Spawning rings of exceptional points out of Dirac cones

Exceptional-point-enhanced phase sensing

Optical sensors, crucial in diverse fields like gravitational wave detection, biomedical imaging, and structural health monitoring, rely on optical phase to convey valuable information. Enhancing sensitivity is important for detecting weak signals. Exceptional points (EPs), identified in non-Hermitian systems, offer great potential for advanced sensors, given their marked response to perturbations. However, strict physical requirements for operating a sensor at EPs limit broader applications. Here, we introduce an EP-enhanced sensing platform featuring plug-in external sensors separated from an EP control unit. EPs are achieved without modifying the sensor, solely through control-unit adjustments. This configuration converts and amplifies optical phase changes into quantifiable spectral features. By separating sensing and control functions, we expand the applicability of EP enhancement to various conventional sensors. As a proof-of-concept, we demonstrate a sixfold reduction in the detection limit of fiber-optic strain sensing using this configuration. This work establishes a universal platform for applying EP enhancement to diverse phase-dependent structures, promising ultrahigh-sensitivity sensing across various applications.

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.

Bulk-Edge Correspondence for Nonlinear Eigenvalue Problems

Although topological phenomena attract growing interest not only in linear systems but also in nonlinear systems, the bulk-edge correspondence under the nonlinearity of eigenvalues has not been established so far. We address this issue by introducing auxiliary eigenvalues. We reveal that the topological edge states of auxiliary eigenstates are topologically inherited as physical edge states when the nonlinearity is weak but finite (i.e., auxiliary eigenvalues are monotonic as for the physical one). This result leads to the bulk-edge correspondence with the nonlinearity of eigenvalues.

Generalized bulk-boundary correspondence in periodically driven non-Hermitian systems

We present a pedagogical review of the periodically driven non-Hermitian systems, particularly on the rich interplay between the non-Hermitian skin effect and the topology. We start by reviewing the non-Bloch band theory of the static non-Hermitian systems and discuss the establishment of its generalized bulk-boundary correspondence (BBC). Ultimately, we focus on the non-Bloch band theory of two typical periodically driven non-Hermitian systems: harmonically driven non-Hermitian system and periodically quenched non-Hermitian system. The non-Bloch topological invariants were defined on the generalized Brillouin zone and the real space wave functions to characterize the Floquet non-Hermtian topological phases. Then, the generalized BBC was established for the two typical periodically driven non-Hermitian systems. Additionally, we review novel phenomena in the higher-dimensional periodically driven non-Hermitian systems, including Floquet non-Hermitian higher-order topological phases and Floquet hybrid skin-topological modes. The experimental realizations and recent advances have also been surveyed. Finally, we end with a summarization and hope this pedagogical review can motivate further research on Floquet non-Hermtian topological physics.

Occupation-Dependent Particle Separation in One-Dimensional Non-Hermitian Lattices

We unveil an exotic phenomenon arising from the intricate interplay between non-Hermiticity and many-body physics, namely, an occupation-dependent particle separation for hardcore bosons in a one-dimensional lattice driven by unidirectional non-Hermitian pumping. Taking hardcore bosons as an example, we find that a pair of particles occupying the same unit cell exhibit an opposite non-Hermitian pumping direction to that of unpaired ones occupying different unit cells. By turning on an intracell interaction, many-body eigenstates split in their real energies, forming separable clusters in the complex energy plane with either left-, right-, or bipolar-types of non-Hermitian skin effect (NHSE). The dependency of skin accumulating directions on particle occupation is further justified with local sublattice correlation and entanglement entropy of many-body eigenstates. Dynamically, this occupation-dependent NHSE manifests as uni- or bidirectional pumping for many-body initial states, allowing for spatially separating paired and unpaired particles. Our results unveil the possibility of designing and exploring novel non-Hermitian phases originated from particle nonconservation in subsystems (e.g., orbitals, sublattices, or spin species) and their spatial configurations.

Observation of Nonlinear Exceptional Points with a Complete Basis in Dynamics

The exotic physics associated with exceptional points (EPs) is always under the scrutiny of theoretical and experimental science. Recently, considerable effort has been invested in the combination of nonlinearity and non-Hermiticity. The concept of nonlinear EPs (NEPs) has been introduced, which can avoid the loss of completeness of the eigenbasis in dynamics while retaining the key features of linear EPs. Here, we present the first direct experimental demonstration of a NEP based on two non-Hermition coupled circuit resonators combined with a nonlinear saturable gain. At the NEP, the response of the eigenfrequency to perturbations demonstrates a third-order root law and the eigenbasis of the Hamiltonian governing the system dynamics is still complete. Our results bring this counterintuitive aspect of the NEP to light and possibly open new avenues for applications.

Topological spin textures in electronic non-Hermitian systems

Non-Hermitian systems have been discussed mostly in the context of open systems and nonequilibrium. Recent experimental progress is much from optical, cold-atomic, and classical platforms due to the vast tunability and clear identification of observables. However, their counterpart in solid-state electronic systems in equilibrium remains unmasked although highly desired, where a variety of materials are available, calculations are solidly founded, and accurate spectroscopic techniques can be applied. We demonstrate that, in the surface state of a topological insulator with spin-dependent relaxation due to magnetic impurities, highly nontrivial topological soliton spin textures appear in momentum space. Such spin-channel phenomena are delicately related to the type of non-Hermiticity and correctly reveal the most robust non-Hermitian features detectable spectroscopically. Moreover, the distinct topological soliton objects can be deformed to each other, mediated by topological transitions driven by tuning across a critical direction of doped magnetism. These results not only open a solid-state avenue to exotic spin patterns via spin- and angle-resolved photoemission spectroscopy, but also inspire non-Hermitian dissipation engineering of spins in solids.

