Topological hybrid silicon microlasers

Topological nanophotonics and artificial neural networks

We propose the use of artificial neural networks to design and characterize photonic topological insulators. As a hallmark, the band structures of these systems show the key feature of the emergence of edge states, with energies lying within the energy gap of the bulk materials and localized at the boundary between regions of distinct topological invariants. We consider different structures such as one-dimensional photonic crystals, [Formula: see text]-symmetric chains and cylindrical systems and show how, through a machine learning application, one can identify the parameters of a complex topological insulator to obtain protected edge states at target frequencies. We show how artificial neural networks can be used to solve the long-standing quest for a solution to inverse problems solution and apply this to the cutting edge topic of topological nanophotonics.

Actively controlled asymmetric edge states for directional wireless power transfer

Wireless power transfer (WPT) has triggered immense research interest in a range of practical applications, including mobile phones, logistic robots, medical-implanted devices and electric vehicles. With the development of WPT devices, efficient long-range and robust WPT is highly desirable but also challenging. In addition, it is also very important to actively control the transmission direction of long-range WPT. Recently, the rise of topological photonics provides a powerful tool for near-field robust control of WPT. Considering the technical requirements of robustness, long-range and directionality, in this work we design and fabricate a one-dimensional quasiperiodic Harper chain and realize the robust directional WPT using asymmetric topological edge states. Specially, by further introducing a power source into the system, we selectively light up two Chinese characters, which are composed of LED lamps at both ends of the chain, to intuitively show the long-range directional WPT. Moreover, by adding variable capacitance diodes into the topological quasiperiodic chain, we present an experimental demonstration of the actively controlled directional WPT based on electrically controllable coil resonators. With the increase in voltage, we measure the transmission at two ends of the chain and observe the change of transmission direction. The realization of an actively tuned topological edge states in the topological quasiperiodic chain will open up a new avenue in the dynamical control of robust long-range WPT.

Wurtzite InP microdisks: from epitaxy to room-temperature lasing

Metastable wurtzite crystal phases of conventional semiconductors comprise enormous potential for high-performance electro-optical devices, owed to their extended tunable direct band gap range. However, synthesizing these materials in good quality and beyond nanowire size constraints has remained elusive. In this work, the epitaxy of wurtzite InP microdisks and related geometries on insulator for advanced optical applications is explored. This is achieved by an elaborate combination of selective area growth of fins and a zipper-induced epitaxial lateral overgrowth, which enables co-integration of diversely shaped crystals at precise position. The grown material possesses high phase purity and excellent optical quality characterized by STEM and µ-PL. Optically pumped lasing at room temperature is achieved in microdisks with a lasing threshold of 365 µJ cm^-2. Our platform could provide novel geometries for photonic applications.

Topological Dynamic Matter

The principles of topology in condensed matter physics have expanded to areas such as photonics, acoustics, electronics, and mechanics. Their extension to dynamic (soft) matter could enable the control and design of topological thermodynamic (micro)states and nonreciprocal dynamics, potentially leading to paradigmatic applications in molecular and thermal waveguiding, logics, and energy management. This Perspective explores distinct topological concepts for dynamic matter and prospective function. Topological tools are exemplified and discussed for the study of nonlocal order parameters or invariants in dynamic molecular matter, toward the engineering of assemblies, reactions, and system chemistry with unconventional global properties-a scope which has the potential to push the frontiers of physical chemistry and transform chemical topology from form to function.

Particle-antiparticle duality and fractionalization of topological chiral solitons

Although a prototypical Su–Schrieffer–Heeger (SSH) soliton exhibits various important topological concepts including particle-antiparticle (PA) symmetry and fractional fermion charges, there have been only few advances in exploring such properties of topological solitons beyond the SSH model. Here, by considering a chirally extended double-Peierls-chain model, we demonstrate novel PA duality and fractional charge e/2 of topological chiral solitons even under the chiral symmetry breaking. This provides a counterexample to the belief that chiral symmetry is necessary for such PA relation and fractionalization of topological solitons in a time-reversal invariant topological system. Furthermore, we discover that topological chiral solitons are re-fractionalized into two subsolitons which also satisfy the PA duality. As a result, such dualities and fractionalizations support the topological \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}_4$$\end{document}Z4 algebraic structures. Our findings will inspire researches seeking feasible and promising topological systems, which may lead to new practical applications such as solitronics.

