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There have been several criteria for the existence of topological edge states in 1D non-Hermitian two-band sublattice-symmetric tight-binding Hamiltonians. The generalized Brillouin zone (GBZ) approach in Ref. [1, 2] uses the integration of the Berry connection over the GBZ contour in the complex wavevector space. The pole-zero approach in Ref. [3] uses algebraic properties of the off-diagonal matrix elements of the sublattice-symmetric Hamiltonian in off-diagonal form. Both correctly predict the presence or absence of edge states, but there has not been an explicit proof of their equivalence. Here we provide such an explicit proof and moreover we extend the pole-zero approach so that it also applies for sublattice-symmetric models when the Hamiltonian is not in off-diagonal form. We give numerical examples for these invariants.

The invention of the laser unleashed the potential of optical metrology, leading to numerous advancements in modern science and technology. This reliance on lasers, however, also sets a bottleneck for precision optical metrology which is complicated by sophisticated photonic infrastructure required for delicate laser-wave control, leading to limited metrology performance and significant system complexity. Here we make a key step towards resolving this challenge, by demonstrating a Pockels laser with multi-functional capability that advances the optical metrology to a new level. The chip-scale laser exhibits a narrow intrinsic linewidth down to 167 Hz and a broad mode-hop-free tuning range up to 24 GHz. In particular, it offers an unprecedented frequency chirping rate up to 20 EHz/s, and an enormous modulation bandwidth >10 GHz, both orders of magnitude larger than any existing lasers. With this laser, we are able to successfully achieve velocimetry of 40 m/s at a short distance of 0.4 m, with a measurable velocity up to the first cosmic velocity at 1 m away, that is inaccessible by conventional ranging approaches, and distance metrology with a ranging resolution of <2 cm. Moreover, for the first time to the best of our knowledge, we are able to realize a dramatically simplified architecture for laser frequency stabilization, by direct locking the laser to an external reference gas cell without any extra external light control. We successfully achieve a long-term laser stability with a frequency fluctuation of only $\pm$ 6.5 MHz over 60 minutes. The demonstrated Pockels laser combines elegantly high laser coherence with ultrafast frequency reconfigurability and superior multifunctional capability that we envision to have profound impacts on many areas including communication, sensing, autonomous driving, quantum information processing, and beyond., Comment: 10 pages, 4 figures

In this paper, we study the following critical fractional Schr\"odinger equation: \begin{equation} (-\Delta)^s u+V(|y'|,y'')u=K(|y'|,y'')u^{\frac{n+2s}{n-2s}},\quad u>0,\quad y =(y',y'') \in \mathbb{R}^3\times\mathbb{R}^{n-3}, \qquad(0.1)\end{equation} where $n\geq 3$, $s\in(0,1)$, $V(|y'|,y'')$ and $K(|y'|,y'')$ are two bounded nonnegative potential functions. Under the conditions that $K(r,y'')$ has a stable critical point $(r_0,y_0'')$ with $r_0>0$, $K(r_0,y_0'')>0$ and $V(r_0,y_0'')>0$, we prove that equation (0.1) has a new type of infinitely many solutions that concentrate at points lying on the top and the bottom of a cylinder. In particular, the bubble solutions can concentrate at a pair of symmetric points with respect to the origin. Our proofs make use of a modified finite-dimensional reduction method and local Pohozaev identities., Comment: arXiv admin note: substantial text overlap with arXiv:2307.02272 by other authors

Non-Abelian gauge fields provide a conceptual framework for the description of particles having spins. The theoretical importance of non-Abelian gauge fields motivates their experimental synthesis and explorations. Here, we demonstrate non-Abelian lattice gauge fields for photons. In the study of gauge fields, lattice models are essential for the understanding of their implications in extended systems. We utilize the platform of synthetic frequency dimensions, which enables the study of lattice physics in a scalable and programmable way. We observe Dirac cones at time-reversal-invariant momenta as well as the direction reversal of eigenstate trajectories associated with such Dirac cones. Both of them are unique signatures of non-Abelian gauge fields in our lattice system. Our results highlight the implications of non-Abelian gauge field in the study of topological physics and suggest opportunities for the control of photon spins and pseudospins.

