Nonlinear Topological Photonics: Capturing Nonlinear Dynamics and Optical Thermodynamics

Combining multiple optical resonators or engineering dispersion of complex media has provided an effective method for demonstrating topological physics controlling photons in unprecedented ways such as unidirectional light propagation and spatially localized modes between an interface or on a corner. Further, adding nonlinear responses to those topological photonic systems has enabled achieving diverse phases of photons in both space and time, allowing for more functionalities in photonic devices that provide a new playground for studying dynamic features of nonlinear topological systems. However, most methods for describing nonlinear topological photonic systems rely on linear topological theories, making it challenging to accurately characterize the topology of nonlinear systems. Thus, substantial efforts have focused on rigorously describing nonlinear topological phases and developing effective tools to analyze nonlinear topological effects. Meanwhile, coupled multimode optical waveguides with nonlinear dynamic responses provide an excellent platform for the statistical description of photons, opening a new paradigm called "optical thermodynamics". This review will introduce the basic concepts of nonlinear topological photonics and the recent development of theoretical approaches focusing on data-driven approaches for creating phase diagrams as well as the spectral localizer framework and the pseudospectrum method for understanding optical nonlinearities in topological systems. In addition, the new concept of optical thermodynamics will be introduced with some recent theoretical works.

Terahertz Metamaterials Inspired by Quantum Phenomena

The study of many phenomena in the terahertz (THz) frequency spectral range has emerged as a promising playground in modern science and technology, with extensive applications in high-speed communication, imaging, sensing, and biosensing. Many THz metamaterial designs explore quantum physics phenomena embedded into a classical framework and exhibiting various unexpected behaviors. For spatial THz waves, the effects inspired by quantum phenomena include electromagnetically induced transparency (EIT), Fano resonance, bound states in the continuum (BICs), and exceptional points (EPs) in non-Hermitian systems. They facilitate the realization of extensive functional metadevices and applications. For on-chip THz waves, quantum physics-inspired topological metamaterials, as photonic analogs of topological insulators, can ensure robust, low-loss propagation with suppressed backscattering. These trends open new pathways for high-speed on-chip data transmission and THz photonic integrated circuits, being crucial for the upcoming 6G and 7G wireless communication technologies. Here, we summarize the underlying principles of quantum physics-inspired metamaterials and highlight the latest advances in their application in the THz frequency band, encompassing both spatial and on-chip metadevice realizations.

Transition from the topological to the chaotic in the nonlinear Su-Schrieffer-Heeger model

Recent studies on topological materials are expanding into the nonlinear regime, while the central principle, namely the bulk-edge correspondence, is yet to be elucidated in the strongly nonlinear regime. Here, we reveal that nonlinear topological edge modes can exhibit the transition to spatial chaos by increasing nonlinearity, which can be a universal mechanism of the breakdown of the bulk-edge correspondence. Specifically, we unveil the underlying dynamical system describing the spatial distribution of zero modes and show the emergence of chaos. We also propose the correspondence between the absolute value of the topological invariant and the dimension of the stable manifold under sufficiently weak nonlinearity. Our results provide a general guiding principle to investigate the nonlinear bulk-edge correspondence that can potentially be extended to arbitrary dimensions.

Realization of Topology-Controlled Photonic Cavities in a Valley Photonic Crystal

We report the experimental realization of a new type of topology-controlled photonic cavities in valley photonic crystals by adopting judiciously oriented mirrors to localize the valley-polarized edge states along their propagation path. By using microwave frequency- and time-domain measurements, we directly observe the strong confinement of electromagnetic energy at the mirror surface due to the extended time delay required for the valley index flipping. Moreover, we experimentally demonstrate that both the degree of energy localization and quality factors of the topology-controlled photonic cavities are determined by the valley-flipping time which is controlled by the topology of the mirror. These results extend and complement the current design paradigm of topological photonic cavities.

Observation of momentum-gap topology of light at temporal interfaces in a time-synthetic lattice

Topological phases have prevailed across diverse disciplines, spanning electronics, photonics, and acoustics. Hitherto, the understanding of these phases has centred on energy (frequency) bandstructures, showcasing topological boundary states at spatial interfaces. Recent strides have uncovered a unique category of bandstructures characterised by gaps in momentum, referred to as momentum bandgaps or k gaps, notably driven by breakthroughs in photonic time crystals. This discovery hints at abundant topological phases defined within momentum bands, alongside a wealth of topological boundary states in the time domain. Here, we report the experimental observation of k-gap topology in a large-scale optical temporal synthetic lattice, manifesting as temporal topological boundary states. These boundary states are uniquely situated at temporal interfaces between two subsystems with distinct k-gap topology. Counterintuitively, despite the exponential amplification of k-gap modes within both subsystems, these topological boundary states exhibit decay in both temporal directions [i.e., with energy growing (decaying) before (after) the temporal interfaces]. Our findings mark a significant pathway for delving into k gaps, temporal topological states, and time-varying physics.

Non-Hermitian Topology in Hermitian Topological Matter

Non-Hermiticity gives rise to distinctive topological phenomena absent in Hermitian systems. However, connection between such intrinsic non-Hermitian topology and Hermitian topology has remained largely elusive. Here, considering the bulk and boundary as an environment and system, respectively, we demonstrate that anomalous boundary states in Hermitian topological insulators exhibit non-Hermitian topology. We study the self-energy capturing the particle exchange between the bulk and boundary, and show that it detects Hermitian topology in the bulk and induces non-Hermitian topology at the boundary. As an illustrative example, we reveal non-Hermitian topology and concomitant skin effect inherently embedded within chiral edge states of Chern insulators. We also identify the emergence of hinge states within effective non-Hermitian Hamiltonians at surfaces of three-dimensional topological insulators. Furthermore, we comprehensively classify our correspondence across all the tenfold symmetry classes of topological insulators and superconductors. Our Letter uncovers hidden connection between Hermitian and non-Hermitian topology, and provides an approach to identifying non-Hermitian topology in quantum matter.