Tailoring topological edge states with photonic crystal nanobeam cavities

The realization of topological edge states (TESs) in photonic systems has provided unprecedented opportunities for manipulating light in novel manners. The Su-Schrieffer-Heeger (SSH) model has recently gained significant attention and has been exploited in a wide range of photonic platforms to create TESs. We develop a photonic topological insulator strategy based on SSH photonic crystal nanobeam cavities. In contrast to the conventional photonic SSH schemes which are based on alternately tuned coupling strength in one-dimensional lattice, our proposal provides higher flexibility and allows tailoring TESs by manipulating mode coupling in a two-dimensional manner. We reveal that the proposed hole-array based nanobeams in a dielectric membrane can selectively tailor single or double TESs in the telecommunication region by controlling the coupling strength of the adjacent SSH nanobeams in both transverse and axial directions. Our finding provides an additional degree of freedom in exploiting the SSH model for integrated topological photonic devices and functionalities based on the well-established photonic crystal nanobeam cavity platforms.

Knots and Non-Hermitian Bloch Bands

Knots have a twisted history in quantum physics. They were abandoned as failed models of atoms. Only much later was the connection between knot invariants and Wilson loops in topological quantum field theory discovered. Here we show that knots tied by the eigenenergy strings provide a complete topological classification of one-dimensional non-Hermitian (NH) Hamiltonians with separable bands. A Z_{2} knot invariant, the global biorthogonal Berry phase Q as the sum of the Wilson loop eigenphases, is proved to be equal to the permutation parity of the NH bands. We show the transition between two phases characterized by distinct knots occur through exceptional points and come in two types. We further develop an algorithm to construct the corresponding tight-binding NH Hamiltonian for any desired knot, and propose a scheme to probe the knot structure via quantum quench. The theory and algorithm are demonstrated by model Hamiltonians that feature, for example, the Hopf link, the trefoil knot, the figure-8 knot, and the Whitehead link.

Coherent transfer of topological interface states

We demonstrate the controlled coherent transfer of topological interface states in a one-dimensional non-Hermitian chain of interacting Bose-Einstein condensates. The topological protection stems from a spatially patterned pump in an open-dissipative system. As a test bed setup of the proposed phenomenon, we consider a chain of coupled micropillars with embedded quantum wells, possessing exciton-polariton resonances. The transfer of an interface state is driven by spatially localised, adiabatic pump modulation in the vicinity of the interface state. The stochastic calculations prove the coherent nature of the interface state transfer. For appropriate system parameters the coherence degree is preserved after multiple transitions, paving the way towards long-range transfer of a coherent quantum state.

Multipolar lasing modes from topological corner states

Topological photonics provides a fundamental framework for robust manipulation of light, including directional transport and localization with built-in immunity to disorder. Combined with an optical gain, active topological cavities hold special promise for a design of light-emitting devices. Most studies to date have focused on lasing at topological edges of finite systems or domain walls. Recently discovered higher-order topological phases enable strong high-quality confinement of light at the corners. Here, we demonstrate lasing action of corner states in nanophotonic topological structures. We identify several multipole corner modes with distinct emission profiles via hyperspectral imaging and discern signatures of non-Hermitian radiative coupling of leaky topological states. In addition, depending on the pump position in a large-size cavity, we generate selectively lasing from either edge or corner states within the topological bandgap. Our studies provide the direct observation of multipolar lasing and engineered collective resonances in active topological nanostructures.

Higher-order photonic topological states, such as corner states, could enable robust and high-quality confinement of light to a small mode volume. Here, the authors demonstrate lasing from topological multipole corner states and investigate their emission profiles via hyperspectral imaging.

Non-Hermitian CT-Symmetric Spectral Protection of Nonlinear Defect Modes

We investigate, using a microwave platform consisting of a non-Hermitian Su-Schrieffer-Heeger array of coupled dielectric resonators, the interplay of a lossy nonlinearity and CT symmetry in the formation of defect modes. The measurements agree with the theory which predicts that, up to moderate pumping, the defect mode is an eigenstate of the CT-symmetric operator and retains its frequency at the center of the gap. At higher pumping values, the system undergoes a self-induced explicit CT-symmetry violation which removes the spectral topological protection and alters the shape of the defect mode.

