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Optics Express

High-efficiency cascaded up and down conversion in nonlinear Kerr cavities

Chao Zhou, Peter A. Bermel

Published Sep 9, 2015• 7 citations
Publisher

Abstract

In this work, we derive general conditions to achieve high efficiency cascaded third harmonic generation and three photon parametric down conversion in Kerr nonlinear resonant cavities. We employ the general yet rapid temporal coupled-mode method, previously shown to accurately predict electromagnetic conversion processes in the time domain. In our study, we find that high-efficiency cascaded third harmonic generation can be achieved in a triply resonant cavity. In contrast, high-efficiency cascaded three-photon parametric down conversion cannot be achieved directly in a triply resonant cavity, although a combination of two doubly resonant cavities and three waveguides is an effective alternative. The stabilities of the calculated steady-state solutions for both processes are revealed by applying Jacobian matrices. Finally, we find that the inclusion of self- and cross- phase modulation introduces multi-stable solutions. Further study is required to find a simple way to reliably achieve stable conversion at the highest possible efficiency.

DOI
10.1364/oe.23.024390

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In this work, we derive general conditions to achieve high efficiency cascaded third harmonic generation and three photon parametric down conversion in Kerr nonlinear resonant cavities. We employ the general yet rapid temporal coupled-mode method, previously shown to accurately predict electromagnetic conversion processes in the time domain. In our study, we find that high-efficiency cascaded third harmonic generation can be achieved in a triply resonant cavity. In contrast, high-efficiency cascaded three-photon parametric down conversion cannot be achieved directly in a triply resonant cavity, although a combination of two doubly resonant cavities and three waveguides is an effective alternative. The stabilities of the calculated steady-state solutions for both processes are revealed by applying Jacobian matrices. Finally, we find that the inclusion of self- and cross- phase modulation introduces multi-stable solutions. Further study is required to find a simple way to reliably achieve stable conversion at the highest possible efficiency.
We describe a theoretical analysis of the nonlinear dynamics of third-harmonic generation $(\ensuremath{\omega}\ensuremath{\rightarrow}3\ensuremath{\omega})$ via Kerr $({\ensuremath{\chi}}^{(3)})$ nonlinearities in a resonant cavity with resonances at both $\ensuremath{\omega}$ and $3\ensuremath{\omega}$. Such a doubly resonant cavity greatly reduces the required power for efficient harmonic generation, by a factor of $\ensuremath{\sim}V∕{Q}^{2}$, where VVV is the modal volume and QQQ is the lifetime, and can even exhibit 100% harmonic conversion efficiency at a critical input power. However, we show that it also exhibits a rich variety of nonlinear dynamics, such as multistable solutions and long-period limit cycles. We describe how to compensate for self- and cross-phase modulation (which otherwise shifts the cavity frequencies out of resonance), and how to excite the different stable solutions (and especially the high-efficiency solutions) by specially modulated input pulses.
We demonstrate theoretical conditions for highly efficient degenerate four-wave mixing in triply resonant nonlinear (Kerr) cavities. We employ a general and accurate temporal coupled-mode analysis in which the interaction of light in arbitrary microcavities is expressed in terms of a set of coupling coefficients that we rigorously derive from the full Maxwell equations. Using the coupled-mode theory, we show that light consisting of an input signal of frequency ${\ensuremath{\omega}}_{0}\ensuremath{-}\ensuremath{\Delta}\ensuremath{\omega}$ can, in the presence of pump light at ${\ensuremath{\omega}}_{0}$, be converted with quantum-limited efficiency into an output shifted signal of frequency ${\ensuremath{\omega}}_{0}+\ensuremath{\Delta}\ensuremath{\omega}$, and we derive expressions for the critical input powers at which this occurs. We find the critical powers in the order of 10 mW, assuming very conservative cavity parameters (modal volumes  10~10 10 cubic wavelengths and quality factors  1000~1000 1000). The standard Manley-Rowe efficiency limits are obtained from the solution of the classical coupled-mode equations, although we also derive them from simple photon-counting ``quantum'' arguments. Finally, using a linear stability analysis, we demonstrate that maximal conversion efficiency can be retained even in the presence of self- and cross-phase modulation effects that generally act to disrupt the resonance condition.
Nonlinear multiphoton conversion processes are theoretically and experimentally challenging systems in quantum optics and related areas. These phenomena bring remarkable quantum properties and can be of fundamental relevance in quantum information and computation applications. Based on recent experimental advances and the possible uses in generalized three-photon squeezing, we investigate degenerate three-photon parametric down-conversion generated inside a cavity fed by a classical pump. In addition, we consider that the process is stimulated by a coherent driving mode and subjected to dissipation by spontaneous emission of the cavity. A treatment using phase-space methods and stochastic differential equations is applied and the stability conditions of its steady states are found. We find that the system does not exhibit multistability, instead it has a limited parameter region leading to a single stable steady branch, alongside multiple unstable branches. Focusing on the stable steady states and linearizing the dynamics around them, we calculate the spectral densities of the quadrature variables. We identify and characterize the squeezing of three-photon down-conversion. Moreover, since the quasi-probability differential equations of the higher-order multiphoton processes have higher-order derivatives, hindering their mapping into stochastic differential equations, we make use of a positive Wigner function approach to calculate two-dimensional spectral densities associated with three-time correlation functions, with the goal of studying non-Gaussian properties of the system.
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.
We analyze an optical parametric oscillator (OPO) in which cascaded down-conversion occurs inside a cavity resonant for all modes but the initial pump. Due to the resonant cascade design, the OPO presents two ${\ensuremath{\chi}}^{(2)}$-level oscillation thresholds that are therefore much lower than for a ${\ensuremath{\chi}}^{(3)}$ OPO. This is promising for reaching the regime of an effective third-order nonlinearity well above both thresholds. Such a ${\ensuremath{\chi}}^{(2)}$ cascaded device also has potential applications in frequency conversion to far-infrared regimes. But, most importantly, it can generate novel multipartite quantum correlations in the output radiation, which represent a step beyond squeezed or entangled light. The output can be highly non-Gaussian and therefore not describable by any semiclassical model. In this paper, we derive quantum stochastic equations in the positive-PPP representation and undertake an analysis of steady-state and dynamical properties of this system.
We analyze an optical parametric oscillator (OPO) in which cascaded down-conversion occurs inside a cavity resonant for all modes but the initial pump. Due to the resonant cascade design, the OPO present two χ^2 level oscillation thresholds that are therefore remarkably lower than for a χ^3 OPO. This is promising for reaching the regime of an effective third-order nonlinearity well above both thresholds. Such a χ^2 cascaded device also has potential applications in frequency conversion to far infra-red regimes. But, most importantly, it can generate novel multi-partite quantum correlations in the output radiation, which represent a step beyond squeezed or entangled light. The output can be highly non-Gaussian, and therefore not describable by any semi-classical model.