Edge and skin effects in rhombus reciprocal photonic crystals

With the development of non-Hermitian physics, the non-Hermitian skin effect (NHSE) has attracted much attention. Existing research highlights the critical roles of the periodic boundary condition (PBC) spectrum, lattice symmetry, and macroscopic symmetry of the lattice in relation to the geometry-dependent skin effect (GDSE). However, the impact of macroscopic edge geometry is frequently neglected. We find that the GDSE is highly sensitive to the edge and cannot be simply determined by the symmetries. Specifically, the GDSE can emerge at trivial interfaces of rhombus photonic crystals (PhCs) with zigzag edge and bearded edge. Furthermore, we analyze the underlying mechanisms from the perspective of point-gap topology. This work underscores important, yet frequently overlooked, aspects in two-dimensional (2D) reciprocal PhC systems and can be used to enhance design flexibility, allowing the NHSE to have better applications in areas such as lasers and highly sensitive sensors.

All-optical switching in nonlinear topological waveguide arrays

Photonic topological states are prospective in integrated optical devices due to their robustness to perturbations and defects. When taking into account the nonlinear effects of the system, the functionality of topological photonics can be further enhanced. Here, we investigated the interplay between topological edge states and nonlinear effects based on the Su-Schrieffer-Heeger (SSH) model. Relying on the theory prediction that topological edge states would shift upward under the action of nonlinearity, two types of optical switching are designed and experimentally realized in femtosecond laser direct-write waveguide arrays. This work provides a new, to the best of our knowledge, approach to preparing all-optical switches and offers a new perspective on the application of nonlinearity in topological optical devices.

Realization of a quadrupole topological insulator phase in a gyromagnetic photonic crystal

The field of topological photonics was initiated with the realization of a Chern insulator phase in a gyromagnetic photonic crystal (PhC) with broken time-reversal symmetry (T), hosting chiral edge states that are topologically protected propagating modes. Along a separate line of research, a quadrupole topological insulator was the first higher-order topological phase supporting localized corner states, but has been so far limited to T-invariant systems, as T is a key ingredient in early models. Here we report the realization of a quadrupole topological insulator phase in a gyromagnetic PhC, as a consequence of topological phase transition from the previously demonstrated Chern insulator phase. The phase transition has been demonstrated with microwave measurements, which characterize the evolution from propagating chiral edge states to localized corner states. We also demonstrate the migration of topological boundary states into the continuum, when the gyromagnetic PhC is magnetically tuned. These results extend the quadrupole topological insulator phase into T-broken systems, and integrate topologically protected propagating and localized modes in a magnetically tunable photonic crystal platform.

Velocity Scanning Tomography for Room-Temperature Quantum Simulation

Quantum simulation offers an analog approach for exploring exotic quantum phenomena using controllable platforms, typically necessitating ultracold temperatures to maintain the quantum coherence. Superradiance lattices (SLs) have been harnessed to simulate coherent topological physics at room temperature, but the thermal motion of atoms remains a notable challenge in accurately measuring the physical quantities. To overcome this obstacle, we implement a velocity scanning tomography technique to discern the responses of atoms with different velocities, allowing cold-atom spectroscopic resolution within room-temperature SLs. By comparing absorption spectra with and without atoms moving at specific velocities, we can derive the Wannier-Stark ladders of the SL across various effective static electric fields, their strengths being proportional to the atomic velocities. We extract the Zak phase of the SL by monitoring the ladder frequency shift as a function of the atomic velocity, effectively demonstrating the topological winding of the energy bands. Our research signifies the feasibility of room-temperature quantum simulation and facilitates their applications in quantum information processing.

Plasmonic Hinge Modes in Metal-Coated Nanolasers

Plasmonic lasers have traditionally been built on flat metal substrates. Here, we introduce substrate-free plasmonic lasers created by coating semiconductor particles with an optically thin layer of noble metal. This architecture supports plasmonic "hinge" modes highly localized along the particle's edges and corners, exhibiting Purcell factors exceeding 100 and Q-factors of 15-20 near the plasmon resonance frequency. We demonstrate hinge-mode lasing in submicron CsPbBr3 perovskite cubes encapsulated with conformal 15-nm-thick gold shells. The lasing is achieved with 480-nm nanosecond pumping at 10 pJ/μm2 through the translucent gold layer, producing a line width of 0.6 at 538 nm. Their rapidly decaying evanescent fields outside the gold coating show distinct sensitivities to long- and short-range external perturbations. Our results suggest the potential of these novel laser modes for sensing and imaging applications.

General Criterion for Non-Hermitian Skin Effects and Application: Fock Space Skin Effects in Many-Body Systems

Non-Hermiticity enables macroscopic accumulation of bulk states, named non-Hermitian skin effects. The non-Hermitian skin effects are well established for single-particle systems, but their proper characterization for general systems is elusive. Here, we propose a general criterion of non-Hermitian skin effects, which works for any finite-dimensional system evolved by a linear operator. The applicable systems include many-body systems and network systems. A system meeting the criterion exhibits enhanced non-normality of the evolution operator, accompanied by exceptional characteristics intrinsic to non-Hermitian systems. Applying the criterion, we discover a new type of non-Hermitian skin effect in many-body systems, which we dub the Fock space skin effect. We also discuss the Fock space skin effect-induced slow dynamics, which gives an experimental signal for the Fock space skin effect.