Observation of supersymmetric pseudo-Landau levels in strained microwave graphene

Using an array of coupled microwave resonators arranged in a deformed honeycomb lattice, we experimentally observe the formation of pseudo-Landau levels in the whole crossover from vanishing to large pseudomagnetic field strengths. This result is achieved by utilising an adaptable setup in a geometry that is compatible with the pseudo-Landau levels at all field strengths. The adopted approach enables us to observe the fully formed flat-band pseudo-Landau levels spectrally as sharp peaks in the photonic density of states and image the associated wavefunctions spatially, where we provide clear evidence for a characteristic nodal structure reflecting the previously elusive supersymmetry in the underlying low-energy theory. In particular, we resolve the full sublattice polarisation of the anomalous 0th pseudo-Landau level, which reveals a deep connection to zigzag edge states in the unstrained case.

Nontrivial coupling of light into a defect: the interplay of nonlinearity and topology

The flourishing of topological photonics in the last decade was achieved mainly due to developments in linear topological photonic structures. However, when nonlinearity is introduced, many intriguing questions arise. For example, are there universal fingerprints of the underlying topology when modes are coupled by nonlinearity, and what can happen to topological invariants during nonlinear propagation? To explore these questions, we experimentally demonstrate nonlinearity-induced coupling of light into topologically protected edge states using a photonic platform and develop a general theoretical framework for interpreting the mode-coupling dynamics in nonlinear topological systems. Performed on laser-written photonic Su-Schrieffer-Heeger lattices, our experiments show the nonlinear coupling of light into a nontrivial edge or interface defect channel that is otherwise not permissible due to topological protection. Our theory explains all the observations well. Furthermore, we introduce the concepts of inherited and emergent nonlinear topological phenomena as well as a protocol capable of revealing the interplay of nonlinearity and topology. These concepts are applicable to other nonlinear topological systems, both in higher dimensions and beyond our photonic platform.

Room-temperature lasing from nanophotonic topological cavities

The study of topological phases of light underpins a promising paradigm for engineering disorder-immune compact photonic devices with unusual properties. Combined with an optical gain, topological photonic structures provide a novel platform for micro- and nanoscale lasers, which could benefit from nontrivial band topology and spatially localized gap states. Here, we propose and demonstrate experimentally active nanophotonic topological cavities incorporating III-V semiconductor quantum wells as a gain medium in the structure. We observe room-temperature lasing with a narrow spectrum, high coherence, and threshold behaviour. The emitted beam hosts a singularity encoded by a triade cavity mode that resides in the bandgap of two interfaced valley-Hall periodic photonic lattices with opposite parity breaking. Our findings make a step towards topologically controlled ultrasmall light sources with nontrivial radiation characteristics.

PT-Symmetric Topological Edge-Gain Effect

We demonstrate a non-Hermitian topological effect that is characterized by having complex eigenvalues only in the edge states of a topological material, despite the fact that the material is completely uniform. Such an effect can be constructed in any topological structure formed by two gapped subsystems, e.g., a quantum spin-Hall system, with a suitable non-Hermitian coupling between the spins. The resulting complex-eigenvalued edge state is robust against defects due to the topological protection. In photonics, such an effect can be used for the implementation of topological lasers, in which a uniform pumping provides gain only in the edge lasing state. Furthermore, such a topological lasing model is reciprocal and is thus compatible with standard photonic platforms.

Topological Anderson phase in quasi-periodic waveguide lattices

The topological trivial band of a lattice can be driven into a topological phase by disorder in the system. This so-called topological Anderson phase has been predicted and observed for uncorrelated static disorder, while in the presence of correlated disorder, conflicting results are found. Here we consider a Su-Schrieffer-Heeger waveguide lattice in the trivial topological phase and show that quasi-periodic disorder in the coupling constants can drive the lattice into a topological nontrivial phase. A method to detect the emergence of the topological Anderson phase, based on light dynamics at the edge of a quasi-periodic waveguide lattice, is suggested.

Topological Phase Transition in the Non-Hermitian Coupled Resonator Array

In a two-dimensional non-Hermitian topological photonic system, the physics of topological states is complicated, which brings great challenges for clarifying the topological phase transitions and achieving precise active control. Here, we prove the topological phase transition exists in a two-dimensional parity-time-symmetric coupled-resonator optical waveguide system. We reveal the inherent condition of the appearance of topological phase transition, which is described by the analytical algebraic relation of coupling strength and the quantity of gain-loss. In this framework, the system can be switched between the topological and trivial states by pumping the site rings. This work provides a new degree of freedom to control topological states and offers a scheme for studying non-Hermitian topological photonics.