Photonic Floquet Skin-Topological Effect

Non-Hermitian skin effect and photonic topological edge states are of great interest in non-Hermitian physics and optics. However, the interplay between them is largely unexplored. Here, we propose and demonstrate experimentally the non-Hermitian skin effect constructed from the nonreciprocal flow of Floquet topological edge states, which can be dubbed "Floquet skin-topological effect." We first show the non-Hermitian skin effect can be induced by structured loss when the one-dimensional (1D) system is periodically driven. Next, based on a two-dimensional (2D) Floquet topological photonic lattice with structured loss, we investigate the interaction between the non-Hermiticity and the topological edge states. We observe that all the one-way edge states are imposed onto specific corners, featuring both the non-Hermitian skin effect and topological edge states. Furthermore, a topological switch for the skin-topological effect is presented by utilizing the phase-transition mechanism. Our experiment paves the way for realizing non-Hermitian topological effects in nonlinear and quantum regimes.

Directive giant upconversion by supercritical bound states in the continuum

Photonic bound states in the continuum (BICs), embedded in the spectrum of free-space waves1,2 with diverging radiative quality factor, are topologically non-trivial dark modes in open-cavity resonators that have enabled important advances in photonics3,4. However, it is particularly challenging to achieve maximum near-field enhancement, as this requires matching radiative and non-radiative losses. Here we propose the concept of supercritical coupling, drawing inspiration from electromagnetically induced transparency in near-field coupled resonances close to the Friedrich-Wintgen condition2. Supercritical coupling occurs when the near-field coupling between dark and bright modes compensates for the negligible direct far-field coupling with the dark mode. This enables a quasi-BIC field to reach maximum enhancement imposed by non-radiative loss, even when the radiative quality factor is divergent. Our experimental design consists of a photonic-crystal nanoslab covered with upconversion nanoparticles. Near-field coupling is finely tuned at the nanostructure edge, in which a coherent upconversion luminescence enhanced by eight orders of magnitude is observed. The emission shows negligible divergence, narrow width at the microscale and controllable directivity through input focusing and polarization. This approach is relevant to various physical processes, with potential applications for light-source development, energy harvesting and photochemical catalysis.

Asymmetric Full-Color Vectorial Meta-holograms Empowered by Pairs of Exceptional Points

Holography holds tremendous promise in applications such as immersive virtual reality and optical communications. With the emergence of optical metasurfaces, planar optical components that have the remarkable ability to precisely manipulate the amplitude, phase, and polarization of light on the subwavelength scale have expanded the potential applications of holography. However, the realization of metasurface-based full-color vectorial holography remains particularly challenging. Here, we report a general approach utilizing a modified Gerchberg-Saxton algorithm to achieve spatially aligned full-color display and incorporating wavelength information with an image compensation strategy. We combine the Pancharatnam-Berry phase and pairs of exceptional points to address the issue of redundant twin images that generally appear for the two orthogonal circular polarizations and to enable full polarization control of the vectorial field. Our results enable the realization of an asymmetric full-color vectorial meta-hologram, paving the way for the development of full-color display, complex beam generation, and secure data storage applications.

Creating pairs of exceptional points for arbitrary polarization control: asymmetric vectorial wavefront modulation

Exceptional points (EPs) can achieve intriguing asymmetric control in non-Hermitian systems due to the degeneracy of eigenstates. Here, we present a general method that extends this specific asymmetric response of EP photonic systems to address any arbitrary fully-polarized light. By rotating the meta-structures at EP, Pancharatnam-Berry (PB) phase can be exclusively encoded on one of the circular polarization-conversion channels. To address any arbitrary wavefront, we superpose the optical signals originating from two orthogonally polarized -yet degenerate- EP eigenmodes. The construction of such orthogonal EP eigenstates pairs is achieved by applying mirror-symmetry to the nanostructure geometry flipping thereby the EP eigenmode handedness from left to right circular polarization. Non-Hermitian reflective PB metasurfaces designed using such EP superposition enable arbitrary, yet unidirectional, vectorial wavefront shaping devices. Our results open new avenues for topological wave control and illustrate the capabilities of topological photonics to distinctively operate on arbitrary polarization-state with enhanced performances.

Low-threshold single-mode nanowire array flat-band photonic-crystal surface-emitting lasers with high-reflectivity bottom mirrors

A Si-based nanowire array photonic-crystal surface-emitting laser based on a flat band is designed and simulated. By introducing an air gap between the nanowire and substrate, the bottom reflectivity is significantly enhanced, resulting in much lower threshold and smaller cutoff diameter. Through adjusting the lattice constant (the distance between neighboring nanowires) and nanowire diameter, a photonic crystal structure with a flat band is achieved, in which strong interaction between light and matter occurs in the flat band mode. For the device with a small size, single-mode lasing is obtained with a side-mode suppression ratio of 21 dB, high quality factor of 3940, low threshold gain of 624 cm-1, and small beam divergency angle of ∼7.5°. This work may pave the way for the development of high-performance Si-based surface-emitting nanolasers and high-density photonic integrated circuits.

