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Physical Review A

Photon conversion processes in dispersive microcavities: Quantum-field model

Alex Hayat, Meir Orenstein

Published Jan 28, 2008• 11 citations
Publisher

Abstract

We present a simple quantum-field model for photon frequency conversion processes in dispersive dielectric microcavities, allowing practical calculations of conversion rates and efficiency optimization. The model incorporates the effect of dispersion into the structure of spatial cavity eigenmodes, while the material nonlinear properties enter the calculations through the coupling coefficients. Similar to the radiative atomic transitions, cavity photon-to-photon decay rates are derived for stimulated and spontaneous photon conversion. The phase matching requirement for a nonlinear process in a cavity is shown to be translated into a nonlinear mode overlap, manifesting the momentum conservation in a $k$ space spanned by the cavity eigenmodes. We illustrate the efficiency of the model by calculating degenerate two-photon and three-photon parametric down-conversion rates and analyze the photon lifetime dependence on various cavity and material parameters.

DOI
10.1103/physreva.77.013830

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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.
Two-photon states are essential for fundamental applications in quantum information. One of the primary methods of two-photon generation is based on parametric down-conversion, but this suffers from low efficiency and a large footprint. This work presents a detailed theoretical investigation of an alternative approach: two-photon generation from an emitter in a doubly resonant cavity. The system is modelled by the Lindblad master Equation, and an approximate analytical solution is derived to determine the experimentally achievable limits on efficiency and brightness. Additionally, the optimal cavity parameters for achieving these limits are also identified. For experimentally feasible parameters, the maximum efficiency is approximately 35%, which is significantly higher than that of parametric down-conversion-based methods. The optimal rate and efficiency for two-photon generation are achieved when the outcoupling rate of the cavity mode at the two-photon emission frequency matches the single-photon atom-field coupling strength. Moreover, the outcoupling rate of the cavity mode at the one-photon emission frequency for single photons should be minimized. The cavity field properties are also examined by studying the second-order correlation function at zero time delay and the Mandel Q parameter, revealing highly bunched two-photon emission and super-Poissonian statistics. The quantum-jump framework, combined with Monte Carlo simulations, is used to characterize the mechanism of two-photon emission and the emission spectra of the cavity. Two-photon emission is demonstrated to be a rapid cascade process of quantum jumps, and its spectrum consists of three prominent peaks corresponding to transitions between the dressed states of the system.
Ultracompact nonlinear optical devices utilizing two-dimensional (2D) materials and nanostructures are emerging as important elements of photonic circuits. Integration of the nonlinear material into a subwavelength cavity or waveguide leads to a strong Purcell enhancement of the nonlinear processes and compensates for a small interaction volume. The generic feature of such devices which makes them especially challenging for analysis is strong dissipation of both the nonlinear polarization and highly confined modes of a subwavelength cavity. Here we solve a quantum-electrodynamic problem of the spontaneous and stimulated parametric down-conversion in a nonlinear quasi-2D waveguide or cavity. We develop a rigorous Heisenberg-Langevin approach which includes dissipation and fluctuations in the electron ensemble and in the electromagnetic field of a cavity on equal footing. Within a relatively simple model, we take into account the nonlinear coupling of the quantized cavity modes, their interaction with a dissipative reservoir and the outside world, amplification of thermal noise and zero-point fluctuations of the electromagnetic field, and other relevant effects. We derive closed-form analytic results for relevant quantities such as the spontaneous parametric signal power and the threshold for parametric instability. We find a strong reduction in the parametric instability threshold for 2D nonlinear materials in a subwavelength cavity and provide a comparison with conventional nonlinear photonic devices.