Spin-Momentum-Locked Edge Mode for Topological Vortex Lasing

Spin-momentum locking is a direct consequence of bulk topological order and provides a basic concept to control a carrier's spin and charge flow for new exotic phenomena in condensed matter physics. However, up to date the research on spin-momentum locking solely focuses on its in-plane transport properties. Here, we report an emerging out-of-plane radiation feature of spin-momentum locking in a non-Hermitian topological photonic system and demonstrate a high performance topological vortex laser based on it. We find that the gain saturation effect lifts the degeneracy of the paired counterpropagating spin-momentum-locked edge modes enabling lasing from a single topological edge mode. The near-field spin and orbital angular momentum of the topological edge mode lasing has a one-to-one far-field radiation correspondence. The methodology of probing the near-field topology feature by far-field lasing emission can be used to study other exotic phenomena. The device can lead to applications in superresolution imaging, optical tweezers, free-space optical sensing, and communication.

Edge mode bifurcations of two-dimensional topological lasers

Topological lasers are of growing interest as a way to achieve disorder-robust single-mode lasing using arrays of coupled resonators. We study lasing in a two-dimensional coupled resonator lattice exhibiting transitions between trivial and topological phases, which allows us to systematically characterize the lasing modes throughout a topological phase. We show that, unlike conventional topological robustness that requires a sufficiently large bulk band gap, bifurcations in topological edge mode lasing can occur even when the band gap is maximized. We show that linear mode bifurcations from single-mode to multi-mode lasing can occur deep within the topological phase, sensitive to both the pump shape and lattice geometry. We suggest ways to suppress these bifurcations and preserve single-edge mode lasing.

Low-threshold topological nanolasers based on the second-order corner state

Topological lasers are immune to imperfections and disorder. They have been recently demonstrated based on many kinds of robust edge states, which are mostly at the microscale. The realization of 2D on-chip topological nanolasers with a small footprint, a low threshold and high energy efficiency has yet to be explored. Here, we report the first experimental demonstration of a topological nanolaser with high performance in a 2D photonic crystal slab. A topological nanocavity is formed utilizing the Wannier-type 0D corner state. Lasing behaviour with a low threshold of approximately 1 µW and a high spontaneous emission coupling factor of 0.25 is observed with quantum dots as the active material. Such performance is much better than that of topological edge lasers and comparable to that of conventional photonic crystal nanolasers. Our experimental demonstration of a low-threshold topological nanolaser will be of great significance to the development of topological nanophotonic circuitry for the manipulation of photons in classical and quantum regimes.

Topological framework for directional amplification in driven-dissipative cavity arrays

Directional amplification, in which signals are selectively amplified depending on their propagation direction, has attracted much attention as key resource for applications, including quantum information processing. Recently, several, physically very different, directional amplifiers have been proposed and realized in the lab. In this work, we present a unifying framework based on topology to understand non-reciprocity and directional amplification in driven-dissipative cavity arrays. Specifically, we unveil a one-to-one correspondence between a non-zero topological invariant defined on the spectrum of the dynamic matrix and regimes of directional amplification, in which the end-to-end gain grows exponentially with the number of cavities. We compute analytically the scattering matrix, the gain and reverse gain, showing their explicit dependence on the value of the topological invariant. Parameter regimes achieving directional amplification can be elegantly obtained from a topological 'phase diagram', which provides a guiding principle for the design of both phase-preserving and phase-sensitive multimode directional amplifiers.

Observation of Protected Photonic Edge States Induced by Real-Space Topological Lattice Defects

Topological defects (TDs) in crystal lattices are elementary lattice imperfections that cannot be removed by local perturbations, due to their real-space topology. In the emerging field of topological photonics, photonic topological edge states arise from the nontrivial topology of the band structure defined in momentum space and are generally protected against defects. Here we show that adding TDs into a valley photonic crystal generates a lattice disclination that acts like a domain wall and hosts photonic topological edge states. Unlike previous topological waveguides, the disclination forms an open arc and functions as a free-form waveguide connecting a pair of TDs of opposite topological charge. This interplay between the real-space topology of lattice defects and momentum-space band topology provides a novel scheme to implement large-scale photonic structures with complex arrangements of robust topological waveguides and resonators.