Experimental Realization of Geometry-Dependent Skin Effect in a Reciprocal Two-Dimensional Lattice

Recent studies of non-Hermitian periodic lattices unveiled the non-Hermitian skin effect (NHSE), in which the bulk modes under the periodic boundary conditions (PBC) become skin modes under open boundary conditions. The NHSE is a topological effect owing to the nontrivial spectral winding, and such spectral behaviors appear naturally in nonreciprocal systems. Hence prevailing approaches rely on nonreciprocity to achieve the NHSE. Here, we report the experimental realization of the geometry-dependent skin effect in a two-dimensional reciprocal system, in which the skin effect occurs only at boundaries whose macroscopic symmetry mismatches with the lattice symmetry. The role of spectral reciprocity and symmetry is revealed by connecting reflective channels at given boundaries with the spectral topology of the PBC spectrum. Our work highlights the vital role of reciprocity, symmetry, and macroscopic geometry on the NHSE in dimensionality larger than one and opens new routes for wave structuring using non-Hermitian effects.

Observation of the geometry-dependent skin effect and dynamical degeneracy splitting

The non-Hermitian skin effect is a distinctive phenomenon in non-Hermitian systems, which manifests as the anomalous localization of bulk states at the boundary. To understand the physical origin of the non-Hermitian skin effect, a bulk band characterization based on the dynamical degeneracy on an equal frequency contour is proposed, which reflects the strong anisotropy of the spectral function. In this paper, we report the experimental observation of a newly-discovered geometry-dependent non-Hermitian skin effect and dynamical degeneracy splitting in a two-dimensional acoustic crystal and reveal their remarkable correspondence by performing single-frequency excitation measurements. Our work not only provides a controllable experimental platform for studying the non-Hermitian physics, but also confirms the unique correspondence between the non-Hermitian skin effect and the dynamical degeneracy splitting, paving a new way to characterize the non-Hermitian skin effect.

Realization and topological properties of third-order exceptional lines embedded in exceptional surfaces

As the counterpart of Hermitian nodal structures, the geometry formed by exceptional points (EPs), such as exceptional lines (ELs), entails intriguing spectral topology. We report the experimental realization of order-3 exceptional lines (EL3) that are entirely embedded in order-2 exceptional surfaces (ES2) in a three-dimensional periodic synthetic momentum space. The EL3 and the concomitant ES2, together with the topology of the underlying space, prohibit the evaluation of their topology in the eigenvalue manifold by prevailing topological characterization methods. We use a winding number associated with the resultants of the Hamiltonian. This resultant winding number can be chosen to detect only the EL3 but ignores the ES2, allowing the diagnosis of the topological currents carried by the EL3, which enables the prediction of their evolution under perturbations. We further reveal the connection between the intersection multiplicity of the resultants and the winding of the resultant field around the EPs and generalize the approach for detecting and topologically characterizing higher-order EPs. Our work exemplifies the unprecedented topology of higher-order exceptional geometries and may inspire new non-Hermitian topological applications.

Spin-dependent properties of optical modes guided by adiabatic trapping potentials in photonic Dirac metasurfaces

The Dirac-like dispersion in photonic systems makes it possible to mimic the dispersion of relativistic spin-1/2 particles, which led to the development of the concept of photonic topological insulators. Despite recent demonstrations of various topological photonic phases, the full potential offered by Dirac photonic systems, specifically their ability to emulate the spin degree of freedom-referred to as pseudo-spin-beyond topological boundary modes has remained underexplored. Here we demonstrate that photonic Dirac metasurfaces with smooth one-dimensional trapping gauge potentials serve as effective waveguides with modes carrying pseudo-spin. We show that spatially varying gauge potentials act unevenly on the two pseudo-spins due to their different field distributions, which enables control of guided modes by their spin, a property that is unattainable with conventional optical waveguides. Silicon nanophotonic metasurfaces are used to experimentally confirm the properties of these guided modes and reveal their distinct spin-dependent radiative character; modes of opposite pseudo-spin exhibit disparate radiative lifetimes and couple differently to incident light. The spin-dependent field distributions and radiative lifetimes of their guided modes indicate that photonic Dirac metasurfaces could be used for spin-multiplexing, controlling the characteristics of optical guided modes, and tuning light-matter interactions with photonic pseudo-spins.

Loss-induced Purcell enhancement in PT-broken whispering gallery microcavities

Parity-time (PT)-symmetry brings various opportunities for electromagnetic field manipulation and light-matter interaction, such as modification of spontaneous emission. However, previous works mainly focused on the behavior of spontaneous emission at exceptional points or in the PT-symmetry situation. Here, we theoretically demonstrate loss-induced Purcell enhancement in PT-broken whispering gallery microcavities. In the PT-broken phase, one of the supermodes decays slowly thereby playing a leading role in spontaneous emission. As the loss increases, the quality factor of this supermode is higher and the mode volume is smaller, so that the Purcell factors will be larger if the emitter is placed near the lossless cavity. Our findings indicate that loss can enhance the interaction between light and matter, which could be applied to single photon emission, non-Hermitian photonic devices, etc.

Observation of geometry-dependent skin effect in non-Hermitian phononic crystals with exceptional points

Exceptional points and skin effect, as the two distinct hallmark features unique to the non-Hermitian physics, have each attracted enormous interests. Recent theoretical works reveal that the topologically nontrivial exceptional points can guarantee the non-Hermitian skin effect, which is geometry-dependent, relating these two unique phenomena. However, such novel relation remains to be confirmed by experiments. Here, we realize a non-Hermitian phononic crystal with exceptional points, which exhibits the geometry-dependent skin effect. The exceptional points connected by the bulk Fermi arcs, and the skin effects with the geometry dependence, are evidenced in simulations and experiments. Our work, building an experimental bridge between the exceptional points and skin effect and uncovering the unconventional geometry-dependent skin effect, expands a horizon in non-Hermitian physics.

