Dirac mass induced by optical gain and loss.

Mass is commonly considered an intrinsic property of matter, but modern physics reveals particle masses to have complex origins1, such as the Higgs mechanism in high-energy physics2,3. In crystal lattices such as graphene, relativistic Dirac particles can exist as low-energy quasiparticles4 with masses imparted by lattice symmetry-breaking perturbations5-8. These mass-generating mechanisms all assume Hermiticity, or the conservation of energy in detail. Using a photonic synthetic lattice, we show experimentally that Dirac masses can be generated by means of non-Hermitian perturbations based on optical gain and loss. We then explore how the spacetime engineering of the gain and loss-induced Dirac mass affects the quasiparticles. As we show, the quasiparticles undergo Klein tunnelling at spatial boundaries, but a local breaking of a non-Hermitian symmetry can produce a new flux non-conservation effect at the domain walls. At a temporal boundary that abruptly flips the sign of the Dirac mass, we observe a variant of the time-reflection phenomenon: in the non-relativistic limit, the Dirac quasiparticle reverses its velocity, whereas in the relativistic limit, the original velocity is retained.

Topological Chern vectors in three-dimensional photonic crystals.

The paradigmatic example of a topological phase of matter, the two-dimensional Chern insulator1-5, is characterized by a topological invariant consisting of a single integer, the scalar Chern number. Extending the Chern insulator phase from two to three dimensions requires generalization of the Chern number to a three-vector6,7, similar to the three-dimensional (3D) quantum Hall effect8-13. Such Chern vectors for 3D Chern insulators have never been explored experimentally. Here we use magnetically tunable 3D photonic crystals to achieve the experimental demonstration of Chern vectors and their topological surface states. We demonstrate Chern vector magnitudes of up to six, higher than all scalar Chern numbers previously realized in topological materials. The isofrequency contours formed by the topological surface states in the surface Brillouin zone form torus knots or links, whose characteristic integers are determined by the Chern vectors. We demonstrate a sample with surface states forming a (2, 2) torus link or Hopf link in the surface Brillouin zone, which is topologically distinct from the surface states of other 3D topological phases. These results establish the Chern vector as an intrinsic bulk topological invariant in 3D topological materials, with surface states possessing unique topological characteristics.

Electrically pumped topological laser with valley edge modes.

Quantum cascade lasers are compact, electrically pumped light sources in the technologically important mid-infrared and terahertz region of the electromagnetic spectrum1,2. Recently, the concept of topology3 has been expanded from condensed matter physics into photonics4, giving rise to a new type of lasing5-8 using topologically protected photonic modes that can efficiently bypass corners and defects4. Previous demonstrations of topological lasers have required an external laser source for optical pumping and have operated in the conventional optical frequency regime5-8. Here we demonstrate an electrically pumped terahertz quantum cascade laser based on topologically protected valley edge states9-11. Unlike topological lasers that rely on large-scale features to impart topological protection, our compact design makes use of the valley degree of freedom in photonic crystals10,11, analogous to two-dimensional gapped valleytronic materials12. Lasing with regularly spaced emission peaks occurs in a sharp-cornered triangular cavity, even if perturbations are introduced into the underlying structure, owing to the existence of topologically protected valley edge states that circulate around the cavity without experiencing localization. We probe the properties of the topological lasing modes by adding different outcouplers to the topological cavity. The laser based on valley edge states may open routes to the practical use of topological protection in electrically driven laser sources.

Interfacial Cherenkov radiation from ultralow-energy electrons.

Cherenkov radiation occurs only when a charged particle moves with a velocity exceeding the phase velocity of light in that matter. This radiation mechanism creates directional light emission at a wide range of frequencies and could facilitate the development of on-chip light sources except for the hard-to-satisfy requirement for high-energy particles. Creating Cherenkov radiation from low-energy electrons that has no momentum mismatch with light in free space is still a long-standing challenge. Here, we report a mechanism to overcome this challenge by exploiting a combined effect of interfacial Cherenkov radiation and umklapp scattering, namely the constructive interference of light emission from sequential particle-interface interactions with specially designed (umklapp) momentum-shifts. We find that this combined effect is able to create the interfacial Cherenkov radiation from ultralow-energy electrons, with kinetic energies down to the electron-volt scale. Due to the umklapp scattering for the excited high-momentum Bloch modes, the resulting interfacial Cherenkov radiation is uniquely featured with spatially separated apexes for its wave cone and group cone.

A programmable topological photonic chip.