Arbitrarily Configurable Nonlinear Topological Modes

Topological modes (TMs) are typically localized at boundaries, interfaces and dislocations, and exponentially decay into the bulk of a large enough lattice. Recently, the non-Hermitian skin effect has been leveraged to delocalize the wave functions of TMs from the boundary and thus to increase the capacity of TMs dramatically. Here, we explore the capability of nonlinearity in designing and configuring the wave functions of TMs. With growing intensity, wave functions of these in-gap nonlinear TMs undergo an initial deviation from exponential decay, gradually merge into arbitrarily designable plateaus, then encompass the entire nonlinear domain, and eventually concentrate at the nonlinear boundary. Intriguingly, such extended nonlinear TMs are still robust against defects and disorders, and stable in dynamics under external excitation. Advancing the conceptual understanding of the nonlinear TMs, our results open new avenues for increasing the capacity of TMs and developing compact and configurable topological devices.

Parity-Frequency-Space Elastic Spin Control of Wave Routing in Topological Phononic Circuits

Topological phononic cavities, such as ring resonators with topological whispering gallery modes (TWGMs), offer a flexible platform for the realization of robust phononic circuits. However, the chiral mechanism governing TWGMs and their selective routing in integrated phononic circuits remain unclear. This work reveals, both experimentally and theoretically, that at a phononic topological interface, the elastic spin texture is intricately linked to, and can be explained through a knowledge of, the phonon eigenmodes inside each unit cell. Furthermore, for paired, counterpropagating TWGMs based on such interfaces in a waveguide resonator, this study demonstrates that the elastic spin exhibits locking at discrete frequencies. Backed up by theory, experiments on kHz TWGMs in thin honeycomb-lattice aluminum plates bored with clover-leaf shaped holes show that together with this spin-texture related angular-momentum locking mechanism at a single topological interface, there are triplicate parity-frequency-space selective wave routing mechanisms. In the future, these mechanisms can be harnessed for the versatile manipulation of elastic-spin based routing in phononic topological insulators.

Localized and delocalized topological modes of heat

The topological physics has sparked intensive investigations into topological lattices in photonic, acoustic, and mechanical systems, powering counterintuitive effects otherwise inaccessible with usual settings. Following the success of these endeavors in classical wave dynamics, there has been a growing interest in establishing their topological counterparts in diffusion. Here, we propose an additional real-space dimension in diffusion, and the system eigenvalues are transformed from "imaginary" to "real." By judiciously tailoring the effective Hamiltonian with coupling networks, localized and delocalized topological modes are realized in heat transfer. Simulations and experiments in active thermal lattices validate the effectiveness of the proposed theoretical strategy. This approach can be applied to establish various topological lattices in diffusion systems, offering insights into engineering topologically protected edge states in dynamic diffusive scenarios.

Vortex solitons in topological disclination lattices

The existence of thresholdless vortex solitons trapped at the core of disclination lattices that realize higher-order topological insulators is reported. The study demonstrates the interplay between nonlinearity and higher-order topology in these systems, as the vortex state in the disclination lattice bifurcates from its linear topological counterpart, while the position of its propagation constant within the bandgap and localization can be controlled by its power. It is shown that vortex solitons are characterized by strong field confinement at the disclination core due to their topological nature, leading to enhanced stability. Simultaneously, the global discrete rotational symmetry of the disclination lattice imposes restrictions on the maximal possible topological charge of such vortex solitons. The results illustrate the strong stabilizing action that topologically nontrivial structures may exert on excited soliton states, opening new prospects for soliton-related applications.

Near-Field Spin Chern Number Quantized by Real-Space Topology of Optical Structures

The Chern number has been widely used to describe the topological properties of periodic structures in momentum space. Here, we introduce a real-space spin Chern number for the optical near fields of finite-sized structures. This new spin Chern number is intrinsically quantized and equal to the structure's Euler characteristic. The relationship is robust against continuous deformation of the structure's geometry and is irrelevant to the specific material constituents or external excitation. Our Letter enriches topological physics by extending the Chern number to real space, opening exciting possibilities for exploring the real-space topological properties of light.

An Electrically Pumped Topological Polariton Laser

With a seminal work of Raghu and Haldane in 2008, concepts of topology have been introduced into optical systems, where some of the most promising routes to an application are efficient and highly coherent topological lasers. While some attempts have been made to excite such structures electrically, the majority of published experiments use a form of laser excitation. In this paper, we use a lattice of vertical resonator polariton micropillars to form an exponentially localized topological Su-Schrieffer-Heeger defect. Upon electrical excitation, the system unequivocally shows polariton lasing from the topological defect using a carefully placed gold contact. Despite the presence of doping and electrical contacts, the polariton band structure clearly preserves its topological properties. At high excitation power the Mott density is exceeded, leading to highly efficient lasing in the weak coupling regime. This work is an important step toward applied topological lasers using vertical resonator microcavity structures.

High-power electrically pumped terahertz topological laser based on a surface metallic Dirac-vortex cavity

Topological lasers (TLs) have attracted widespread attention due to their mode robustness against perturbations or defects. Among them, electrically pumped TLs have gained extensive research interest due to their advantages of compact size and easy integration. Nevertheless, limited studies on electrically pumped TLs have been reported in the terahertz (THz) and telecom wavelength ranges with relatively low output powers, causing a wide gap between practical applications. Here, we introduce a surface metallic Dirac-vortex cavity (SMDC) design to solve the difficulty of increasing power for electrically pumped TLs in the THz spectral range. Due to the strong coupling between the SMDC and the active region, robust 2D topological defect lasing modes are obtained. More importantly, enough gain and large radiative efficiency provided by the SMDC bring in the increase of the output power to a maximum peak power of 150 mW which demonstrates the practical application potential of electrically pumped TLs.