Loss Difference Induced Localization in a Non-Hermitian Honeycomb Photonic Lattice

Non-Hermitian systems with complex-valued energy spectra provide an extraordinary platform for manipulating unconventional dynamics of light. Here, we demonstrate the localization of light in an instantaneously reconfigurable non-Hermitian honeycomb photonic lattice that is established in a coherently prepared atomic system. One set of the sublattices is optically modulated to introduce the absorptive difference between neighboring lattice sites, where the Dirac points in reciprocal space are extended into dispersionless local flat bands, with two shared eigenstates: low-loss (high-loss) one with fields confined at sublattice B (A). When these local flat bands are broad enough due to larger loss difference, the incident beam with its tangential wave vector being at the K point in reciprocal space is effectively localized at sublattice B with weaker absorption, namely, the commonly seen power exchange between adjacent channels in photonic lattices is effectively prohibited. The current work unlocks a new capability from non-Hermitian two-dimensional photonic lattices and provides an alternative route for engineering tunable local flat bands in photonic structures.

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.

Higher-order hybrid topological bound states in a non-Hermitian system

Higher-order topological states, such as the corner and pseudo-hinge states, have been discovered in both Hermitian and non-Hermitian systems. These states have inherent high-quality factors that make them useful in the application of photonic devices. In this work, we design a non-Hermiticity solely induced Su-Schrieffer-Heeger (SSH) lattice and demonstrate the existence of diverse higher-order topological bound states in the continuum (BICs). In particular, we first uncover some hybrid topological states that occur in the form of BICs in the non-Hermitian system. Furthermore, these hybrid states with an amplified and localized field have been demonstrated to excite nonlinear harmonic generation with high efficiency. The appearance of these topological bound states will advance the study of the interplay of topology, BICs, and non-Hermitian optics.

Nonlinearity-enabled higher-order exceptional singularities with ultra-enhanced signal-to-noise ratio

Higher-order exceptional points (HOEPs) with extraordinary responsivity are expected to exhibit a vastly improved performance in detection-related applications. However, over the past few years, such an approach has been questioned due to several potential drawbacks, including the stringent parameter requirements, fundamental resolution limits and noise. Here, exploring the consequence of nonlinear gain saturation in exceptional singularities of non-Hermitian systems, we offer a feasible scheme to overcome all the above difficulties. We provide a simple and intuitive example by demonstrating with both theory and circuit experiments an 'exceptional nexus' ('EX'), a HOEP with an ultra-enhanced signal-to-noise ratio (SNR), in only two coupled resonators with the aid of nonlinear gain. The tedious parameter tuning in a six-dimensional hyper-dimensional space is reduced to two dimensions. The feedback mechanism of nonlinear saturable gain can give a solution to the ongoing debate on the SNR of EPs in other linear systems. Our findings advance the fundamental understanding of the peculiar topology of nonlinear non-Hermitian systems, significantly reduce the practical difficulty in EP sensing and possibly open new avenues for applications.

Observation of Exceptional Points in Thermal Atomic Ensembles

Exceptional points (EPs) in non-Hermitian systems have recently attracted wide interest and spawned intriguing prospects for enhanced sensing. However, EPs have not yet been realized in thermal atomic ensembles, which is one of the most important platforms for quantum sensing. Here we experimentally observe EPs in multilevel thermal atomic ensembles and realize enhanced sensing of the magnetic field for 1 order of magnitude. We take advantage of the rich energy levels of atoms and construct effective decays for selected energy levels by employing laser coupling with the excited state, yielding unbalanced decay rates for different energy levels, which finally results in the existence of EPs. Furthermore, we propose the optical polarization rotation measurement scheme to detect the splitting of the resonance peaks, which makes use of both the absorption and dispersion properties and shows an advantage with enhanced splitting compared with the conventional transmission measurement scheme. Additionally, in our system both the effective coupling strength and decay rates are flexibly adjustable, and thus the position of the EPs are tunable, which expands the measurement range. Our Letter not only provides a new controllable platform for studying EPs and non-Hermitian physics, but also provide new ideas for the design of EP-enhanced sensors and opens up realistic opportunities for practical applications in the high-precision sensing of magnetic field and other physical quantities.

Nonlinear Exceptional Points with a Complete Basis in Dynamics

Exceptional points (EPs) are special spectral singularities at which two or more eigenvalues, and their corresponding eigenvectors, coalesce and become identical. In conventional wisdom, the coalescence of eigenvectors inevitably leads to the loss of completeness of the eigenbasis. Here, we show that this scenario breaks down in general at nonlinear EPs (NEPs). As an example, we realize a fifth-order NEP (NEP_{5}) within only three coupled resonators with both a theoretical model and simulations in circuits. One stable and another four auxiliary steady eigenstates of the nonlinear Hamiltonian coalesce at the NEP_{5}, and the response of eigenfrequency to perturbations demonstrates a fifth-order root law. Intriguingly, the biorthogonal eigenbasis of the Hamiltonian governing the system dynamics is still complete, and this fact is corroborated by a finite Petermann factor instead of a divergent one at conventional EPs. Consequently, the amplification of noise, which diverges at other EPs, converges at our NEP_{5}. Our finding transforms the understanding of EPs and shows potential for miniaturizing various key applications operating near EPs.

Collective excitations of a bound-in-the-continuum condensate

Spectra of low-lying elementary excitations are critical to characterize properties of bosonic quantum fluids. Usually these spectra are difficult to observe, due to low occupation of non-condensate states compared to the ground state. Recently, low-threshold Bose-Einstein condensation was realised in a symmetry-protected bound state in the continuum, at a saddle point, thanks to coupling of this electromagnetic resonance to semiconductor excitons. While it has opened the door to long-living polariton condensates, their intrinsic collective properties are still unexplored. Here we unveil the peculiar features of the Bogoliubov spectrum of excitations in this system. Thanks to the dark nature of the bound-in-the-continuum state, collective excitations lying directly above the condensate become observable in enhanced detail. We reveal interesting aspects, such as energy-flat parts of the dispersion characterized by two parallel stripes in photoluminescence pattern, pronounced linearization at non-zero momenta in one of the directions, and a strongly anisotropic velocity of sound.

