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Scientific Reports

Kerr nonlinearity, self-injection locking and correlation in a microresonator

Andrey B. Matsko, Abdelkrim El Amili, Lute Maleki

Published Nov 6, 2025• 1 citation
Publisher

Abstract

Production of entangled photon pairs is important in secure communication systems, quantum computing, and fundamental physics experiments. Achieving efficient generation of such photon pairs with low-loss parametric oscillators is a key objective in advancing integrated quantum technologies. However, spatially separating the generated photons while preserving their entanglement represents a significant technical challenge. In this work, we demonstrate nonlinear generation of correlated optical harmonics based on non-degenerate four-wave mixing with an optimally pumped optical microcavity with Kerr nonlinearity. The phase matching of the process is achieved with self-injection locked lasers producing parametric oscillation while locked to two different modes of the microresonator. This condition is reminiscent of slow-light technique developed for coherent atomic systems. The experimental design, utilizing counterpropagating light from two self-injection locked lasers, also effectively addresses the challenge of spatial separation of the generated harmonics. Additionally, we demonstrate correlation mediated by self injection locking and Kerr nonlinearity between the two lasers. We validate the theoretical predictions using two self-injection locked semiconductor lasers integrated with a crystalline whispering gallery mode resonator with optimized spectral structure. ©2025 All Rights Reserved.