We prove some results on the density and multiplicity of positive solutions to the prescribed Webster scalar curvature problem on the $(2n+1)$-dimensional standard unit CR sphere $(\mathbb{S} ^{2n+1},\theta_0)$. Specifically, we construct arbitrarily many multi-bump solutions via the variational gluing method. In particular, we show the Webster scalar curvature functions of contact forms conformal to $\theta_0$ are $C^{0}$-dense among bounded functions which are positive somewhere. Existence results of infinitely many positive solutions to the related equation $-\Delta_{\mathbb{H}} u=R(\xi) u^{(n+2) /n}$ on the Heisenberg group $\Hn $ with $R(\xi)$ being asymptotically periodic with respect to left translation are also obtained. Our proofs make use of a refined analysis of bubbling behavior, gradient flow, Pohozaev identity, as well as blow up arguments.

Optical frequency division based on bulk or fiber optics provides unprecedented spectral purity for microwave oscillators. To extend the applications of this approach, the big challenges are to develop miniaturized optical frequency division oscillators without trading off phase noise performance. In this paper, we report a chip-scale electro-optical frequency division microwave oscillator with ultra-low phase noise performance. Dual laser sources are co-self-injection-locked to a single silicon nitride spiral resonator to provide a record high-stability, fully on-chip optical reference. An integrated electro-optical frequency comb based on a novel thin-film lithium niobate phase modulator chip is incorporated for the first time to perform optical-to-microwave frequency division. The resulting chip-scale photonic microwave oscillator achieves a phase noise level of -129 dBc/Hz at 10 kHz offset for 37.7 GHz carrier. The results represent a major advance in high performance, integrated photonic microwave oscillators for applications including signal processing, radar, timing, and coherent communications.

Speech super-resolution (SR) is the task that restores high-resolution speech from low-resolution input. Existing models employ simulated data and constrained experimental settings, which limit generalization to real-world SR. Predictive models are known to perform well in fixed experimental settings, but can introduce artifacts in adverse conditions. On the other hand, generative models learn the distribution of target data and have a better capacity to perform well on unseen conditions. In this study, we propose a novel two-stage approach that combines the strengths of predictive and generative models. Specifically, we employ a diffusion-based model that is conditioned on the output of a predictive model. Our experiments demonstrate that the model significantly outperforms single-stage counterparts and existing strong baselines on benchmark SR datasets. Furthermore, we introduce a repainting technique during the inference of the diffusion process, enabling the proposed model to regenerate high-frequency components even in mismatched conditions. An additional contribution is the collection of and evaluation on real SR recordings, using the same microphone at different native sampling rates. We make this dataset freely accessible, to accelerate progress towards real-world speech super-resolution.

We present the viewpoint of treating one-dimensional band structures as Riemann surfaces, linking the unique properties of non-Hermiticity to the geometry and topology of the Riemann surface. Branch cuts and branch points play a significant role when this viewpoint is applied to both the open-boundary spectrum and the braiding structure. An open-boundary spectrum is interpreted as branch cuts connecting certain branch points, and its consistency with the monodromy representation severely limits its possible morphology. A braid word for the Brillouin zone can be read off from its intersections with branch cuts, and its crossing number is given by the winding number of the discriminant. These results open new avenues to generate important insights into the physical behaviors of non-Hermitian systems., Comment: 13 pages, 6 figures

Ultra-low-noise laser sources are crucial for a variety of applications, including microwave synthesizers, optical gyroscopes, and the manipulation of quantum systems. Silicon photonics has emerged as a promising solution for high-coherence applications due to its ability to reduce system size, weight, power consumption, and cost (SWaP-C). Semiconductor lasers based on self-injection locking (SIL) have reached fiber laser coherence, but typically require a high-Q external cavity to suppress coherence collapse through frequency-selective feedback. Lasers based on external-cavity locking (ECL) are a low-cost and turnkey operation option, but their coherence is generally inferior to SIL lasers. In this work, we demonstrate quantum-dot (QD) lasers grown directly on Si that achieve SIL laser coherence under turnkey ECL. The high-performance QD laser offers a scalable and low-cost heteroepitaxial integration platform. Moreover, the QD laser's chaos-free nature enables a 16 Hz Lorentzian linewidth under ECL using a low-Q external cavity, and improves the frequency noise by an additional order of magnitude compared to conventional quantum-well lasers.