Lasing in Non-Hermitian Flat Bands: Quantum Geometry, Coherence, and the Fate of Kardar-Parisi-Zhang Physics

We show that lasing in flat-band lattices can be stabilized by means of the geometrical properties of the Bloch states, in settings where the single-particle dispersion is flat in both its real and imaginary parts. We illustrate a general projection method and compute the collective excitations, which display a diffusive behavior ruled by quantum geometry through a peculiar coefficient involving gain, losses and interactions, and entailing resilience against modulational instabilities. Then, we derive an equation of motion for the phase dynamics and identify a Kardar-Parisi-Zhang term of geometric origin. This term is shown to exactly cancel whenever the real and imaginary parts of the laser nonlinearity are proportional to each other, or when the uniform-pairing condition is satisfied. We confirm our results through numerical studies of the π-flux diamond chain. This Letter highlights the key role of Bloch geometric effects in nonlinear dissipative systems and KPZ physics, with direct implications for the design of laser arrays with enhanced coherence.

On the spectral response of a taiji-CROW device

Physical systems with topological properties are robust against disorder. However, implementing them in integrated photonic devices is challenging because of the various fabrication imperfections and/or limitations that affect the spectral response of their building blocks. One such feature is strong backscattering due to the surface wall roughness of the waveguides, which can flip the propagating modes to counterpropagating modes and destroy the desired topological behavior. Here, we report a study on modeling, designing and testing an integrated photonic structure based on a sequence of two taiji microresonators coupled with a middle link microresonator (a taiji-CROW device, where CROW stands for coupled resonator optical waveguides). Our study provides design constraints to preserve the ideal operation of the structure by quantifying a minimum ratio between the coupling coefficients and the backscattering coefficients. This ratio is valuable to avoid surface roughness problems in designing topological integrated photonic devices based on arrays of microresonators.

Pseudo-spin switches and Aharonov-Bohm effect for topological boundary modes

Topological boundary modes in electronic and classical-wave systems exhibit fascinating properties. In photonics, topological nature of boundary modes can make them robust and endows them with an additional internal structure-pseudo-spins. Here, we introduce heterogeneous boundary modes, which are based on mixing two of the most widely used topological photonics platforms-the pseudo-spin-Hall-like and valley-Hall photonic topological insulators. We predict and confirm experimentally that transformation between the two, realized by altering the lattice geometry, enables a continuum of boundary states carrying both pseudo-spin and valley degrees of freedom (DoFs). When applied adiabatically, this leads to conversion between pseudo-spin and valley polarization. We show that such evolution gives rise to a geometrical phase associated with the synthetic gauge fields, which is confirmed via an Aharonov-Bohm type experiment on a silicon chip. Our results unveil a versatile approach to manipulating properties of topological photonic states and envision topological photonics as a powerful platform for devices based on synthetic DoFs.

On-Chip Photonic Localization in Aharonov-Bohm Cages Composed of Microring Lattices

Flatband localization endowed with robustness holds great promise for disorder-immune light transport, particularly in the advancement of optical communication and signal processing. However, effectively harnessing these principles for practical applications in nanophotonic devices remains a significant challenge. Herein, we delve into the investigation of on-chip photonic localization in AB cages composed of indirectly coupled microring lattices. By strategically vertically shifting the auxiliary rings, we successfully introduce a magnetic flux of π into the microring lattice, thereby facilitating versatile control over the localization and delocalization of light. Remarkably, the compact edge modes of this structure exhibit intriguing topological properties, rendering them strongly robust against disorders, regardless of the size of the system. Our findings open up new avenues for exploring the interaction between flatbands and topological photonics on integrated platforms.

Bulk-Boundary Correspondence in Point-Gap Topological Phases

A striking feature of non-Hermitian systems is the presence of two different types of topology. One generalizes Hermitian topological phases, and the other is intrinsic to non-Hermitian systems, which are called line-gap topology and point-gap topology, respectively. Whereas the bulk-boundary correspondence is a fundamental principle in the former topology, its role in the latter has not been clear yet. This Letter establishes the bulk-boundary correspondence in the point-gap topology in non-Hermitian systems. After revealing the requirement for point-gap topology in the open boundary conditions, we clarify that the bulk point-gap topology in open boundary conditions can be different from that in periodic boundary conditions. On the basis of real space topological invariants and the K theory, we give a complete classification of the open boundary point-gap topology with symmetry and show that the nontrivial open boundary topology results in robust and exotic surface states.

Hyperbolic photonic topological insulators

Topological photonics provides a new degree of freedom to robustly control electromagnetic fields. To date, most of established topological states in photonics have been employed in Euclidean space. Motivated by unique properties of hyperbolic lattices, which are regular tessellations in non-Euclidean space with a constant negative curvature, the boundary-dominated hyperbolic topological states have been proposed. However, limited by highly crowded boundary resonators and complicated site couplings, the hyperbolic topological insulator has only been experimentally constructed in electric circuits. How to achieve hyperbolic photonic topological insulators is still an open question. Here, we report the experimental realization of hyperbolic photonic topological insulators using coupled ring resonators on silicon chips. Boundary-dominated one-way edge states with pseudospin-dependent propagation directions have been observed. Furthermore, the robustness of edge states in hyperbolic photonic topological insulators is also verified. Our findings have potential applications in the field of designing high-efficient topological photonic devices with enhanced boundary responses.

Dynamic and Polarization-Independent Wavefront Control Based on Hybrid Topological Metasurfaces

Exceptional points (EPs), known as non-Hermitian singularities, have been observed and investigated in parity-time symmetric metasurfaces. However, the chirality and tunability in non-Hermitian metasurfaces still need to be explored. Here, we propose a dynamic topological metasurface with the meta-atom consisting of two orthogonally oriented nanorods, which are placed on the phase change material Ge2Sb2Te5 (GST) and SiO2 dielectric layer, respectively. When GST is converted from the amorphous state (a-GST) to the crystalline state (c-GST), an EP can be dynamically switched from the "ON" state to the "OFF" state in a parameter space. Moreover, based on the topologically protected phase and amplitude modulations of the cross-polarization component, the phase-only hologram and amplitude-only hologram are engineered in the a-GST case and concealed in the c-GST case. Finally, we explore the 2D-chiral symmetry of meta-atoms and further propose two spin-selective meta-deflectors and a hybrid meta-deflector operating with arbitrary polarizations. The GST-based hybrid metasurface offers richer possibilities to realize various wavefront controls.