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.

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.

Parity-time symmetry breaking optical nanocircuit

Gain and loss balanced parity-time (PT) inversion symmetry has been achieved across multiple platforms including acoustics, electronics, and photonics. Tunable subwavelength asymmetric transmission based on PT symmetry breaking has attracted great interest. However, due to the diffraction limit, the geometric size of an optical PT symmetric system is much larger than the resonant wavelength, which limits the device miniaturization. Here, we theoretically studied a subwavelength optical PT symmetry breaking nanocircuit based on the similarity between a plasmonic system and an RLC circuit. Firstly, the asymmetric coupling of an input signal is observed by varying the coupling strength and gain-loss ratio between the nanocircuits. Furthermore, a subwavelength modulator is proposed by modulating the gain of the amplified nanocircuit. Notably, the modulation effect near the exceptional point is remarkable. Finally, we introduce a four-level atomic model modified by the Pauli exclusion principle to simulate the nonlinear dynamics of a PT symmetry broken laser. The asymmetric emission of a coherent laser is realized by full-wave simulation with a contrast of about 50. This subwavelength optical nanocircuit with broken PT symmetry is of great significance for realizing directional guided light, modulator and asymmetric-emission laser at subwavelength scales.

Probing Complex-Energy Topology via Non-Hermitian Absorption Spectroscopy in a Trapped Ion Simulator

Non-Hermitian systems generically have complex energies, which may host topological structures, such as links or knots. While there has been great progress in experimentally engineering non-Hermitian models in quantum simulators, it remains a significant challenge to experimentally probe complex energies in these systems, thereby making it difficult to directly diagnose complex-energy topology. Here, we experimentally realize a two-band non-Hermitian model with a single trapped ion whose complex eigenenergies exhibit the unlink, unknot, or Hopf link topological structures. Based on non-Hermitian absorption spectroscopy, we couple one system level to an auxiliary level through a laser beam and then experimentally measure the population of the ion on the auxiliary level after a long period of time. Complex eigenenergies are then extracted, illustrating the unlink, unknot, or Hopf link topological structure. Our work demonstrates that complex energies can be experimentally measured in quantum simulators via non-Hermitian absorption spectroscopy, thereby opening the door for exploring various complex-energy properties in non-Hermitian quantum systems, such as trapped ions, cold atoms, superconducting circuits, or solid-state spin systems.

Non-Hermitian chiral degeneracy of gated graphene metasurfaces

Non-Hermitian degeneracies, also known as exceptional points (EPs), have been the focus of much attention due to their singular eigenvalue surface structure. Nevertheless, as pertaining to a non-Hermitian metasurface platform, the reduction of an eigenspace dimensionality at the EP has been investigated mostly in a passive repetitive manner. Here, we propose an electrical and spectral way of resolving chiral EPs and clarifying the consequences of chiral mode collapsing of a non-Hermitian gated graphene metasurface. More specifically, the measured non-Hermitian Jones matrix in parameter space enables the quantification of nonorthogonality of polarisation eigenstates and half-integer topological charges associated with a chiral EP. Interestingly, the output polarisation state can be made orthogonal to the coalesced polarisation eigenstate of the metasurface, revealing the missing dimension at the chiral EP. In addition, the maximal nonorthogonality at the chiral EP leads to a blocking of one of the cross-polarised transmission pathways and, consequently, the observation of enhanced asymmetric polarisation conversion. We anticipate that electrically controllable non-Hermitian metasurface platforms can serve as an interesting framework for the investigation of rich non-Hermitian polarisation dynamics around chiral EPs.

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.

Topological Unidirectional Guided Resonances Emerged from Interband Coupling

Unidirectional guided resonances (UGRs) are optical modes in photonic crystal slabs that radiate toward one side without the need for mirrors on the other. In this Letter, we report a mechanism to realize UGRs by tuning the interband coupling effect originating from up-down symmetry breaking. We theoretically find that UGRs that reside along high-symmetric lines correspond to phase singularities of far-field radiation, depicted by phase winding numbers as a type of topological indices. We investigate the phase dislocation lines in three-dimensional parameter space and elaborate on the interplay between UGRs and non-Hermitian degeneracies accordingly. Our findings reveal the topological nature of UGRs about their generation, evolution, and annihilation in general parameter spaces, thus paving the way to new possibilities of light manipulation.

The Promise of Soft-Matter-Enabled Quantum Materials

The field of quantum materials has experienced rapid growth over the past decade, driven by exciting new discoveries with immense transformative potential. Traditional synthetic methods to quantum materials have, however, limited the exploration of architectural control beyond the atomic scale. By contrast, soft matter self-assembly can be used to tailor material structure over a large range of length scales, with a vast array of possible form factors, promising emerging quantum material properties at the mesoscale. This review explores opportunities for soft matter science to impact the synthesis of quantum materials with advanced properties. Existing work at the interface of these two fields is highlighted, and perspectives are provided on possible future directions by discussing the potential benefits and challenges which can arise from their bridging.