Accurately detecting voiced intervals in speech signals is a critical step in pitch tracking and has numerous applications. While conventional signal processing methods and deep learning algorithms have been proposed for this task, their need to fine-tune threshold parameters for different datasets and limited generalization restrict their utility in real-world applications. To address these challenges, this study proposes a supervised voicing detection model that leverages recorded laryngograph data. The model is based on a densely-connected convolutional recurrent neural network (DC-CRN), and trained on data with reference voicing decisions extracted from laryngograph data sets. Pretraining is also investigated to improve the generalization ability of the model. The proposed model produces robust voicing detection results, outperforming other strong baseline methods, and generalizes well to unseen datasets. The source code of the proposed model with pretraining is provided along with the list of used laryngograph datasets to facilitate further research in this area.

Speech enhancement aims to improve the quality of speech signals in terms of quality and intelligibility, and speech editing refers to the process of editing the speech according to specific user needs. In this paper, we propose a Unified Speech Enhancement and Editing (uSee) model with conditional diffusion models to handle various tasks at the same time in a generative manner. Specifically, by providing multiple types of conditions including self-supervised learning embeddings and proper text prompts to the score-based diffusion model, we can enable controllable generation of the unified speech enhancement and editing model to perform corresponding actions on the source speech. Our experiments show that our proposed uSee model can achieve superior performance in both speech denoising and dereverberation compared to other related generative speech enhancement models, and can perform speech editing given desired environmental sound text description, signal-to-noise ratios (SNR), and room impulse responses (RIR). Demos of the generated speech are available at https://muqiaoy.github.io/usee.

Diffusion models have gained attention in speech enhancement tasks, providing an alternative to conventional discriminative methods. However, research on target speech extraction under multi-speaker noisy conditions remains relatively unexplored. Moreover, the superior quality of diffusion methods typically comes at the cost of slower inference speed. In this paper, we introduce the Discriminative Diffusion model for Target Speech Extraction (DDTSE). We apply the same forward process as diffusion models and utilize the reconstruction loss similar to discriminative methods. Furthermore, we devise a two-stage training strategy to emulate the inference process during model training. DDTSE not only works as a standalone system, but also can further improve the performance of discriminative models without additional retraining. Experimental results demonstrate that DDTSE not only achieves higher perceptual quality but also accelerates the inference process by 3 times compared to the conventional diffusion model., Comment: Accepted by SLT2024

Enhancing speech signal quality in adverse acoustic environments is a persistent challenge in speech processing. Existing deep learning based enhancement methods often struggle to effectively remove background noise and reverberation in real-world scenarios, hampering listening experiences. To address these challenges, we propose a novel approach that uses pre-trained generative methods to resynthesize clean, anechoic speech from degraded inputs. This study leverages pre-trained vocoder or codec models to synthesize high-quality speech while enhancing robustness in challenging scenarios. Generative methods effectively handle information loss in speech signals, resulting in regenerated speech that has improved fidelity and reduced artifacts. By harnessing the capabilities of pre-trained models, we achieve faithful reproduction of the original speech in adverse conditions. Experimental evaluations on both simulated datasets and realistic samples demonstrate the effectiveness and robustness of our proposed methods. Especially by leveraging codec, we achieve superior subjective scores for both simulated and realistic recordings. The generated speech exhibits enhanced audio quality, reduced background noise, and reverberation. Our findings highlight the potential of pre-trained generative techniques in speech processing, particularly in scenarios where traditional methods falter. Demos are available at https://whmrtm.github.io/SoundResynthesis., Comment: Paper in submission