Topological edge and corner states in biphenylene photonic crystal

The biphenylene network (BPN) has a unique two-dimensional atomic structure, where hexagonal unit cells are arranged on a square lattice. Inspired by such a BPN structure, we design a counterpart in the fashion of photonic crystals (PhCs), which we refer to as the BPN PhC. We study the photonic band structure using the finite element method and characterize the topological properties of the BPN PhC through the use of the Wilson loop. Our findings reveal the emergence of topological edge states in the BPN PhC, specifically in the zigzag edge and the chiral edge, as a consequence of the nontrivial Zak phase in the corresponding directions. In addition, we find the localization of electromagnetic waves at the corners formed by the chiral edges, which can be considered as second-order topological states, i.e., topological corner states.

Kardar-Parisi-Zhang universality in the coherence time of nonequilibrium one-dimensional quasicondensates

We investigate the finite-size origin of the coherence time (or equivalently of its inverse, the emission linewidth) of a spatially extended, one-dimensional nonequilibrium condensate. We show that the well-known Schawlow-Townes scaling of laser theory, possibly including the Henry broadening factor, only holds for small system sizes, while in larger systems the linewidth displays a novel scaling determined by Kardar-Parisi-Zhang physics. This is shown to lead to an opposite dependence of the coherence time on the optical nonlinearity in the two cases. We then study how subuniversal properties of the phase dynamics such as the higher moments of the phase-phase correlator are affected by the finite size and discuss the relation between the field coherence and the exponential of the phase-phase correlator. We finally identify a configuration with enhanced open boundary conditions, which supports a spatially uniform steady state and facilitates experimental studies of the coherence time scaling.

Perovskite Topological Lasers: A Brand New Combination

Nanolasers are the essential components of modern photonic chips due to their low power consumption, high energy efficiency and fast modulation. As nanotechnology has advanced, researchers have proposed a number of nanolasers operating at both wavelength and sub-wavelength scales for application as light sources in photonic chips. Despite the advances in chip technology, the quality of the optical cavity, the operating threshold and the mode of operation of the light source still limit its advanced development. Ensuring high-performance laser operation has become a challenge as device size has been significantly reduced. A potential solution to this problem is the emergence of a novel optical confinement mechanism using photonic topological insulator lasers. In addition, gain media materials with perovskite-like properties have shown great potential for lasers, a role that many other gain materials cannot fulfil. When combined with topological laser modes, perovskite materials offer new possibilities for the operation and emission mechanism of nanolasers. This study introduces the operating mechanism of topological lasers and the optical properties of perovskite materials. It then outlines the key features of their combination and discusses the principles, structures, applications and prospects of perovskite topological lasers, including the scientific hurdles they face. Finally, the future development of low-dimensional perovskite topological lasers is explored.

Nonlinear topological laser on the non-Hermitian Haldane model with higher-order corner states

The non-Hermitian skin effect (NHSE) on the non-Hermitian Haldane model with gain and loss on the honeycomb lattice with the outline of a triangle is discussed. The NHSE only occurs on the edge of the lattice, transforming the edge modes into the higher-order corner modes. The NHSE can also occur on a lattice with only loss, which can be treated as a lattice with gain and loss as well as a global loss added to it. When the saturated gain is added to the three corner sites of the dissipative lattice, a single-mode laser system is obtained. When any one site is stimulated initially, the system will reach a saturated state depending on the distribution of the corner modes, and the stable laser light is emitted by sites at the corners.

Bulk-local-density-of-state correspondence in topological insulators

In the quest to connect bulk topological quantum numbers to measurable parameters in real materials, current established approaches often necessitate specific conditions, limiting their applicability. Here we propose and demonstrate an approach to link the non-trivial hierarchical bulk topology to the multidimensional partition of local density of states (LDOS), denoted as the bulk-LDOS correspondence. In finite-size topologically nontrivial photonic crystals, we observe the LDOS partitioned into three distinct regions: a two-dimensional interior bulk area, a one-dimensional edge region, and zero-dimensional corner sites. Contrarily, topologically trivial cases exhibit uniform LDOS distribution across the entire two-dimensional bulk area. Our findings provide a general framework for distinguishing topological insulators and uncovering novel aspects of topological directional band-gap materials, even in the absence of in-gap states.

Photonic skin-topological effects in microring lattices

We investigate the non-Hermitian Hofstadter-Harper model composed of microring resonators, in which the non-Hermitian skin effect (NHSE) is particularly analyzed. The effect is achieved through the interaction between well-designed gain-loss layouts and artificial gauge fields. Remarkably, we reveal the emergence of a hybrid skin-topological effect (HSTE), where only the original topological edge modes convert to skin modes while bulk modes remain extended. By changing the distributions of gauge fields, we show the NHSE can manifest itself in bulk modes and be localized at specific edges. Using the equivalence of sites in the bulk or at boundaries to 1D SSH chains, we analyze the potential cancellation of NHSE in these configurations. Additionally, we demonstrate a new, to the best of our knowledge, type of HSTE in topological insulators which emerge at any gain-loss interfaces. The study may improve the understanding of the NHSE behavior in 2D topological systems and provide a promising avenue for tuning light propagation and localization.

