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arXiv:1909.06170 · Physical Review A

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

Jacob G. Koefoed, Karsten K. Rottwitt

Published Dec 6, 2019• 8 citations
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Abstract

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.

DOI
10.1103/physreva.100.063813

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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 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.
We consider integrated photon pair sources based on spontaneous four-wave mixing and derive expressions for the pump powers at which various nonlinear processes become relevant for a variety of source materials and structures. These expressions serve as rules of thumb in identifying reasonable parameter regimes for the design of such sources. We demonstrate that if pump powers are kept low enough to suppress cross-phase modulation, multi-pair events as well as many other nonlinear effects are often also constrained to negligible levels.
We provide an efficient method for the calculation of high-gain, twin-beam generation in waveguides derived from a canonical treatment of Maxwell's equations. Equations of motion are derived that naturally accommodate photon generation via spontaneous parametric down-conversion (SPDC) or spontaneous four-wave mixing and, also, include the effects both of self-phase modulation of the pump and of cross-phase modulation of the twin beams by the pump. The equations we solve involve fields that evolve in space and are labeled by a frequency. We provide a proof that these fields satisfy bona fide commutation relations and that in the distant past and future they reduce to standard time-evolving Heisenberg operators. Having solved for the input-output relations of these Heisenberg operators we also show how to construct the ket describing the quantum state of the twin beams. Finally, we consider the example of high-gain SPDC in a waveguide with a flat nonlinearity profile, for which our approach provides an explicit solution that requires only a single matrix exponentiation.
We study the preparation of heralded single-photon states using dual-pump spontaneous four-wave mixing. The dual-pump configuration, which in our case employs cross-polarized pumps, allows for a gradual variation of the nonlinear interaction strength enabled by a birefringence-induced walk-off between the pump pulses. The scheme enables the preparation of highly pure heralded single-photon states, and proves to be extremely robust against the effect of nonlinear phase modulation at the required photon-pair production rates.
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.