In this work, we address the challenge of encoding speech captured by a microphone array using deep learning techniques with the aim of preserving and accurately reconstructing crucial spatial cues embedded in multi-channel recordings. We propose a neural spatial audio coding framework that achieves a high compression ratio, leveraging single-channel neural sub-band codec and SpatialCodec. Our approach encompasses two phases: (i) a neural sub-band codec is designed to encode the reference channel with low bit rates, and (ii), a SpatialCodec captures relative spatial information for accurate multi-channel reconstruction at the decoder end. In addition, we also propose novel evaluation metrics to assess the spatial cue preservation: (i) spatial similarity, which calculates cosine similarity on a spatially intuitive beamspace, and (ii), beamformed audio quality. Our system shows superior spatial performance compared with high bitrate baselines and black-box neural architecture. Demos are available at https://xzwy.github.io/SpatialCodecDemo. Codes and models are available at https://github.com/XZWY/SpatialCodec., Comment: Accepted by ICASSP2024

Frequency-modulated (FM) laser combs, which offer a periodic quasi-continuous-wave output and a flat-topped optical spectrum, are emerging as a promising solution for wavelength-division multiplexing applications, precision metrology, and ultrafast optical ranging. The generation of FM combs relies on spatial hole burning, group velocity dispersion (GVD), Kerr nonlinearity, and four-wave mixing (FWM). While FM combs have been widely observed in quantum cascade Fabry-Perot (FP) lasers, the requirement for a low-dispersion FP cavity can be a challenge in platforms where the waveguide dispersion is mainly determined by the material. Here we report a 60 GHz quantum-dot (QD) mode-locked laser in which both the amplitude-modulated (AM) and the FM comb can be generated independently. The high FWM efficiency of -5 dB allows the QD laser to generate an FM comb efficiently. We also demonstrate that the Kerr nonlinearity can be practically engineered to improve the FM comb bandwidth without the need for GVD engineering. The maximum 3-dB bandwidth that our QD platform can deliver is as large as 2.2 THz. This study gives novel insights into the improvement of FM combs and paves the way for small-footprint, electrically-pumped, and energy-efficient frequency combs for silicon photonic integrated circuits (PICs).

Convolution is an essential operation in signal and image processing and consumes most of the computing power in convolutional neural networks. Photonic convolution has the promise of addressing computational bottlenecks and outperforming electronic implementations. Performing photonic convolution in the synthetic frequency dimension, which harnesses the dynamics of light in the spectral degrees of freedom for photons, can lead to highly compact devices. Here we experimentally realize convolution operations in the synthetic frequency dimension. Using a modulated ring resonator, we synthesize arbitrary convolution kernels using a pre-determined modulation waveform with high accuracy. We demonstrate the convolution computation between input frequency combs and synthesized kernels. We also introduce the idea of an additive offset to broaden the kinds of kernels that can be implemented experimentally when the modulation strength is limited. Our work demonstrate the use of synthetic frequency dimension to efficiently encode data and implement computation tasks, leading to a compact and scalable photonic computation architecture., Comment: Science Advances, in press

Soliton microcombs are helping to advance the miniaturization of a range of comb systems. These combs mode lock through the formation of short temporal pulses in anomalous dispersion resonators. Here, a new microcomb is demonstrated that mode locks through the formation of pulse pairs in normal-dispersion coupled-ring resonators. Unlike conventional microcombs, pulses in this system cannot exist alone, and instead must phase lock in pairs to form a bright soliton comb. Also, the pulses can form at recurring spectral windows and the pulses in each pair feature different optical spectra. This pairwise mode-locking modality extends to higher dimensions and we demonstrate 3-ring systems in which 3 pulses mode lock through alternating pairwise pulse coupling. The results are demonstrated using the new CMOS-foundry platform that has not previously produced bright solitons on account of its inherent normal dispersion. The ability to generate multi-color pulse pairs over multiple rings is an important new feature for microcombs. It can extend the concept of all-optical soliton buffers and memories to multiple storage rings that multiplex pulses with respect to soliton color and that are spatially addressable. The results also suggest a new platform for the study of quantum combs and topological photonics.