Realization of exceptional points along a synthetic orbital angular momentum dimension

Exceptional points (EPs), at which more than one eigenvalue and eigenvector coalesce, are unique spectral features of non-Hermiticity (NH) systems. They exist widely in open systems with complex energy spectra. We experimentally demonstrate the appearance of paired EPs in a periodical-driven degenerate optical cavity along the synthetic orbital angular momentum dimension with a tunable parameter. The complex-energy band structures and the key features of EPs, i.e., their bulk Fermi arcs, parity-time symmetry breaking transition, energy swapping, and half-integer band windings, are directly observed by detecting the wavefront angle-resolved transmission spectrum. Our results demonstrate the flexibility of using the photonic synthetic dimensions to implement NH systems beyond their geometric dimension and EP-based sensing.

On-chip mechanical exceptional points based on an optomechanical zipper cavity

Exceptional points (EPs) represent a distinct type of spectral singularity in non-Hermitian systems, and intriguing physics concepts have been studied with optical EPs recently. As a system beyond photonics, the mechanical oscillators coupling with many physical systems are expected to be further exploited EPs for mechanical sensing, topology energy transfer, nonreciprocal dynamics, etc. In this study, we demonstrated on-chip mechanical EPs with a silicon optomechanical zipper cavity, wherein two near-degenerate mechanical breathing modes are coupled via a single colocalized optical mode. By tailoring the dissipative and coherent couplings between two mechanical oscillators, the spectral splitting with 1/2 order response, a distinctive feature of EP, was observed successfully. Our work provides an integrated platform for investigating the physics related to mechanical EPs on silicon chips and suggests their possible applications for ultrasensitive measurements.

Stable Atomic Magnetometer in Parity-Time Symmetry Broken Phase

Random motion of spins is usually detrimental in magnetic resonance experiments. The spin diffusion in nonuniform magnetic fields causes broadening of the resonance and limits the sensitivity and the spectral resolution in applications like magnetic resonance spectroscopy. Here, by observation of the parity-time (PT) phase transition of diffusive spins in gradient magnetic fields, we show that the spatial degrees of freedom of atoms could become a resource, rather than harmful, for high-precision measurement of weak signals. In the normal phase with zero or low gradient fields, the diffusion results in dissipation of spin precession. However, by increasing the field gradient, the spin system undergoes a PT transition, and enters the PT symmetry broken phase. In this novel phase, the spin precession frequency splits due to spatial localization of the eigenmodes. We demonstrate that, using these spatial-motion-induced split frequencies, the spin system can serve as a stable magnetometer, whose output is insensitive to the inevitable long-term drift of control parameters. This opens a door to detect extremely weak signals in imperfectly controlled environments.

Properties of the non-Hermitian SSH model: role ofPTsymmetry

The present work addresses the distinction between the topological properties ofPTsymmetric and non-PTsymmetric scenarios for the non-Hermitian Su-Schrieffer-Heeger model. The non-PTsymmetric case is represented by non-reciprocity in both the inter- and the intra-cell hopping amplitudes, while the one withPTsymmetry is modeled by a complex on-site staggered potential. In particular, we study the loci of the exceptional points, the winding numbers, band structures, and explore the breakdown of bulk-boundary correspondence (BBC). We further study the interplay of the dimerization strengths on the observables for these cases. The non-PTsymmetric case denotes a more familiar situation, where the winding number abruptly changes by half-integer through tuning of the non-reciprocity parameters, and demonstrates a complete breakdown of BBC, thereby showing non-Hermitian skin effect. The topological nature of thePTsymmetric case appears to follow closely to its Hermitian analogue, except that it shows unbroken (broken) regions with complex (purely real) energy spectra, while another variant of the winding number exhibits a continuous behavior as a function of the strength of the potential, while the conventional BBC is preserved.

Diverse lateral shifts of beams in non-Hermitian waveguide arrays

Non-Hermitian systems have attracted considerable attention in optics due to the rich physics introduced by the existence of real spectra and exceptional points (EPs), which is exploited in lasers, optical sensors and on-chip manipulations of light. Here, focusing on the dynamics of beams in non-Hermitian waveguide arrays supporting a ring of EPs (exceptional ring) and 3rd-order EPs, we theoretically demonstrate that the center of energy of a beam prepared around an eigenstate of the waveguide array near EPs could exhibit non-zero shifts in the lateral direction during its propagation. When the initial state of the beam prepared around an eigenstate inside (outside) the exceptional ring with the imaginary (real) eigenvalue, the lateral shifts of the beams are manifested by the non-oscillating (Zitterbewegung-like) motions, which are robust to the perturbations of coupling coefficients between waveguides. Remarkably, the amplitude of the non-oscillating shift is dependent on a non-Hermitian Berry connection (U(1) gauge invariance). It contradicts the conventional wisdom that the Berry connection cannot induce the dynamic effect. Furthermore, near the high-order EPs, the initial-state-dependent lateral shifts of the beams present diversity, such as multifrequencies and destructive interferences. The counterintuitive lateral shifts of the beams stem from the non-orthogonal nature of eigenstate of the non-Hermitian systems, which may open a gateway towards the non-Hermitian beam dynamics and manipulations of beams.

Universal characteristics of one-dimensional non-Hermitian superconductors

We establish a non-Bloch band theory for one-dimensional(1D) non-Hermitian topological superconductors. The universal physical properties of non-Hermitian topological superconductors are revealed based on the theory. According to the particle-hole symmetry, there exist reciprocal particle and hole loops of generalized Brillouin zone. The critical point of quantum phase transition, where the energy gap closes, appears when the particle and hole loops intersect at Bloch points. If the non-Hermitian system has non-Hermitian skin effects, the non-Hermitian skin effect should be theZ₂skin effect: the corresponding eigenstates of particle and hole localize at opposite ends of an open chain, respectively. The non-Bloch band theory is applied to two examples, non-Hermitianp- ands-wave topological superconductors. In terms of Majorana Pfaffian, aZ₂non-Bloch topological invariant is defined to establish the non-Hermitian bulk-boundary correspondence for the non-Hermitian topological superconductors.

