Parametric processes in nonlinear structures with reflections: an asymptotic-field approach

TL;DR

The paper addresses the challenge of modeling nonlinear quantum optical processes, such as SPDC and SFWM, inside resonant structures while properly accounting for reflections. It extends the asymptotic-fields formalism to Fabry-Pérot cavities, derives the nonlinear interaction Hamiltonian in terms of asymptotic-in/out modes, and computes photon-pair generation rates via perturbation theory, illustrating the approach with SPDC in a flat-cavity, counter-propagating generation in periodically-poled media, and SFWM in Bragg-cavity structures. By unifying transfer-matrix mode matching with nonlinear quantum optics, the method naturally includes reflections and multi-stack materials and remains applicable beyond undepleted-pump and CW limits. The work provides a flexible, extensible toolkit for designing and analyzing cavity-enhanced quantum light sources, with open-source numerical codes available for broader use.

Abstract

The generation of engineered quantum states of light via nonlinear processes is fundamental for quantum technologies based on photons. Although embedding nonlinear materials within resonant structures allows for the enhancement and tailoring of photon properties, accurately modeling these quantum interactions remains a challenge. In this work, we apply the asymptotic-fields formalism, an approach based on scattering theory, to describe nonlinear optical processes within a Fabry-Pérot cavity. Unlike previous applications of this formalism, we explicitly account for reflections in the system. We derive the interaction Hamiltonian and calculate photon-pair generation rates using perturbation theory. The versatility of this model is illustrated through three examples: (i) spontaneous parametric down-conversion in an idealized cavity with flat-response mirrors; (ii) the generation of counter-propagating photon pairs in a periodically-poled material; and (iii) spontaneous four-wave mixing in a cavity built with Bragg reflectors.

Figures (8)

• Figure 1: Sketch of a scattering problem in quantum nonlinear optics illustrating the idea of asymptotic-in/out fields. In this example, the system is composed of four channels connected by an interaction region. a) Asymptotic-in field for channel 1. b) Asymptotic-out field for channel 4.
• Figure 2: Illustration of a Fabry-Pérot cavity. The left (L) and right (R) channels are connected to a cavity having semi-reflective surfaces at $z = \pm \ell / 2$. The field amplitudes $f_{\pm}$ and $e_{\pm}$ are depicted showing the direction of propagation of the associated fields.
• Figure 3: Sketch of the asymptotic conditions on the mode amplitudes. a) Asymptotic-in mode from the left; b) asymptotic-out mode from the right; c) asymptotic-out mode from the left.
• Figure 4: Spectral distribution of signal photons as a function of the signal wavenumber for different output-channel configurations. The results were normalized by the maximum rate value. The parameters used were: $\lambda_P = 750\mathrm{nm}$, $r_1 = -r_2 = 0.3$, $\ell = 10.15 \mu \mathrm{m}$, $(\bar{n}_P, \bar{n}_S, \bar{n}_I) = (2.18, 2.14, 2.22)$, $(\bar{n}_{G,P}, \bar{n}_{G,S}, \bar{n}_{G,I}) = (2.28, 2.18, 2.27)$.
• Figure 5: Total rate of pair generation as a function of the cavity length for different reflection amplitudes. The results were normalized by the maximum rate value. Each panel presents the results for different output-channel configuration. The parameters used were: $\lambda_P = 750\mathrm{nm}$, $(\bar{n}_P, \bar{n}_S, \bar{n}_I) = (2.18, 2.14, 2.22)$, $(\bar{n}_{G,P}, \bar{n}_{G,S}, \bar{n}_{G,I}) = (2.28, 2.18, 2.27)$.