Normal group velocity dispersion (GVD) microcombs offer high comb line power and high pumping efficiency compared to bright pulse microcombs. The recent demonstration of normal GVD microcombs using CMOS-foundry-produced microresonators is an important step towards scalable production. However, the chromatic dispersion of CMOS devices is large and impairs generation of broadband microcombs. Here, we report the development of a microresonator in which GVD is reduced due to a couple-ring resonator configuration. Operating in the turnkey self-injection-locking mode, the resonator is hybridly integrated with a semiconductor laser pump to produce high-power-efficiency combs spanning a bandwidth of 9.9 nm (1.22 THz) centered at 1560 nm, corresponding to 62 comb lines. Fast, linear optical sampling of the comb waveform is used to observe the rich set of near-zero GVD comb behaviors, including soliton molecules, switching waves (platicons) and their hybrids. Tuning of the 20 GHz repetition rate by electrical actuation enables servo locking to a microwave reference, which simultaneously stabilizes the comb repetition rate, offset frequency and temporal waveform. This hybridly integrated system could be used in coherent communications or for ultra-stable microwave signal generation by two-point optical frequency division., Comment: 8 pages, 4 figures

While photonic integrated circuits (PICs) are being widely used in applications such as telecommunications and datacenter interconnects, PICs capable of replacing bulk optics and fibers in high-precision, highly-coherent applications will require ultra-low-noise laser sources to be integrated with other photonic components in a compact and robustly aligned format -- that is, on a single chip. Such PICs could offer superior scalability for complex functionalities and volume production, as well as improved stability and reliability over time. However, there are two major issues preventing the realization of such envisioned PICs: the high phase noise of semiconductor lasers, and the difficulty of integrating optical isolators directly on chip. PICs are still considered as inferior solutions in optical systems such as microwave synthesizers, optical gyroscopes and atomic clocks, despite their advantages in size, weight, power consumption and cost (SWaPC). Here, we challenge this convention by introducing three-dimensional (3D) integration in silicon photonics that results in ultra-low-noise, isolator-free PICs. Through multiple monolithic and heterogeneous processing sequences, direct on-chip integration of III-V gain and ultra-low-loss (ULL) silicon nitride (SiN) waveguides with optical loss around 0.5 dB/m are demonstrated. Consequently, the demonstrated PIC enters a new regime, such that an integrated ultra-high-Q cavity reduces the laser noise close to that of fiber lasers. Moreover, the cavity acts as an effective block for any downstream on-chip or off-chip reflection-induced destabilization, thus eliminating the need for optical isolators. We further showcase isolator-free, widely-tunable, low-noise, heterodyne microwave generation using two ultra-low-noise lasers on the same silicon chip.

We present a novel single-shot interferometric ToF camera targeted for precise 3D measurements of dynamic objects. The camera concept is based on Synthetic Wavelength Interferometry, a technique that allows retrieval of depth maps of objects with optically rough surfaces at submillimeter depth precision. In contrast to conventional ToF cameras, our device uses only off-the-shelf CCD/CMOS detectors and works at their native chip resolution (as of today, theoretically up to 20 Mp and beyond). Moreover, we can obtain a full 3D model of the object in single-shot, meaning that no temporal sequence of exposures or temporal illumination modulation (such as amplitude or frequency modulation) is necessary, which makes our camera robust against object motion. In this paper, we introduce the novel camera concept and show first measurements that demonstrate the capabilities of our system. We present 3D measurements of small (cm-sized) objects with > 2 Mp point cloud resolution (the resolution of our used detector) and up to sub-mm depth precision. We also report a "single-shot 3D video" acquisition and a first single-shot "Non-Line-of-Sight" measurement. Our technique has great potential for high-precision applications with dynamic object movement, e.g., in AR/VR, industrial inspection, medical imaging, and imaging through scattering media like fog or human tissue.