Multi-orbital tight binding model for cavity-polariton lattices

In this work we present a tight-binding model that allows to describe with a minimal amount of parameters the band structure of exciton-polariton lattices. This model based on s and p non-orthogonal photonic orbitals faithfully reproduces experimental results reported for polariton graphene ribbons. We analyze in particular the influence of the non-orthogonality, the inter-orbitals interaction and the photonic spin-orbit coupling on the polarization and dispersion of bulk bands and edge states.

Topological photonic crystals: a review

The field of topological photonic crystals has attracted growing interest since the inception of optical analog of quantum Hall effect proposed in 2008. Photonic band structures embraced topological phases of matter, have spawned a novel platform for studying topological phase transitions and designing topological optical devices. Here, we present a brief review of topological photonic crystals based on different material platforms, including all-dielectric systems, metallic materials, optical resonators, coupled waveguide systems, and other platforms. Furthermore, this review summarizes recent progress on topological photonic crystals, such as higherorder topological photonic crystals, non-Hermitian photonic crystals, and nonlinear photonic crystals. These studies indicate that topological photonic crystals as versatile platforms have enormous potential applications in maneuvering the flow of light.

Electrically pumped topological laser with valley edge modes

Quantum cascade lasers are compact, electrically pumped light sources in the technologically important mid-infrared and terahertz region of the electromagnetic spectrum^1,2. Recently, the concept of topology³ has been expanded from condensed matter physics into photonics⁴, giving rise to a new type of lasing^5-8 using topologically protected photonic modes that can efficiently bypass corners and defects⁴. Previous demonstrations of topological lasers have required an external laser source for optical pumping and have operated in the conventional optical frequency regime^5-8. Here we demonstrate an electrically pumped terahertz quantum cascade laser based on topologically protected valley edge states^9-11. Unlike topological lasers that rely on large-scale features to impart topological protection, our compact design makes use of the valley degree of freedom in photonic crystals^10,11, analogous to two-dimensional gapped valleytronic materials¹². Lasing with regularly spaced emission peaks occurs in a sharp-cornered triangular cavity, even if perturbations are introduced into the underlying structure, owing to the existence of topologically protected valley edge states that circulate around the cavity without experiencing localization. We probe the properties of the topological lasing modes by adding different outcouplers to the topological cavity. The laser based on valley edge states may open routes to the practical use of topological protection in electrically driven laser sources.

Non-Bloch ${\cal P}{\cal T}$PT symmetry breaking in non-Hermitian photonic quantum walks

A hallmark of topological band theory in periodic media is that bulk properties are not affected by boundary conditions. Remarkably, in certain non-Hermitian lattices, the bulk properties are affected largely by boundaries, leading to such major effects as the non-Hermitian skin effect and violation of the bulk-boundary correspondence. Here, we unveil that non-unitary discrete-time quantum walks of photons in systems involving gain and loss show rather generally non-Bloch parity-time symmetry-breaking phase transitions and suggest a bulk probing method to detect such boundary-driven phase transitions.

Breakup and Recovery of Topological Zero Modes in Finite Non-Hermitian Optical Lattices

The topological edge state (TES) in a one-dimensional optical lattice has exhibited robust field localization or waveguiding against the structural perturbations that would give rise to fault-tolerant photonic integrations. However, the zero mode as a kind of TES usually deviates from the exact zero-energy state in a finite Hermitian lattice due to the coupling between these edge states, which inevitably weaken the topological protection. Here, we first show such a breakup of zero modes in finite Su-Schriffer-Heeger optical lattices and then reveal their recovery by introducing non-Hermitian degeneracies with parity-time (PT) symmetry. We carry out experiments in a finite silicon waveguide lattice, where a passive-PT symmetry was implemented with carefully controlled lossy silicon waveguides. The experimental results are fully compatible with the theoretical prediction. Our results show that the topological property of an open system can be tuned by non-Hermitian lattice engineering, which offers a route to enhance the topological protection in a finite system.