Topological Inverse Band Theory in Waveguide Quantum Electrodynamics

Topological phases play a crucial role in the fundamental physics of light-matter interaction and emerging applications of quantum technologies. However, the topological band theory of waveguide QED systems is known to break down, because the energy bands become disconnected. Here, we introduce a concept of the inverse energy band and explore analytically topological scattering in a waveguide with an array of quantum emitters. We uncover a rich structure of topological phase transitions, symmetric scale-free localization, completely flat bands, and the corresponding dark Wannier states. Although bulk-edge correspondence is partially broken because of radiative decay, we prove analytically that the scale-free localized states are distributed in a single inverse energy band in the topological phase and in two inverse bands in the trivial phase. Surprisingly, the winding number of the scattering textures depends on both the topological phase of inverse subradiant band and the odevity of the cell number. Our Letter uncovers the field of the topological inverse bands, and it brings a novel vision to topological phases in light-matter interactions.

Topological photonic crystal nanowire array laser with edge states

A topological photonic crystal InGaAsP/InP core-shell nanowire array laser operating in the 1550 nm wavelength band is proposed and simulated. The structure is composed of an inner topological nontrivial photonic crystal and outer topological trivial photonic crystal. For a nanowire with height of 8 µm, high quality factor of 4.7 × 104 and side-mode suppression ratio of 11 dB are obtained, approximately 32.9 and 5.5 times that of the uniform photonic crystal nanowire array, respectively. Under optical pumping, the topological nanowire array laser exhibits a threshold 27.3% lower than that of the uniform nanowire array laser, due to the smaller nanowire slit width and stronger optical confinement. Moreover, the topological NW laser exhibits high tolerence to manufacturing errors. This work may pave the way for the development of low-threshold single-mode high-robustness nanolasers.

Turning whispering-gallery-mode responses through Fano interferences in coupled all-dielectric block-disk cavities

Here, we theoretically demonstrate a strategy for efficiently turning whispering-gallery-mode (WGM) responses of a subwavelength dielectric disk through their near-field couplings with common low-order electromagnetic resonances of a dielectric block. Both simulations and an analytical coupled oscillator model show that the couplings are Fano interferences between dark high-quality WGMs and bright modes of the block. The responses of a WGM in the coupled system are highly dependent on the strengths and the relative phases of the block modes, the coupling strength, and the decay rate of the WGM. The WGM responses of coupled systems can exceed that of the individual disk. In addition, such a configuration will also facilitate the excitation of WGMs by a normal incident plane wave in experiments. These results could enable new applications for enhancing light-matter interactions.

Topological quantum devices: a review

The introduction of the concept of topology into condensed matter physics has greatly deepened our fundamental understanding of transport properties of electrons as well as all other forms of quasi particles in solid materials. It has also fostered a paradigm shift from conventional electronic/optoelectronic devices to novel quantum devices based on topology-enabled quantum device functionalities that transfer energy and information with unprecedented precision, robustness, and efficiency. In this article, the recent research progress in topological quantum devices is reviewed. We first outline the topological spintronic devices underlined by the spin-momentum locking property of topology. We then highlight the topological electronic devices based on quantized electron and dissipationless spin conductivity protected by topology. Finally, we discuss quantum optoelectronic devices with topology-redefined photoexcitation and emission. The field of topological quantum devices is only in its infancy, we envision many significant advances in the near future.

Adiabatic topological photonic interfaces

Topological phases of matter have been attracting significant attention across diverse fields, from inherently quantum systems to classical photonic and acoustic metamaterials. In photonics, topological phases offer resilience and bring novel opportunities to control light with pseudo-spins. However, topological photonic systems can suffer from limitations, such as breakdown of topological properties due to their symmetry-protected origin and radiative leakage. Here we introduce adiabatic topological photonic interfaces, which help to overcome these issues. We predict and experimentally confirm that topological metasurfaces with slowly varying synthetic gauge fields significantly improve the guiding features of spin-Hall and valley-Hall topological structures commonly used in the design of topological photonic devices. Adiabatic variation in the domain wall profiles leads to the delocalization of topological boundary modes, making them less sensitive to details of the lattice, perceiving the structure as an effectively homogeneous Dirac metasurface. As a result, the modes showcase improved bandgap crossing, longer radiative lifetimes and propagation distances.

Unconventional quantum criticality in a non-Hermitian extended Kitaev chain

We investigate the nature of quantum criticality and topological phase transitions near the critical lines obtained for the extended Kitaev chain with next nearest neighbor hopping parameters and non-Hermitian chemical potential. We surprisingly find multiple gap-less points, the locations of which in the momentum space can change along the critical line unlike the Hermitian counterpart. The interesting simultaneous occurrences of vanishing and sign flipping behavior by real and imaginary components, respectively of the lowest excitation is observed near the topological phase transition. Introduction of non-Hermitian factor leads to an isolated critical point instead of a critical line and hence, reduced number of multi-critical points as compared to the Hermitian case. The critical exponents obtained for the multi-critical and critical points show a very distinct behavior from the Hermitian case.

Hybrid topological photonic crystals

Topologically protected photonic edge states offer unprecedented robust propagation of photons that are promising for waveguiding, lasing, and quantum information processing. Here, we report on the discovery of a class of hybrid topological photonic crystals that host simultaneously quantum anomalous Hall and valley Hall phases in different photonic band gaps. The underlying hybrid topology manifests itself in the edge channels as the coexistence of the dual-band chiral edge states and unbalanced valley Hall edge states. We experimentally realize the hybrid topological photonic crystal, unveil its unique topological transitions, and verify its unconventional dual-band gap topological edge states using pump-probe techniques. Furthermore, we demonstrate that the dual-band photonic topological edge channels can serve as frequency-multiplexing devices that function as both beam splitters and combiners. Our study unveils hybrid topological insulators as an exotic topological state of photons as well as a promising route toward future applications in topological photonics.