This paper is devoted to establishing some results on the density and multiplicity of solutions to the fractional Nirenberg problem which is equivalent to studying the conformally invariant equation $P_\sigma(v)=K v^{\frac{n+2\sigma}{n-2\sigma}}$ on the standard unit sphere $(\mathbb{S}^n,g_0)$ with $\sigma\in (0,1)$ and $n\geq 2$, where $P_\sigma$ is the intertwining operator of order $2\sigma$ and $K$ is the prescribed curvature function. More specifically, by using the variational gluing method, refined analysis of bubbling behavior, extension formula, as well as the blow up analysis arguments, we obtain the existence of infinitely many multi-bump solutions. In particular, we show the smooth curvature functions of metrics conformal to $g_0$ are dense in the $C^{0}$ topology. Moreover, the related fractional Laplacian equations $(-\Delta)^{\sigma} u=K(x) u^{\frac{n+2\sigma}{n-2\sigma}}$ in $\mathbb{R}^n$, with $K(x)$ being asymptotically periodic in one of the variables, are also studied and infinitely many solutions are obtained under natural flatness assumptions.

In this paper we study the problem of prescribing fractional $Q$-curvature of order $2\sigma$ for a conformal metric on the standard sphere $\Sn$ with $\sigma\in (0,n/2)$ and $n\geq2$. Compactness and existence results are obtained in terms of the flatness order $\beta$ of the prescribed curvature function $K$. Making use of integral representations and perturbation result, we develop a unified approach to obtain these results when $\beta\in [n-2\sigma,n)$ for all $\sigma\in (0,n/2)$. This work generalizes the corresponding results of Jin-Li-Xiong [Math. Ann. 369: 109--151, 2017] for $\beta\in (n-2\sigma,n)$., Comment: 30 pages

High coherence visible and near-visible laser sources are centrally important to the operation of advanced position/navigation/timing systems as well as classical/quantum sensing systems. However, the complexity and size of these bench-top lasers is an impediment to their transitioning beyond the laboratory. Here, a system-on-a-chip that emits high-coherence visible and near-visible lightwaves is demonstrated. The devices rely upon a new approach wherein wavelength conversion and coherence increase by self-injection-locking are combined within in a single nonlinear resonator. This simplified approach is demonstrated in a hybridly-integrated device and provides a short-term linewidth around 10-30 kHz. On-chip, converted optical power over 2 mW is also obtained. Moreover, measurements show that heterogeneous integration can result in conversion efficiency higher than 25% with output power over 11 mW. Because the approach uses mature III-V pump lasers in combination with thin-film lithium niobate, it can be scaled for low-cost manufacturing of high-coherence visible emitters. Also, the coherence generation process can be transferred to other frequency conversion processes including optical parametric oscillation, sum/difference frequency generation, and third-harmonic generation.

Stimulated Brillouin scattering provides optical gain for efficient and narrow-linewidth lasers in high-Q microresonator systems. However, the thermal dependence of the Brillouin process, as well as the microresonator, impose strict temperature control requirements for long-term frequency-stable operation. Here, we study Brillouin back action and use it to both measure and phase-sensitively lock modal temperature to a reference temperature defined by the Brillouin phase-matching condition. At a specific lasing wavelength, the reference temperature can be precisely set by adjusting resonator free spectral range. This backaction control method is demonstrated in a chip-based Brillouin laser, but can be applied in all Brillouin laser platforms. It offers a new approach for frequency-stable operation of Brillouin lasers in atomic clock, frequency metrology, and gyroscope applications.

The development of integrated semiconductor lasers has miniaturized traditional bulky laser systems, enabling a wide range of photonic applications. A progression from pure III-V based lasers to III-V/external cavity structures has harnessed low-loss waveguides in different material systems, leading to significant improvements in laser coherence and stability. Despite these successes, however, key functions remain absent. In this work, we address a critical missing function by integrating the Pockels effect into a semiconductor laser. Using a hybrid integrated III-V/Lithium Niobate structure, we demonstrate several essential capabilities that have not existed in previous integrated lasers. These include a record-high frequency modulation speed of 2 exahertz/s (2.0$\times$10$^{18}$ Hz/s) and fast switching at 50 MHz, both of which are made possible by integration of the electro-optic effect. Moreover, the device co-lases at infrared and visible frequencies via the second-harmonic frequency conversion process, the first such integrated multi-color laser. Combined with its narrow linewidth and wide tunability, this new type of integrated laser holds promise for many applications including LiDAR, microwave photonics, atomic physics, and AR/VR.