Controlling topological phases of light allows the observation of abundant topological phenomena and the development of robust photonic devices. The prospect of more sophisticated control with topological photonic devices for practical implementations requires high-level programmability. Here we demonstrate a fully programmable topological photonic chip with large-scale integration of silicon photonic nanocircuits and microresonators. Photonic artificial atoms and their interactions in our compound system can be individually addressed and controlled, allowing the arbitrary adjustment of structural parameters and geometrical configurations for the observation of dynamic topological phase transitions and diverse photonic topological insulators. Individual programming of artificial atoms on the generic chip enables the comprehensive statistical characterization of topological robustness against relatively weak disorders, and counterintuitive topological Anderson phase transitions induced by strong disorders. This generic topological photonic chip can be rapidly reprogrammed to implement multifunctionalities, providing a flexible and versatile platform for applications across fundamental science and topological technologies.

Realization of a three-dimensional photonic topological insulator.

Confining photons in a finite volume is highly desirable in modern photonic devices, such as waveguides, lasers and cavities. Decades ago, this motivated the study and application of photonic crystals, which have a photonic bandgap that forbids light propagation in all directions1-3. Recently, inspired by the discoveries of topological insulators4,5, the confinement of photons with topological protection has been demonstrated in two-dimensional (2D) photonic structures known as photonic topological insulators6-8, with promising applications in topological lasers9,10 and robust optical delay lines11. However, a fully three-dimensional (3D) topological photonic bandgap has not been achieved. Here we experimentally demonstrate a 3D photonic topological insulator with an extremely wide (more than 25 per cent bandwidth) 3D topological bandgap. The composite material (metallic patterns on printed circuit boards) consists of split-ring resonators (classical electromagnetic artificial atoms) with strong magneto-electric coupling and behaves like a 'weak' topological insulator (that is, with an even number of surface Dirac cones), or a stack of 2D quantum spin Hall insulators. Using direct field measurements, we map out both the gapped bulk band structure and the Dirac-like dispersion of the photonic surface states, and demonstrate robust photonic propagation along a non-planar surface. Our work extends the family of 3D topological insulators from fermions to bosons and paves the way for applications in topological photonic cavities, circuits and lasers in 3D geometries.

Near-field directionality governed by asymmetric dipole-matter interactions.

Directionally molding the near-field and far-field radiation lies at the heart of nanophotonics and is crucial for applications such as on-chip information processing and chiral quantum networks. The most fundamental model for radiating structures is a dipolar source located inside homogeneous matter. However, the influence of matter on the directionality of dipolar radiation is oftentimes overlooked, especially for the near-field radiation. As background, the dipole-matter interaction is intrinsically asymmetric and does not fulfill the duality principle, originating from the inherent asymmetry of Maxwell's equations, i.e., electric charge and current density are ubiquitous but their magnetic counterparts are non-existent to elusive. We find that the asymmetric dipole-matter interaction could offer an enticing route to reshape the directionality of not only the near-field radiation but also the far-field radiation. As an example, both the near-field and far-field radiation directionality of the Huygens dipole (located close to a dielectric-metal interface) would be reversed if the dipolar position is changed from the dielectric region to the metal region.

Non-Abelian Braiding of Topological Edge Bands.

Braiding is a geometric concept that manifests itself in a variety of scientific contexts from biology to physics, and has been employed to classify bulk band topology in topological materials. Topological edge states can also form braiding structures, as demonstrated recently in a type of topological insulators known as Möbius insulators, whose topological edge states form two braided bands exhibiting a Möbius twist. While the formation of Möbius twist is inspiring, it belongs to the simple Abelian braid group B_{2}. The most fascinating features about topological braids rely on the non-Abelianness in the higher-order braid group B_{N} (N≥3), which necessitates multiple edge bands, but so far it has not been discussed. Here, based on the gauge enriched symmetry, we develop a scheme to realize non-Abelian braiding of multiple topological edge bands. We propose tight-binding models of topological insulators that are able to generate topological edge states forming non-Abelian braiding structures. Experimental demonstrations are conducted in two acoustic crystals, which carry three and four braided acoustic edge bands, respectively. The observed braiding structure can correspond to the topological winding in the complex eigenvalue space of projective translation operator, akin to the previously established point-gap winding topology in the bulk of the Hatano-Nelson model. Our Letter also constitutes the realization of non-Abelian braiding topology on an actual crystal platform, but not based on the "virtual" synthetic dimensions.

Localization of Chiral Edge States by the Non-Hermitian Skin Effect.