Imaginary Gauge Transformation in Momentum Space and Dirac Exceptional Point

An imaginary gauge transformation is at the core of the non-Hermitian skin effect. Here, we show that such a transformation can be performed in momentum space as well, which reveals that certain gain- and loss-modulated systems in their parity-time (PT) symmetric phases are equivalent to Hermitian systems with real potentials. Our analysis in momentum space also distinguishes two types of exceptional points (EPs) in the same system. Besides the conventional type that leads to a PT transition upon the continuous increase of gain and loss, we find real-valued energy bands connected at a Dirac EP in hybrid dimensions, consisting of a spatial dimension and a synthetic dimension for the gain and loss strength.

Topological beaming of light

Nanophotonic light emitters are key components in numerous application areas because of their compactness and versatility. Here, we propose a topological beam emitter structure that takes advantage of submicrometer footprint size, small divergence angle, high efficiency, and adaptable beam shaping capability. The proposed structure consists of a topological junction of two guided-mode resonance gratings inducing a leaky Jackiw-Rebbi state resonance. The leaky Jackiw-Rebbi state leads to in-plane optical confinement with funnel-like energy flow and enhanced emission probability, resulting in highly efficient optical beam emission. In addition, the structure allows adaptable beam shaping for any desired positive definite profiles by means of Dirac mass distribution control, which can be directly encoded in lattice geometry parameters. Therefore, the proposed approach provides highly desirable properties for efficient micro-light emitters and detectors in various applications including display, solid-state light detection and ranging, laser machining, label-free sensors, optical interconnects, and telecommunications.

Designing non-Hermitian real spectra through electrostatics

Non-hermiticity presents a vast newly opened territory that harbors new physics and applications such as lasing and sensing. However, only non-Hermitian systems with real eigenenergies are stable, and great efforts have been devoted in designing them through enforcing parity-time (PT) symmetry. In this work, we exploit a lesser-known dynamical mechanism for enforcing real-spectra, and develop a comprehensive and versatile approach for designing new classes of parent Hamiltonians with real spectra. Our design approach is based on a new electrostatics analogy for modified non-Hermitian bulk-boundary correspondence, where electrostatic charge corresponds to density of states and electric fields correspond to complex spectral flow. As such, Hamiltonians of any desired spectra and state localization profile can be reverse-engineered, particularly those without any guiding symmetry principles. By recasting the diagonalization of non-Hermitian Hamiltonians as a Poisson boundary value problem, our electrostatics analogy also transcends the gain/loss-induced compounding of floating-point errors in traditional numerical methods, thereby allowing access to far larger system sizes.

Stabilized Dirac points in one-dimensional non-Hermitian optical lattices

We demonstrate stable Dirac points (DPs) in low dimensions by taking advantage of non-Hermiticity in an optical lattice composed of two coupled Su-Schrieffer-Heeger chains. The occurrence of DPs stems from the constraints of pseudo-Hermiticity and charge-conjugation parity symmetry, which force the system to support both real bands and orthogonal eigenmodes despite its non-Hermitian nature. The two characteristics hold even at spectral degeneracies of zero energy, giving rise to non-Hermitian DPs. We show that DPs are stable with the variation of dissipation since they are topological charges and can develop into nodal rings in two dimensions. Moreover, we investigate the beam dynamics around DPs and observe beam splitting with stable power evolution. The study paves the way for controlling the flow of light to aid dissipation together with high stability of energy.

Annihilation of exceptional points from different Dirac valleys in a 2D photonic system

Topological physics relies on Hamiltonian’s eigenstate singularities carrying topological charges, such as Dirac points, and – in non-Hermitian systems – exceptional points (EPs), lines or surfaces. So far, the reported non-Hermitian topological transitions were related to the creation of a pair of EPs connected by a Fermi arc out of a single Dirac point by increasing non-Hermiticity. Such EPs can annihilate by reducing non-Hermiticity. Here, we demonstrate experimentally that an increase of non-Hermiticity can lead to the annihilation of EPs issued from different Dirac points (valleys). The studied platform is a liquid crystal microcavity with voltage-controlled birefringence and TE-TM photonic spin-orbit-coupling. Non-Hermiticity is provided by polarization-dependent losses. By increasing the non-Hermiticity degree, we control the position of the EPs. After the intervalley annihilation, the system becomes free of any band singularity. Our results open the field of non-Hermitian valley-physics and illustrate connections between Hermitian topology and non-Hermitian phase transitions.

The authors study a liquid crystal microcavity with polarization-dependent absorption, a source of non-Hermiticity. The transition in the Hermitian topology of the spin-orbit coupling makes possible the annihilation of exceptional points.

Non-Hermitian Absorption Spectroscopy

While non-Hermitian Hamiltonians have been experimentally realized in cold atom systems, it remains an outstanding open question of how to experimentally measure their complex energy spectra in momentum space for a realistic system with boundaries. The existence of non-Hermitian skin effects may make the question even more difficult to address given the fact that energy spectra for a system with open boundaries are dramatically different from those in momentum space; the fact may even lead to the notion that momentum-space band structures are not experimentally accessible for a system with open boundaries. Here, we generalize the widely used radio-frequency spectroscopy to measure both real and imaginary parts of complex energy spectra of a non-Hermitian quantum system for either bosonic or fermionic atoms. By weakly coupling the energy levels of a non-Hermitian system to auxiliary energy levels, we theoretically derive a formula showing that the decay of atoms on the auxiliary energy levels reflects the real and imaginary parts of energy spectra in momentum space. We further prove that measurement outcomes are independent of boundary conditions in the thermodynamic limit, providing strong evidence that the energy spectrum in momentum space is experimentally measurable. We finally apply our non-Hermitian absorption spectroscopy protocol to the Hatano-Nelson model and non-Hermitian Weyl semimetals to demonstrate its feasibility.

