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We have demonstrated, for the first time to our knowledge, self-injection locking of a distributed feedback diode laser to a multimode 4H-silicon carbide (4H-SiC) microring resonator, which is also used for the observation of resonant opto-mechanical oscillation in the cavity modes. While the fundamental transverse-electric mode family of the silicon carbide microring was optically pumped, Stokes light was generated in the adjacent fundamental transverse-magnetic resonant mode. The threshold of the process did not exceed 5 mW of light entering the cavity characterized by a loaded optical quality factor of 2 × 106. These results mark a significant milestone in unlocking the potential of 4H-SiC through turnkey soliton microcomb generation and empowering future advancements in areas such as cavity optomechanics using this versatile and quantum-friendly material platform. 

Dichromatic coherent pumping of optical microresonators possessing broadband Kerr nonlinearity reveals a rich variety of nonlinear and critical phenomena, including time‐translation symmetry breaking, multistability, and chaos. This pumping scheme has garnered significant interest owing to its practical applications arising from optical‐to‐microwave frequency stability transfer and frequency‐division phase noise reduction. In this work, a generalized analytical model is developed to elucidate the dynamics of dissipative Kerr cavity solitons as particles in dual‐frequency pumped Kerr cavities. The model offers quantitative and intuitive insight into the behavior of the system both within and beyond the locking range wherein solitons synchronize to the two driving lasers. It also predicts new phenomena and expounds hitherto unexplained experimental observations covering bistability, hysteresis, and Arnold tongues. The unified framework simultaneously delineates the parameter space for the existence of dissipative discrete time crystals recently reported in dually pumped optical solitons. Bridging a fundamental study with practical applications, this work offers valuable understanding of two‐point injection‐locked microcombs for realizing chip‐scale optical clocks and superior microwave photonic oscillators. 

Optical microresonators possessing Kerr-type nonlinearity have emerged over the past decade as reliable and versatile sources of optical frequency combs, with varied applications including in the generation of low-phasenoise radio frequency (RF) signals, small-footprint precision timekeeping, and LiDAR. One of the key parameters affecting Kerr microcomb generation in different wavelength ranges is cavity modal dispersion. Dispersion effects such as avoided mode crossings (AMCs) have been shown to greatly limit mode-locked microcomb generation, especially when pumping in close proximity to such disruptions. We present numerical modeling and experimental evidence demonstrating that using an auxiliary laser pump can suppress the detrimental impact of near-pump AMCs. We also report, for the first time to our knowledge, the possibility of the breaking of characteristic soliton steps into two stable branches corresponding to different stable pulse trains arising from the interplay of dichromatic pumping and AMCs. These findings bear significance, particularly for the generation of frequency combs in larger resonators or at smaller wavelengths, such as the visible range, where the cavities become overmoded. 

We study convex empirical risk minimization for high-dimensional inference in binary linear classification under both discriminative binary linear models, as well as generative Gaussian-mixture models. Our first result sharply predicts the statistical performance of such estimators in the proportional asymptotic regime under isotropic Gaussian features. Importantly, the predictions hold for a wide class of convex loss functions, which we exploit to prove bounds on the best achievable performance. Notably, we show that the proposed bounds are tight for popular binary models (such as signed and logistic) and for the Gaussian-mixture model by constructing appropriate loss functions that achieve it. Our numerical simulations suggest that the theory is accurate even for relatively small problem dimensions and that it enjoys a certain universality property.