Topological Microlaser with a Non-Hermitian Topological Bulk

Bulk-edge correspondence, with quantized bulk topology leading to protected edge states, is a hallmark of topological states of matter and has been experimentally observed in electronic, atomic, photonic, and many other systems. While bulk-edge correspondence has been extensively studied in Hermitian systems, a non-Hermitian bulk could drastically modify the Hermitian topological band theory due to the interplay between non-Hermiticity and topology, and its effect on bulk-edge correspondence is still an ongoing pursuit. Importantly, including non-Hermicity can significantly expand the horizon of topological states of matter and lead to a plethora of unique properties and device applications, an example of which is a topological laser. However, the bulk topology, and thereby the bulk-edge correspondence, in existing topological edge-mode lasers is not well defined. Here, we propose and experimentally probe topological edge-mode lasing with a well-defined non-Hermitian bulk topology in a one-dimensional (1D) array of coupled ring resonators. By modeling the Hamiltonian with an additional degree of freedom (referred to as synthetic dimension), our 1D structure is equivalent to a 2D non-Hermitian Chern insulator with precise mapping. Our Letter may open a new pathway for probing non-Hermitian topological effects and exploring non-Hermitian topological device applications.

Exceptional points and non-Hermitian photonics at the nanoscale

Exceptional points (EPs) arising in non-Hermitian systems have led to a variety of intriguing wave phenomena, and have been attracting increased interest in various physical platforms. In this Review, we highlight the latest fundamental advances in the context of EPs in various nanoscale systems, and overview the theoretical progress related to EPs, including higher-order EPs, bulk Fermi arcs and Weyl exceptional rings. We peek into EP-associated emerging technologies, in particular focusing on the influence of noise for sensing near EPs, improving the efficiency in asymmetric transmission based on EPs, optical isolators in nonlinear EP systems and novel concepts to implement EPs in topological photonics. We also discuss the constraints and limitations of the applications relying on EPs, and offer parting thoughts about promising ways to tackle them for advanced nanophotonic applications.

Asymmetric transmission in nanophotonics

In a reciprocal medium, transmission of electromagnetic (EM) waves is symmetric along opposite directions which restrict design and implementation of various systems in optics and photonics. Asymmetric transmission (AT) is essential for designing isolators and circulators in optics and photonics, and it benefits other applications such as photovoltaic systems, lasers, cloaking, and EM shielding. While bulky nonreciprocal devices based on magnetic field biases have been well known, creating AT in subwavelength structures is more challenging, and structures with a subwavelength thickness that show AT have drawn a lot of attention over the last decade. Various approaches have been reported to create metasurfaces featuring nonreciprocal transmission, such as plasmonic and dielectric metasurfaces that enhance Faraday rotation, nonlinear metasurfaces with intensity-dependent refractive indices, and implementing spatiotemporal modulation in a metasurface. On the other hand, AT has also been reported in reciprocal structures by creating multiple paths for the transmission of EM waves by changing the polarization of light or redirecting light to higher-order diffraction orders. Here, we present a review of various approaches implemented for realizing AT in subwavelength structures in both reciprocal and nonreciprocal systems. We also discuss the main design principles and limitations of AT achieved in various approaches.

Vector valley Hall edge solitons in distorted type-II Dirac photonic lattices

Topological edge states have recently garnered a lot of attention across various fields of physics. The topological edge soliton is a hybrid edge state that is both topologically protected and immune to defects or disorders, and a localized bound state that is diffraction-free, owing to the self-balance of diffraction by nonlinearity. Topological edge solitons hold great potential for on-chip optical functional device fabrication. In this report, we present the discovery of vector valley Hall edge (VHE) solitons in type-II Dirac photonic lattices, formed by breaking lattice inversion symmetry with distortion operations. The distorted lattice features a two-layer domain wall that supports both in-phase and out-of-phase VHE states, appearing in two different band gaps. Superposing soliton envelopes onto VHE states generates bright-bright and bright-dipole vector VHE solitons. The propagation dynamics of such vector solitons reveal a periodic change in their profiles, accompanied by the energy periodically transferring between the layers of the domain wall. The reported vector VHE solitons are found to be metastable.

Electrically-pumped compact topological bulk lasers driven by band-inverted bound states in the continuum

One of the most exciting breakthroughs in physics is the concept of topology that was recently introduced to photonics, achieving robust functionalities, as manifested in the recently demonstrated topological lasers. However, so far almost all attention was focused on lasing from topological edge states. Bulk bands that reflect the topological bulk-edge correspondence have been largely missed. Here, we demonstrate an electrically pumped topological bulk quantum cascade laser (QCL) operating in the terahertz (THz) frequency range. In addition to the band-inversion induced in-plane reflection due to topological nontrivial cavity surrounded by a trivial domain, we further illustrate the band edges of such topological bulk lasers are recognized as the bound states in the continuum (BICs) due to their nonradiative characteristics and robust topological polarization charges in the momentum space. Therefore, the lasing modes show both in-plane and out-of-plane tight confinements in a compact laser cavity (lateral size ~3λlaser). Experimentally, we realize a miniaturized THz QCL that shows single-mode lasing with a side-mode suppression ratio (SMSR) around 20 dB. We also observe a cylindrical vector beam for the far-field emission, which is evidence for topological bulk BIC lasers. Our demonstration on miniaturization of single-mode beam-engineered THz lasers is promising for many applications including imaging, sensing, and communications.