Experimental Realization of Weyl Exceptional Rings in a Synthetic Three-Dimensional Non-Hermitian Phononic Crystal

Weyl points-topological monopoles of quantized Berry flux-are predicted to spread to Weyl exceptional rings in the presence of non-Hermiticity. Here, we use a one-dimensional Aubry-Andre-Harper model to construct a Weyl semimetal in a three-dimensional parameter space comprising one reciprocal dimension and two synthetic dimensions. The inclusion of non-Hermiticity in the form of gain and loss produces a synthetic Weyl exceptional ring (SWER). The topology of the SWER is characterized by both its topological charge and non-Hermitian winding numbers. We experimentally observe the SWER and synthetic Fermi arc in a one-dimensional phononic crystal with the non-Hermiticity introduced by active acoustic components. Our findings pave the way for studying the high-dimensional non-Hermitian topological physics in acoustics.

Unveiling the Enhancement of Spontaneous Emission at Exceptional Points

Exceptional points (EPs), singularities of non-Hermitian physics where complex spectral resonances degenerate, are one of the most exotic features of nonequilibrium open systems with unique properties. For instance, the emission rate of quantum emitters placed near resonators with EPs is enhanced (compared to the free-space emission rate) by a factor that scales quadratically with the resonance quality factor. Here, we verify the theory of spontaneous emission at EPs by measuring photoluminescence from photonic-crystal slabs that are embedded with a high-quantum-yield active material. While our experimental results verify the theoretically predicted enhancement, they also highlight the practical limitations on the enhancement due to material loss. Our designed structures can be used in applications that require enhanced and controlled emission, such as quantum sensing and imaging.

A non-Hermitian optical atomic mirror

Explorations of symmetry and topology have led to important breakthroughs in quantum optics, but much richer behaviors arise from the non-Hermitian nature of light-matter interactions. A high-reflectivity, non-Hermitian optical mirror can be realized by a two-dimensional subwavelength array of neutral atoms near the cooperative resonance associated with the collective dipole modes. Here we show that exceptional points develop from a nondefective degeneracy by lowering the crystal symmetry of a square atomic lattice, and dispersive bulk Fermi arcs that originate from exceptional points are truncated by the light cone. From its nontrivial energy spectra topology, we demonstrate that the geometry-dependent non-Hermitian skin effect emerges in a ribbon geometry. Furthermore, skin modes localized at a boundary show a scale-free behavior that stems from the long-range interaction and whose mechanism goes beyond the framework of non-Bloch band theory. Our work opens the door to the study of the interplay among non-Hermiticity, topology, and long-range interaction.

General recipe to realize photonic-crystal surface-emitting lasers with 100-W-to-1-kW single-mode operation

Realization of one-chip, ultra-large-area, coherent semiconductor lasers has been one of the ultimate goals of laser physics and photonics for decades. Surface-emitting lasers with two-dimensional photonic crystal resonators, referred to as photonic-crystal surface-emitting lasers (PCSELs), are expected to show promise for this purpose. However, neither the general conditions nor the concrete photonic crystal structures to realize 100-W-to-1-kW-class single-mode operation in PCSELs have yet to be clarified. Here, we analytically derive the general conditions for ultra-large-area (3~10 mm) single-mode operation in PCSELs. By considering not only the Hermitian but also the non-Hermitian optical couplings inside PCSELs, we mathematically derive the complex eigenfrequencies of the four photonic bands around the Γ point as well as the radiation constant difference between the fundamental and higher-order modes in a finite-size device. We then reveal concrete photonic crystal structures which allow the control of both Hermitian and non-Hermitian coupling coefficients to achieve 100-W-to-1-kW-class single-mode lasing.

Here, the authors analytically derive the general conditions for 100-W-to-1-kW-class single-mode operation in ultra-large-area (3~10 mm) photonic crystal lasers. Such high power single-mode semiconductor lasers will bring innovation to a wide variety of fields.

Measuring the knot of non-Hermitian degeneracies and non-commuting braids

Any system of coupled oscillators may be characterized by its spectrum of resonance frequencies (or eigenfrequencies), which can be tuned by varying the system's parameters. The relationship between control parameters and the eigenfrequency spectrum is central to a range of applications^1-3. However, fundamental aspects of this relationship remain poorly understood. For example, if the controls are varied along a path that returns to its starting point (that is, around a 'loop'), the system's spectrum must return to itself. In systems that are Hermitian (that is, lossless and reciprocal), this process is trivial and each resonance frequency returns to its original value. However, in non-Hermitian systems, where the eigenfrequencies are complex, the spectrum may return to itself in a topologically non-trivial manner, a phenomenon known as spectral flow. The spectral flow is determined by how the control loop encircles degeneracies, and this relationship is well understood for [Formula: see text] (where [Formula: see text] is the number of oscillators in the system)^4,5. Here we extend this description to arbitrary [Formula: see text]. We show that control loops generically produce braids of eigenfrequencies, and for [Formula: see text] these braids form a non-Abelian group that reflects the non-trivial geometry of the space of degeneracies. We demonstrate these features experimentally for [Formula: see text] using a cavity optomechanical system.