Quantum Hall systems host chiral edge states extending along the one-dimensional boundary of any two-dimensional sample. In solid state materials, the edge states serve as perfectly robust transport channels that produce a quantized Hall conductance; due to their chirality, and the topological protection by the Chern number of the bulk band structure, they cannot be spatially localized by defects or disorder. Here, we show experimentally that the chiral edge states of a lossy quantum Hall system can be localized. In a gyromagnetic photonic crystal exhibiting the quantum Hall topological phase, an appropriately structured loss configuration imparts the edge states' complex energy spectrum with a feature known as point-gap winding. This intrinsically non-Hermitian topological invariant is distinct from the Chern number invariant of the bulk (which remains intact) and induces mode localization via the "non-Hermitian skin effect." The interplay of the two topological phenomena-the Chern number and point-gap winding-gives rise to a non-Hermitian generalization of the paradigmatic Chern-type bulk-boundary correspondence principle. Compared to previous realizations of the non-Hermitian skin effect, the skin modes in this system have superior robustness against local defects and disorders.

Multiple Brillouin Zone Winding of Topological Chiral Edge States for Slow Light Applications.

Photonic Chern insulators are known for their topological chiral edge states (CESs), whose absolute existence is determined by the bulk band topology, but concrete dispersion can be engineered to exhibit various properties. For example, the previous theory suggested that the edge dispersion can wind many times around the Brillouin zone to slow down light, which can potentially overcome fundamental limitations in conventional slow-light devices: narrow bandwidth and keen sensitivity to fabrication imperfection. Here, we report the first experimental demonstration of this idea, achieved by coupling CESs with resonance-induced nearly flat bands. We show that the backscattering-immune hybridized CESs are significantly slowed down over a relatively broad bandwidth. Our work thus paves an avenue to broadband topological slow-light devices.

Unsupervised Data-Driven Classification of Topological Gapped Systems with Symmetries.

A remarkable breakthrough in topological phase classification is the establishment of the topological periodic table, which is mainly based on the classifying space analysis or K theory, but not based on concrete Hamiltonians that possess finite bands or arise in a lattice. As a result, it is still difficult to identify the topological phase of an arbitrary Hamiltonian; the common practice is, instead, to check the incomplete and still growing list of topological invariants one by one, very often by trial and error. Here, we develop unsupervised classifications of topological gapped systems with symmetries, and demonstrate the data-driven construction of the topological periodic table without a priori knowledge of topological invariants. This unsupervised data-driven strategy can take into account spatial symmetries, and further classify phases that were previously classified as trivial in the past. Our Letter introduces machine learning into topological phase classification and paves the way for intelligent explorations of new phases of topological matter.

Continuum of Bound States in a Non-Hermitian Model.

In a Hermitian system, bound states must have quantized energies, whereas free states can form a continuum. We demonstrate how this principle fails for non-Hermitian systems, by analyzing non-Hermitian continuous Hamiltonians with an imaginary momentum and Landau-type vector potential. The eigenstates, which we call "continuum Landau modes" (CLMs), have Gaussian spatial envelopes and form a continuum filling the complex energy plane. We present experimentally realizable 1D and 2D lattice models that host CLMs; the lattice eigenstates are localized and have other features matching the continuous model. One of these lattices can serve as a rainbow trap, whereby the response to an excitation is concentrated at a position proportional to the frequency. Another lattice can act a wave funnel, concentrating an input excitation onto a boundary over a wide frequency bandwidth. Unlike recent funneling schemes based on the non-Hermitian skin effect, this requires a simple lattice design with reciprocal couplings.

Erratum: Observation of π/2 Modes in an Acoustic Floquet System [Phys. Rev. Lett. 129, 254301 (2022)].

This corrects the article DOI: 10.1103/PhysRevLett.129.254301.

Low-Velocity-Favored Transition Radiation.

When a charged particle penetrates through an optical interface, photon emissions emerge-a phenomenon known as transition radiation. Being paramount to fundamental physics, transition radiation has enabled many applications from high-energy particle identification to novel light sources. A rule of thumb in transition radiation is that the radiation intensity generally decreases with the decrease of particle velocity v; as a result, low-energy particles are not favored in practice. Here, we find that there exist situations where transition radiation from particles with extremely low velocities (e.g., v/c<10^{-3}) exhibits comparable intensity as that from high-energy particles (e.g., v/c=0.999), where c is the light speed in free space. The comparable radiation intensity implies an extremely high photon extraction efficiency from low-energy particles, up to 8 orders of magnitude larger than that from high-energy particles. This exotic phenomenon of low-velocity-favored transition radiation originates from the interference of the excited Ferrell-Berreman modes in an ultrathin epsilon-near-zero slab. Our findings may provide a promising route toward the design of integrated light sources based on low-energy electrons and specialized detectors for beyond-standard-model particles.

Observation of π/2 Modes in an Acoustic Floquet System.

