Color symmetry breaking in a nonlinear optical microcavity

TL;DR

The paper demonstrates color symmetry breaking in bichromatically driven Kerr microresonators, revealing a Kerr-XPM–driven power imbalance between nondegenerate cavity modes with a threshold near $19$ mW. It combines a slow-time mean-field model, including a pitchfork bifurcation ($P$), with experiments in Si$_3$N$_4$ microrings to show both symmetric and symmetry-broken states and the influence of intrinsic asymmetries. Beyond fundamental interest, the work shows Kerr-based activation functions—sigmoid, quadratic, and LeakyReLU—that can be tuned via an auxiliary control pump, enabling high-speed, on-chip neuromorphic processing and multi-channel (wavelength-division) computing. The results point toward broadband, multicolor SSB and functional photonic circuits leveraging Kerr nonlinearities for integrated optical computation.

Abstract

Spontaneous symmetry breaking leads to diverse phenomena across the natural sciences, from the Higgs mechanism in particle physics to superconductors and collective animal behavior. In photonic systems, the symmetry of light states can be broken when two optical fields interact through the Kerr nonlinearity, as shown in early demonstrations with counterpropagating and cross-polarized modes. Here, we report the first observation of color symmetry breaking in an integrated silicon nitride microring, where spontaneous power imbalance arises between optical mode at different wavelengths, mediated by the Kerr effect. The threshold power for this effect is as low as 19 mW. By examining the system's homogeneous states, we further demonstrate a Kerr-based nonlinear activation-function generator that produces sigmoid-, quadratic-, and leaky-ReLU-like responses. These findings reveal previously unexplored nonlinear dynamics in dual-pumped Kerr resonators and establish new pathways towards compact, all-optical neuromorphic circuits.

Figures (4)

• Figure 1: (a)Mechanisms of light-field spontaneous symmetry breaking (SSB) in Kerr microresonators. (a.i) Counterpropagating SSB between clockwise (CW) and counter-clockwise (CCW) modes. (a.ii) Cross-polarized SSB between orthogonal polarization modes $E_+$ and $E_-$. (a.iii) Color SSB between two modes at different frequencies, where dual pumping leads to an intracavity power imbalance. Inset: Experimental observation of color SSB. Red (blue) traces show the measured transmission of the low- (high-) frequency pumps when both lasers are simultaneously scanned across their respective resonances. (b)Broadband SSB: symmetry breaking can arise from the interaction of light in any pair of non-degenrate cavity resonances. No SSB occurs when both pumps excite the same resonance (diagonal terms). (c)Multicolor SSB: Numerical simulations of eight simultaneously pumped resonances showing high-dimensional SSB when the pump detunings are synchronously scanned, giving rise to a symmetry-broken comb of frequencies. In the top-left panel, the black curve represents the symmetric solution branch. Once the SSB initiates, individual frequency modes (colored curves) undergo either an increase or a reduction in intracavity intensity, as depicted in the heatmap in the bottom panel. The right panel displays the intracavity-intensity profiles across the spectrum, extracted at selected detuning values indicated by the dashed black lines in the colormap.
• Figure 2: (a) Schematic of the experimental setup. EDFA: erbium-doped fiber amplifier, PD: photodetector. (b) Color symmetry breaking threshold. Experimental (i) and theoretical (ii, iii) power imbalance between the two propagating modes ($a_\mathrm{blue}$ corresponding to a pump at 1571 and $a_\mathrm{red}$ corresponding to a pump at 1577) as a function of input power. The shaded region marks the regime dominated by the cavity’s linear response, while the white region highlights the region of Kerr-induced rapid enhancement of power imbalance. In (i), scattered data points are extracted from the measured transmission, and the dashed line indicates the linear asymptote of the high-power response. In (ii) and (iii), numerical results for the symmetric and asymmetric models, respectively, illustrate the contributions from the linear response alone and from Kerr-induced XPM. (c) Experimental color symmetry breaking traces as a function of detuning for increasing optical power coupled into the bus waveguide ($P_{in})$ (i–iv), as indicated. The intensity of symmetry breaking increases with power. (d) Numerical reproduction of (c) based on the asymmetric model. The normalized input powers $F$ are shown in the plots (see Methods).
• Figure 3: (a) Bifurcation diagram of color symmetry breaking. Intracavity power is plotted as a function of the laser detuning. Black: symmetric solutions. Red and blue: symmetry-broken solutions, where either $a_\mathrm{red}$ or $a_\mathrm{blue}$ dominates. P: pitchfork bifurcation. S: saddle-node bifurcation. Solid (dashed) lines denote stable (unstable) branches. Insets: (i) Symmetric regime, where both lasers are equally detuned from their respective resonances; (ii) symmetry-broken regime, where one resonance is pulled closer to its laser while the other is shifted away. (b, c) Experimental observation of color symmetry breaking, showing deterministic flipping of the dominant mode when the red laser frequency is reduced by 195. (d, e) Numerical reproductions of (b) and (c) based on the asymmetric model (see Methods).
• Figure 4: (a) Experimental transmission traces of the probe laser show a variety of nonlinear responses as a function of the detuning. Different colors correspond to responses obtained by changing the control laser detuning. For clarity, the detuning axis of each trace is offset so that the end of the optical-bistability region aligns across all curves, highlighting the distinct nonlinear behaviors. The inset illustrates the Kerr-mediated interaction through which the control laser modifies the probe laser response. (b) Examples of selected responses fitted with Sigmoid, Quadratic, and LeakyReLU-type functions. (c, d) Experimental (c) and numerical (d) heatmaps of the probe laser transmission as a function of its own relative detuning and the control laser detuning offset.