optimizing the fiber design we show that generation of single photons with very high spectral purity is feasible even in the presence of large core-radius fluctuations. At the same time, contamination from spontaneous Raman scattering is greatly mitigated by separating the single-photon frequency by more than 32 THz from the pump frequency.
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.
We model the spectral quantum-mechanical purity of heralded single photons from a photon-pair source based on nondegenerate spontaneous four-wave mixing taking the impact of distributed dispersion fluctuations into account. The considered photon-pair-generation scheme utilizes pump-pulse walk-off to produce pure heralded photons and phase matching is achieved through the dispersion properties of distinct spatial modes in a few-mode silica step-index fiber. We show that fiber-core-radius fluctuations in general severely impact the single-photon purity. Furthermore, by optimizing the fiber design we show that generation of single photons with very high spectral purity is feasible even in the presence of large core-radius fluctuations. At the same time, contamination from spontaneous Raman scattering is greatly mitigated by separating the single-photon frequency by more than 32 THz from the pump frequency. (C) 2017 Optical Society of America