A second wave of topological phenomena in photonics and acoustics

Light and sound are the most ubiquitous forms of waves, associated with a variety of phenomena and physical effects such as rainbows and echoes. Light and sound, both categorized as classical waves, have lately been brought into unexpected connections with exotic topological phases of matter. We are currently witnessing the onset of a second wave of active research into this topic. The past decade has been marked by fundamental advances comprising two-dimensional quantum Hall insulators and quantum spin and valley Hall insulators, whose topological properties are characterized using linear band topology. Here, going beyond these conventional topological systems, we focus on the latest frontiers, including non-Hermitian, nonlinear and non-Abelian topology as well as topological defects, for which the characterization of the topological features goes beyond the standard band-topology language. In addition to an overview of the current state of the art, we also survey future research directions for valuable applications.

Terahertz topological photonic crystals with dual edge states for efficient routing

Topological photonic crystals with robust pseudo-spin and valley edge states have shown promising and wide applications in topological waveguides, lasers, and antennas. However, the limited bandwidth and intrinsic coupling properties of a single pseudo-spin or valley edge state have imposed restrictions on their multifunctional applications in integrated photonic circuits. Here, we propose a topological photonic crystal that can support pseudo-spin and valley edge states simultaneously in a single waveguiding channel, which effectively broadens the bandwidth and enables a multipath routing solution for terahertz information processing and broadcasting. We show that distorted Kekulé lattices can open two types of bandgaps with different topological properties simultaneously by molding the inter- and intra-unit cell coupling of the tight-binding model. The distinct topological origins of the edge states provide versatile signal routing paths toward free space radiation or on-chip self-localized edge modes by virtue of their intrinsic coupling properties. Such a powerful platform could function as an integrated photonic chip with capabilities of broadband on-chip signal processing and distributions that will especially benefit terahertz wireless communications.

Topological Metamaterials

The topological properties of an object, associated with an integer called the topological invariant, are global features that cannot change continuously but only through abrupt variations, hence granting them intrinsic robustness. Engineered metamaterials (MMs) can be tailored to support highly nontrivial topological properties of their band structure, relative to their electronic, electromagnetic, acoustic and mechanical response, representing one of the major breakthroughs in physics over the past decade. Here, we review the foundations and the latest advances of topological photonic and phononic MMs, whose nontrivial wave interactions have become of great interest to a broad range of science disciplines, such as classical and quantum chemistry. We first introduce the basic concepts, including the notion of topological charge and geometric phase. We then discuss the topology of natural electronic materials, before reviewing their photonic/phononic topological MM analogues, including 2D topological MMs with and without time-reversal symmetry, Floquet topological insulators, 3D, higher-order, non-Hermitian and nonlinear topological MMs. We also discuss the topological aspects of scattering anomalies, chemical reactions and polaritons. This work aims at connecting the recent advances of topological concepts throughout a broad range of scientific areas and it highlights opportunities offered by topological MMs for the chemistry community and beyond.

Intrinsic Triple Degeneracy Point Bounded by Nodal Surfaces in Chiral Photonic Crystal

In periodic systems, band degeneracies are typically protected and classified by spatial symmetries. However, in photonic systems, the Γ point at zero frequency is an intrinsic degeneracy due to the polarization degree of freedom of electromagnetic waves. For chiral photonic crystals, such an intrinsic degeneracy carries ±2 chiral topological charge while having linear band dispersions, different from the general perception of charge-2 nodes being associated with quadratic dispersions. Here, we show that these topological characters originate from the spin-1 Weyl point at zero frequency node of triple degeneracy, due to the existence of an electrostatic flat band. Such a topological charge at zero frequency is usually buried in bulk band projections and has never been experimentally observed. To address this challenge, we introduce space-group screw symmetries in the design of chiral photonic crystal, which makes the Brillouin zone boundary an oppositely charged nodal surface enclosing the Γ point. As a result, the emergent Fermi arcs are forced to connect the projections of these topological singularities, enabling their experimental observation. The number of Fermi arcs then directly reveals the embedded topological charge at zero frequency.

Subwavelength control of light transport at the exceptional point by non-Hermitian metagratings

The concept of non-Hermitian physics, originally developed in the context of quantum field theory, has been investigated on distinct photonic platforms and created a plethora of counterintuitive phenomena. Interfacing non-Hermitian photonics and nanoplasmonics, here, we demonstrate unidirectional excitation and reflection of surface plasmon polaritons by elaborately designing the permittivity profile of non-Hermitian metagratings, in which the eigenstates of the system can coalesce at an exceptional point. Continuous tuning of the excitation or reflection ratios is also possible through altering the geometry of the metagrating. The controllable directionality and robust performance are attributed to the phase transition near the exceptional point, which is fully confirmed by the theoretic calculation, numerical simulation, and experimental characterization. Our work pushes non-Hermitian photonics to the nanoscale regime and paves the way toward high-performance plasmonic devices with superior controllability, performance, and robustness by using the topological effect associated with non-Hermitian systems.

Non-Hermitian Topolectrical Circuit Sensor with High Sensitivity

Electronic sensors play important roles in various applications, such as industry and environmental monitoring, biomedical sample ingredient analysis, wireless networks and so on. However, the sensitivity and robustness of current schemes are often limited by the low quality-factors of resonators and fabrication disorders. Hence, exploring new mechanisms of the electronic sensor with a high-level sensitivity and a strong robustness is of great significance. Here, a new way to design electronic sensors with superior performances based on exotic properties of non-Hermitian topological physics is proposed. Owing to the extreme boundary-sensitivity of non-Hermitian topological zero modes, the frequency shift induced by boundary perturbations can show an exponential growth trend with respect to the size of non-Hermitian topolectrical circuit sensors. Moreover, such an exponential growth sensitivity is also robust against disorders of circuit elements. Using designed non-Hermitian topolectrical circuit sensors, the ultrasensitive identification of the distance, rotation angle, and liquid level is further experimentally verified with the designed capacitive devices. The proposed non-Hermitian topolectrical circuit sensors can possess a wide range of applications in ultrasensitive environmental monitoring and show an exciting prospect for next-generation sensing technologies.