Higher-Order Topological Corner States Induced by Gain and Loss

Higher-order topological insulators and superconductors are topological phases that exhibit novel boundary states on corners or hinges. Recent experimental advances in controlling dissipation such as gain and loss in atomic and optical systems provide a powerful tool for exploring non-Hermitian topological phases. Here we show that higher-order topological corner states can emerge by introducing staggered on-site gain and loss to a Hermitian system in a trivial phase. For such a non-Hermitian system, we establish a general bulk-corner correspondence by developing a biorthogonal nested-Wilson-loop and edge-polarization theory, which can be applied to a wide class of non-Hermitian systems with higher-order topological orders. The theory gives rise to topological invariants characterizing the non-Hermitian topological multipole moments (i.e., corner states) that are protected by reflection or chiral symmetry. Such gain- and loss-induced higher-order topological corner states can be experimentally realized using photons in coupled cavities or cold atoms in optical lattices.

Non-Bloch Band Theory of Non-Hermitian Systems

In spatially periodic Hermitian systems, such as electronic systems in crystals, the band structure is described by the band theory in terms of the Bloch wave functions, which reproduce energy levels for large systems with open boundaries. In this paper, we establish a generalized Bloch band theory in one-dimensional spatially periodic tight-binding models. We show how to define the Brillouin zone in non-Hermitian systems. From this Brillouin zone, one can calculate continuum bands, which reproduce the band structure in an open chain. As an example, we apply our theory to the non-Hermitian Su-Schrieffer-Heeger model. We also show the bulk-edge correspondence between the winding number and existence of the topological edge states.

New topological invariants in non-Hermitian systems

Both theoretical and experimental studies of topological phases in non-Hermitian systems have made a remarkable progress in the last few years of research. In this article, we review the key concepts pertaining to topological phases in non-Hermitian Hamiltonians with relevant examples and realistic model setups. Discussions are devoted to both the adaptations of topological invariants from Hermitian to non-Hermitian systems, as well as origins of new topological invariants in the latter setup. Unique properties such as exceptional points and complex energy landscapes lead to new topological invariants including winding number/vorticity defined solely in the complex energy plane, and half-integer winding/Chern numbers. New forms of Kramers degeneracy appear here rendering distinct topological invariants. Modifications of adiabatic theory, time-evolution operator, biorthogonal bulk-boundary correspondence lead to unique features such as topological displacement of particles, 'skin-effect', and edge-selective attenuated and amplified topological polarizations without chiral symmetry. Extension and realization of topological ideas in photonic systems are mentioned. We conclude with discussions on relevant future directions, and highlight potential applications of some of these unique topological features of the non-Hermitian Hamiltonians.

Unconventional quantum optics in topological waveguide QED

The discovery of topological materials has motivated recent developments to export topological concepts into photonics to make light behave in exotic ways. Here, we predict several unconventional quantum optical phenomena that occur when quantum emitters interact with a topological waveguide quantum electrodynamics bath, namely, the photonic analog of the Su-Schrieffer-Heeger model. When the emitters' frequency lies within the topological bandgap, a chiral bound state emerges, which is located on just one side (right or left) of the emitter. In the presence of several emitters, this bound state mediates topological, tunable interactions between them, which can give rise to exotic many-body phases such as double Néel ordered states. Furthermore, when the emitters' optical transition is resonant with the bands, we find unconventional scattering properties and different super/subradiant states depending on the band topology. Last, we propose several implementations where these phenomena can be observed with state-of-the-art technology.

Topological bound modes in anti-PT-symmetric optical waveguide arrays

We investigate the topological bound modes in a binary optical waveguide array with anti-parity-time (PT) symmetry. The anti-PT-symmetric arrays are realized by incorporating additional waveguides to the bare arrays, such that the effective coupling coefficients are imaginary. The systems experience two kinds of phase transition, including global topological order transition and quantum phase transition. As a result, the system supports two kinds of robust bound modes, which are protected by the global topological order and the quantum phase, respectively. The study provides a promising approach to realizing robust light transport by utilizing mediating components.

Lasing at topological edge states in a photonic crystal L3 nanocavity dimer array

Topological photonics have provided new insights for the manipulation of light. Analogous to electrons in topological insulators, photons travelling through the surface of a topological photonic structure or the interface of two photonic structures with different topological phases are free from backscattering caused by structural imperfections or disorder. This exotic nature of the topological edge state (TES) is truly beneficial for nanophotonic devices that suffer from structural irregularities generated during device fabrication. Although various topological states and device concepts have been demonstrated in photonic systems, lasers based on a topological photonic crystal (PhC) cavity array with a wavelength-scale modal volume have not been explored. We investigated TESs in a PhC nanocavity array in the Su-Schrieffer-Heeger model. Upon optical excitation, the topological PhC cavity array realised using an InP-based multiple-quantum-well epilayer spontaneously exhibits lasing peaks at the topological edge and bulk states. TES characteristics, including the modal robustness caused by immunity to scattering, are confirmed from the emission spectra and near-field imaging and by theoretical simulations and calculations.