Topological phases of matter have remained an active area of research in the last few decades. Periodic driving is a powerful tool for enriching such exotic phases, leading to various phenomena with no static analogs. One such phenomenon is the emergence of the elusive π/2 modes, i.e., a type of topological boundary state pinned at a quarter of the driving frequency. The latter may lead to the formation of Floquet parafermions in the presence of interaction, which is known to support more computational power than Majorana particles. In this Letter, we experimentally verify the signature of π/2 modes in an acoustic waveguide array, which is designed to simulate a square-root periodically driven Su-Schrieffer-Heeger model. This is accomplished by confirming the 4T-periodicity (T being the driving period) profile of an initial-boundary excitation, which we also show theoretically to be the smoking gun evidence of π/2 modes. Our findings are expected to motivate further studies of π/2 modes in quantum systems for potential technological applications.

Projectively Enriched Symmetry and Topology in Acoustic Crystals.

Symmetry plays a key role in modern physics, as manifested in the revolutionary topological classification of matter in the past decade. So far, we seem to have a complete theory of topological phases from internal symmetries as well as crystallographic symmetry groups. However, an intrinsic element, i.e., the gauge symmetry in physical systems, has been overlooked in the current framework. Here, we show that the algebraic structure of crystal symmetries can be projectively enriched due to the gauge symmetry, which subsequently gives rise to new topological physics never witnessed under ordinary symmetries. We demonstrate the idea by theoretical analysis, numerical simulation, and experimental realization of a topological acoustic lattice with projective translation symmetries under a Z_{2} gauge field, which exhibits unique features of rich topologies, including a single Dirac point, Möbius topological insulator, and graphenelike semimetal phases on a rectangular lattice. Our work reveals the impact when gauge and crystal symmetries meet together with topology and opens the door to a vast unexplored land of topological states by projective symmetries.

Giant Enhancement of Unconventional Photon Blockade in a Dimer Chain.

Unconventional photon blockade refers to the suppression of multiphoton states in weakly nonlinear optical resonators via the destructive interference of different excitation pathways. It has been studied in a pair of coupled nonlinear resonators and other few-mode systems. Here, we show that unconventional photon blockade can be greatly enhanced in a chain of coupled resonators. The strength of the nonlinearity in each resonator needed to achieve unconventional photon blockade is suppressed exponentially with lattice size. The analytic derivation, based on a weak drive approximation, is validated by wave function Monte Carlo simulations. These findings show that customized lattices of coupled resonators can be powerful tools for controlling multiphoton quantum states.

Observation of Dislocation-Induced Topological Modes in a Three-Dimensional Acoustic Topological Insulator.

The interplay between real-space topological lattice defects and the reciprocal-space topology of energy bands can give rise to novel phenomena, such as one-dimensional topological modes bound to screw dislocations in three-dimensional topological insulators. We obtain direct experimental observations of dislocation-induced helical modes in an acoustic analog of a weak three-dimensional topological insulator. The spatial distribution of the helical modes is found through spin-resolved field mapping, and verified numerically by tight-binding and finite-element calculations. These one-dimensional helical channels can serve as robust waveguides in three-dimensional media. Our experiment paves the way to studying novel physical modes and functionalities enabled by topological lattice defects in three-dimensional classical topological materials.

Acoustic higher-order topological insulator on a kagome lattice.

Higher-order topological insulators1-5 are a family of recently predicted topological phases of matter that obey an extended topological bulk-boundary correspondence principle. For example, a two-dimensional (2D) second-order topological insulator does not exhibit gapless one-dimensional (1D) topological edge states, like a standard 2D topological insulator, but instead has topologically protected zero-dimensional (0D) corner states. The first prediction of a second-order topological insulator1, based on quantized quadrupole polarization, was demonstrated in classical mechanical6 and electromagnetic7,8 metamaterials. Here we experimentally realize a second-order topological insulator in an acoustic metamaterial, based on a 'breathing' kagome lattice9 that has zero quadrupole polarization but a non-trivial bulk topology characterized by quantized Wannier centres2,9,10. Unlike previous higher-order topological insulator realizations, the corner states depend not only on the bulk topology but also on the corner shape; we show experimentally that they exist at acute-angled corners of the kagome lattice, but not at obtuse-angled corners. This shape dependence allows corner states to act as topologically protected but reconfigurable local resonances.

Topology-Controlled Photonic Cavity Based on the Near-Conservation of the Valley Degree of Freedom.

We demonstrate a novel path to localizing topologically nontrivial photonic edge modes along their propagation direction. Our approach is based on the near-conservation of the photonic valley degree of freedom associated with valley-polarized edge states. When the edge state is reflected from a judiciously oriented mirror, its optical energy is localized at the mirror surface because of an extended time delay required for valley index flipping. The degree of energy localization at the resulting topology-controlled photonic cavity is determined by the valley-flipping time, which is in turn controlled by the geometry of the mirror. Intuitive analytic descriptions of the "leaky" and closed topology-controlled photonic cavities are presented, and two specific designs-one for the microwave and the other for the optical spectral ranges-are proposed.