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arXiv:2606.12749

Discrete phase symmetry of stationary states in bichromatically pumped Kerr microresonators

Boulat N. Nougmanov

Published Jun 10, 2026• 0 citations
arXivHTMLPDF

Abstract

Bichromatically pumped Kerr microresonators exhibit phase bistability and multistability arising from four-wave mixing between pump and generated modes. Here we analyze the symmetry structure of stationary solutions in the coupled-mode description of such systems. We show that the two-pump coupled-mode equations are a particular case of a more general model with pumps placed at equally spaced modes. For this model, any stationary solution generates a finite family of stationary solutions through a discrete phase transformation. This transformation leaves the equations invariant and preserves the stability type of the corresponding stationary states. As a consequence, the possible stationary values of each individual mode form a regular polygon in the complex plane. The number of vertices is determined by the order of the mode index $μ$ in the group $\mathbb{Z}_n$, where $n$ is the separation between the pumped modes. Thus, the phase multistability structure depends not only on nonlinear dynamics but also on the arithmetic relation between the mode index and the pump separation. These results provide a simple symmetry-based explanation for the discrete phase structure observed in multimode Kerr resonators under bichromatic pumping.

DOI
10.48550/arxiv.2606.12749

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We elaborate a comprehensive theory of the sharp variations of the four-wave-mixing (FWM) threshold happening with the tuning of the pump frequency along the dispersive tails of the nonlinear resonance in the driven Kerr microresonators with high finesses and high finesse dispersions. Our theory leads to the explicit estimates for the difference in the pump powers required for the excitation of one sideband pair with momenta $\ifmmode\pm\else\textpm\fi{}\ensuremath{\mu}$ and for the simultaneous excitation of the two neighboring pairs with momenta $\ifmmode\pm\else\textpm\fi{}\ensuremath{\mu}, \ifmmode\pm\else\textpm\fi{}(\ensuremath{\mu}+1)$. This power difference also measures the depth of the Arnold tongues in the pump-frequency and pump-power parameter space, associated with the aforementioned power variations. A set of select pump frequencies and powers, where the instabilities of the two sideband pairs come first, forms the threshold of complexity, which is followed by a sequence of conditions specifying critical powers for the simultaneous instabilities of three, four and so on pairs. We formally link finesse dispersion and the density of states notion borrowed from the condensed matter context, and report a value of the finesse dispersion and a critical mode number signaling the transition from the low- to the high-contrast tongue structure. We also demonstrate that the large finesse dispersion makes possible a surprising for the multimode resonators regime of the bistability without FWM.
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.
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