Two-Dimensional Topological Polariton Laser

We provide proof-of-principle illustration of lasing in a two-dimensional polariton topological insulator. Topological edge states may arise in a structured polariton microcavity under the combined action of spin-orbit coupling and Zeeman splitting in the magnetic field. Their properties and lifetime are strongly affected by gain. Thus, gain concentrated along the edge of the insulator can counteract intrinsic losses in such a selective way that the topologically protected edge states become amplified, while bulk modes remain damped. When gain is compensated by nonlinear absorption the metastable nonlinear edge states are formed. Taking a triangular structure instead of an infinite edge we observed persistent topological currents accompanied by the time-periodic oscillations of the polariton density.

Parity anomaly laser

We propose a novel supersymmetry-inspired scheme for achieving robust single-mode lasing in arrays of coupled microcavities, based on factorizing a given array Hamiltonian into its "supercharge" partner array. Pumping a single sublattice of the partner array preferentially induces lasing of an unpaired zero mode. A chiral symmetry protects the zero mode similar to 1D topological arrays, but it need not be localized to domain walls or edges. We demonstrate single-mode lasing over a wider parameter regime by designing the zero mode to have a uniform intensity profile.

Controlled Outcoupling of Whispering-Gallery-Mode Lasers Based on Self-Assembled Organic Single-Crystalline Microrings

The outcoupling of whispering-gallery-mode (WGM) lasers is crucial for the realization of various photonic functionalities, yet present material structures still suffered the unexpected surface damages or contaminations in the multistep micro/nanofabrications. Here, we propose a strategy to achieve controlled outcoupling of WGM lasers in self-assembled organic single-crystalline microrings. The microrings with molecular-smooth surfaces functioned as organic crystalline whispering-gallery-mode microlasers with a lasing threshold of ∼14.2 μJ cm^-2 and a quality factor on the order of 10³ to 10⁴. The circular self-assembly allowed us to design different derived ring-based structures toward desired outcoupling of the WGM lasers, including unidirectional laser output from wire-ring coupled structures, and single-mode lasing in double-ring coupled systems. The results would provide an alternative avenue to construct versatile organic nanoscale lasers and related components with specific photonic applications.

Photonic flat-band laser

Flat-band photonic lattices, i.e., arrays of waveguides or resonators displaying a flat Bloch band, offer new routes for light trapping and distortion-free imaging. Here it is shown that flat-band lattices can show stable and cooperative laser emission when optical gain is supplied to the system, despite the large degree of degeneracy of flat-band supermodes. By considering a quasi one-dimensional rhombic lattice of coupled semiconductor microrings, selective pumping of the outer sublattices can induce cooperative lasing in a supermode of the flat band.

Photonic Topological Insulating Phase Induced Solely by Gain and Loss

We reveal a one-dimensional topological insulating phase induced solely by gain and loss control in non-Hermitian optical lattices. The system comprises units of four uniformly coupled cavities, where the successive two have loss; the others experience gain, and they are balanced under two magnitudes. The gain and loss parts are effectively dimerized, and a bulk band gap, topological transition, midgap topological edge, and interface states in finite systems can all be achieved by controlled pumping. We also clarify non-Hermitian topological invariants and edge states in gapless conditions.

Zak phase and topological plasmonic Tamm states in one-dimensional plasmonic crystals

The Zak phase and topological plasmonic Tamm states in plasmonic crystals based on periodic metal-insulator-metal waveguides are systematically investigated. We reveal that robust topological interfacial states against structural defects exist when the Zak phase between two adjoining plasmonic lattices are different in a common band gap. A kind of efficient admittance-based transfer matrix method is proposed to calculate and optimize the configuration with inverse symmetry. The topologically protected states are favorable for the spatial confinement and enhancement of electromagnetic fields, which open a new avenue for topological photonic applications.