DOI
10.1038/s41598-025-22664-1

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Production of entangled photon pairs is important in secure communication systems, quantum computing, and fundamental physics experiments. Achieving efficient generation of such photon pairs with low-loss parametric oscillators is a key objective in advancing integrated quantum technologies. However, spatially separating the generated photons while preserving their entanglement represents a significant technical challenge. In this work, we demonstrate nonlinear generation of correlated optical harmonics based on non-degenerate four-wave mixing with an optimally pumped optical microcavity with Kerr nonlinearity. The phase matching of the process is achieved with self-injection locked lasers producing parametric oscillation while locked to two different modes of the microresonator. This condition is reminiscent of slow-light technique developed for coherent atomic systems. The experimental design, utilizing counterpropagating light from two self-injection locked lasers, also effectively addresses the challenge of spatial separation of the generated harmonics. Additionally, we demonstrate correlation mediated by self injection locking and Kerr nonlinearity between the two lasers. We validate the theoretical predictions using two self-injection locked semiconductor lasers integrated with a crystalline whispering gallery mode resonator with optimized spectral structure. ©2025 All Rights Reserved.
Production of entangled photon pairs is important in secure communication systems, quantum computing, and fundamental physics experiments. Achieving efficient generation of such photon pairs with low-loss parametric oscillators is a key objective in advancing integrated quantum technologies. However, spatially separating the generated photons while preserving their entanglement represents a significant technical challenge. In this work, we demonstrate nonlinear generation of correlated optical harmonics based on non-degenerate four-wave mixing with an optimally pumped optical microcavity with Kerr nonlinearity. The phase matching of the process is achieved with self-injection locked lasers producing parametric oscillation while locked to two different modes of the microresonator. This condition is reminiscent of slow-light technique developed for coherent atomic systems. The experimental design, utilizing counterpropagating light from two self-injection locked lasers, also effectively addresses the challenge of spatial separation of the generated harmonics. We validate the theoretical predictions using two self-injection locked semiconductor lasers integrated with a crystalline whispering gallery mode resonator with optimized spectral structure.
The dynamical behavior of Kerr optical frequency combs is very well understood today from the perspective of the semiclassical approximation. These combs are obtained by pumping an ultrahigh-QQQ whispering-gallery mode resonator with a continuous-wave laser. The long-lifetime photons are trapped within the toruslike eigenmodes of the resonator, where they interact nonlinearly via the Kerr effect. In this article, we use quantum Langevin equations to provide a theoretical understanding of the nonclassical behavior of these combs when pumped below and above threshold. In the configuration where the system is under threshold, the pump field is the unique oscillating mode inside the resonator, and it triggers the phenomenon of spontaneous four-wave mixing, where two photons from the pump are symmetrically up- and down-converted in the Fourier domain. This phenomenon, also referred to as parametric fluorescence, can only be understood and analyzed from a fully quantum perspective as a consequence of the coupling between the field of the central (pumped) mode and the vacuum fluctuations of the various side modes. We analytically calculate the power spectra of the spontaneous emission noise, and we show that these spectra can be either single- or double-peaked depending on the value of the laser frequency, chromatic dispersion, pump power, and spectral distance between the central mode and the side mode of interest. We also calculate as well the overall spontaneous noise power per side mode and propose simplified analytical expressions for some particular cases. In the configuration where the system is pumped above threshold, we investigate the phenomena of quantum correlations and multimode squeezed states of light that can occur in the Kerr frequency combs originating from stimulated four-wave mixing. We show that for all stationary spatiotemporal patterns, the side modes that are symmetrical relative to the pumped mode in the frequency domain display quantum correlations that can lead to squeezed states of light under some optimal conditions that are analytically determined. These quantum correlations can persist regardless the dynamical state of the system (rolls or solitons), regardless of the spectral extension of the comb (number side modes) and regardless of the dispersion regime (normal or anomalous). We also explicitly determine the phase quadratures leading to photon entanglement and analytically calculate their quantum-noise spectra. For both the below- and above-threshold cases, we study with particular emphasis the two principal architectures for Kerr comb generation, namely the add-through and add-drop configurations. It is found that regardless of the configuration, an essential parameter is the ratio between out-coupling and total losses, which plays a key role as it directly determines the efficiency of the detected fluorescence or squeezing spectra. We finally discuss the relevance of Kerr combs for quantum information systems at optical telecommunication wavelengths below and above threshold.
In this thesis I study in detail the quantum dynamics of several nonlinear optical processes in microresonator systems. A Heisenberg-picture input-output formalism is developed from first principles that includes the effects of scattering losses and independent quality factors and coupling ratios for different resonances. The task of calculating the device output is then reduced to solving a set of driven, damped, ordinary differential equations for the resonator mode operators alone. This theoretical framework is used to study photon pair generation via spontaneous four-wave mixing in the weakly pumped regime, on which the effects of scattering losses are appraised. A more strongly driven regime is studied for continuous wave pumps, demonstrating when self- and cross-phase modulation and multi-photon pair generation become important, and their effects on the spectral and power scaling properties of the system are examined; A detuning strategy is presented that compensates for some of these effects. The results of the weak-pump regime are applied to study microresonator-based her- alded single photon sources. The impact of scattering losses is studied, revealing that typical systems suffer from low heralding efficiency due to these losses. A technique to improve heralding efficiency is presented through over-coupling the resonator-channel system, and a resultant trade-off between heralding rate and heralding efficiency is un- covered. Limitations to the spectral purity of the heralded single photon output for conventional microresonator systems are also analysed, and a more sophisticated cou- pling scheme presented to overcome the upper bound for spectral purity of 93% that exists in typical systems, permitting the generation of single photons with spectral pu- rity arbitrarily close to 100% without spectral filtering or sophisticated phase-matching techniques. The theory of quantum frequency conversion in microresonators using four-wave mix- ing is then developed in detail, and the spectral conversion probability and conversion efficiency studied. Efficiencies exceeding 90% using less than 100 mW of pump power are predicted to be achievable with current technology. A dressed mode picture is developed to better understand the conversion dynamics. Rabi-like spectral splitting and temporal oscillations of the intraresonator mean photon number are predicted, exhibiting a novel regime of strongly coupled photonic modes.
Type II optical parametric oscillators are amongst the highest-quality sources of quantum-correlated light. In particular, when pumped above threshold, such devices generate a pair of bright orthogonally-polarized beams with strong continuous-variable entanglement. However, these sources are of limited practical use, because the entangled beams emerge with different frequencies and a diffusing phase difference. It has been proven that the use of an internal wave-plate coupling the modes with orthogonal polarization is capable of locking the frequencies of the emerging beams to half the pump frequency, as well as reducing the phase-difference diffusion, at the expense of reducing the entanglement levels. In this work we characterize theoretically an alternative locking mechanism: the injection of a laser at half the pump frequency. Apart from being less invasive, this method should allow for an easier real-time experimental control. We show that such an injection is capable of generating the desired phase locking between the emerging beams, while still allowing for large levels of entanglement. Moreover, we find an additional region of the parameter space (at relatively large injections) where a mode with well defined polarization is in a highly amplitude-squeezed state.
Type II optical parametric oscillators are amongst the highest-quality sources of quantum-correlated light. In particular, when pumped above threshold, such devices generate a pair of bright orthogonally-polarized beams with strong continuous-variable entanglement. However, these sources are of limited practical use, because the entangled beams emerge with different frequencies and a diffusing phase difference. It has been proven that the use of an internal wave-plate coupling the modes with orthogonal polarization is capable of locking the frequencies of the emerging beams to half the pump frequency, as well as reducing the phase-difference diffusion, at the expense of reducing the entanglement levels. In this work we characterize theoretically an alternative locking mechanism: the injection of a laser at half the pump frequency. Apart from being less invasive, this method should allow for an easier real-time experimental control. We show that such an injection is capable of generating the desired phase locking between the emerging beams, while still allowing for large levels of entanglement. Moreover, we find an additional region of the parameter space (at relatively large injections) where a mode with well defined polarization is in a highly amplitude-squeezed state.