Asymmetric topological edge states in a quasiperiodic Harper chain composed of split-ring resonators

Quasiperiodic structures have recently been found to possess salient topological properties. In this Letter, we theoretically and experimentally demonstrate the topological edge states in a quasiperiodic Harper chain composed of split-ring resonators. In a finite-size Harper chain, we find that there is always a pair of edge states in the bandgaps, independent of the number of the resonators. Different from the topological edge states localized at both ends of a periodic chain, this pair of edge states is selectively localized at the left or right end of the quasiperiodic chain at different frequencies. These paired-edge states with asymmetric field distribution can be used for unidirectional propagation. Our results not only provide a versatile platform to study the topological physics beyond the periodic structures but also may be useful in some applications such as selective unidirectional power transfer.

Supercharge optical arrays

We introduce the notion of a supercharge optical array synthesized according to supersymmetric charge operators. Starting from an arbitrary array, mathematical supersymmetry transformation can be used systematically to create a zero-energy physical state below the ground state of the super-partner array. This zero mode, which is pinned deep in the mid-gap of the corresponding supercharge array owing to the square-root spectral relationship between a supercharge and a super-Hamiltonian array, is shown to be protected because of the chiral symmetry inherent to a supercharge array. A supercharge array can be used in practical applications to design a discrete optical system of waveguides or coupled resonators where the mid-gap zero mode facilitates robust light dynamics in either spatial or time domain.

Non-Hermitian Chern Bands

The relation between chiral edge modes and bulk Chern numbers of quantum Hall insulators is a paradigmatic example of bulk-boundary correspondence. We show that the chiral edge modes are not strictly tied to the Chern numbers defined by a non-Hermitian Bloch Hamiltonian. This breakdown of conventional bulk-boundary correspondence stems from the non-Bloch-wave behavior of eigenstates (non-Hermitian skin effect), which generates pronounced deviations of phase diagrams from the Bloch theory. We introduce non-Bloch Chern numbers that faithfully predict the numbers of chiral edge modes. The theory is backed up by the open-boundary energy spectra, dynamics, and phase diagram of representative lattice models. Our results highlight a unique feature of non-Hermitian bands and suggest a non-Bloch framework to characterize their topology.

Feature issue introduction: Topological Photonics and Materials

Photonic crystals have become a very common and powerful concept in optics since its introduction in the 1980s by Eli Yablonovitch and Sajeev John. It is in fact a concept borrowed from condensed matter physics. The discussion of photonic bands and bandgaps allows us to manipulate light on an optical chip, along a photonic crystal fiber and even in the quantum optics regime. Now, we are witnessing another round of concept translation again from condensed matter physics to optics about topology. Describing photonic bands by using their topology in the reciprocal space gives us a new tool to understand wave propagation and to design optical components. Topology is also an important aspect in light-matter interaction in the field of metamaterials and 2D materials.

Edge States and Topological Invariants of Non-Hermitian Systems

The bulk-boundary correspondence is among the central issues of non-Hermitian topological states. We show that a previously overlooked "non-Hermitian skin effect" necessitates redefinition of topological invariants in a generalized Brillouin zone. The resultant phase diagrams dramatically differ from the usual Bloch theory. Specifically, we obtain the phase diagram of the non-Hermitian Su-Schrieffer-Heeger model, whose topological zero modes are determined by the non-Bloch winding number instead of the Bloch-Hamiltonian-based topological number. Our work settles the issue of the breakdown of conventional bulk-boundary correspondence and introduces the non-Bloch bulk-boundary correspondence.

Topological dynamics and excitations in lasers and condensates with saturable gain or loss

We classify symmetry-protected and symmetry-breaking dynamical solutions for nonlinear saturable bosonic systems that display a non-hermitian charge-conjugation symmetry, as realized in a series of recent groundbreaking experiments with lasers and exciton polaritons. In particular, we show that these systems support stable symmetry-protected modes that mirror the concept of zero-modes in topological quantum systems, as well as symmetry-protected power-oscillations with no counterpart in the linear case. In analogy to topological phases in linear systems, the number and nature of symmetry-protected solutions can change. The spectral degeneracies signalling phase transitions in linear counterparts extend to bifurcations in the nonlinear context. As bifurcations relate to qualitative changes in the linear stability against changes of the initial conditions, the symmetry-protected solutions and phase transitions can also be characterized by topological excitations, which set them apart from symmetry-breaking solutions. The stipulated symmetry appears naturally when one introduces nonlinear gain or loss into spectrally symmetric bosonic systems, as we illustrate for one-dimensional topological laser arrays with saturable gain and two-dimensional flat-band polariton condensates with density-dependent loss.