Unidirectional Edge States Research Articles

Anomalous-Chern Steering of Topological Nonreciprocal Guided Waves.

Nonreciprocal topological edge states based on external magnetic bias have been regarded as the last resort for genuine unidirectional wave transport, showing superior robustness over topological states with preserved time-reversal symmetry. However, fast and efficient reconfigurability of their trajectory has remained a formidable challenge due to the difficulty in controlling the spatial distribution of magnetic fields over large areas and short times. Here, thispersistent issue is solved by leveraging the rich topology of unitary scattering networks, and achieve fast steering of nonreciprocal topological transport at an interface between a Chern and an anomalous topological insulator, without having to control a magnetic field. Such interface can be drawn by doping the network with scatterers located at the center of each link, whose level of reflection is electrically tuned. With experiments in the GHz range, the possibility to actively steer the way of unidirectional edge states is demonstrated, switching the transmission path thousands of times per second in a fully-robust topological heterostructure. The approach represents a significant step towards the realization of practical reconfigurable topological meta-devices with broken time-reversal symmetry, and their application to future robust communicationtechnologies.

Molecular and solid-state topological polaritons induced by population imbalance

Abstract Strong coupling between electronic excitations in materials and photon modes results in the formation of polaritons, which display larger nonlinearities than their photonic counterparts due to their material component. We theoretically investigate how to optically control the topological properties of molecular and solid-state exciton–polariton systems by exploiting one such nonlinearity: saturation of electronic transitions. We demonstrate modification of the Berry curvature of three different materials when placed within a Fabry–Perot cavity and pumped with circularly polarized light, illustrating the broad applicability of our scheme. Importantly, while optical pumping leads to nonzero Chern invariants, unidirectional edge states do not emerge in our system as the bulk-boundary correspondence is not applicable. This work demonstrates a versatile approach to control topological properties of novel optoelectronic materials.

An electromagnetic scattering approach to identifying topological and non-topological unidirectional edge states at gyrotropic plasma interfaces

We present an approach to identifying topological edge states in two dimensional (2D) problems. Such scattering-immune surface waves are typically identified using the Berry concept, which evaluates Chern numbers of bulk transmission bands. Instead, here, we approach the task from an analytical electromagnetic scattering perspective, which enables the study of a wide class of configurations in which waves are confined to propagate at the interface between a topologically non-trivial magnetized plasma and a topologically trivial medium such as air. Although it is of theoretical significance to classify interfacial states by their topological invariants, we believe that this approach enables the engagement of a broader range of researchers, particularly experimentalists who seek to design devices that exploit the ensuing non-reciprocal and scattering-immune properties.

Programmable Polariton Topological Insulators All-Optically Controlled by the Stark Effect.

Efficiently and flexibly manipulating unidirectional edge states is key to developing topological insulators as functional devices. In this work, we propose an all-optical method that utilizes the valley-selective optical Stark effect to realize programmable topological insulators. We pattern a two-dimensional honeycomb structure in an exciton-polariton platform resulting from a strong light-matter coupling in a monolayer transition metal dichalcogenide. The optical Stark effect is induced to generate a pseudo magnetic field, combined with spin-orbit coupling to form the topological one-way edge states of the polariton. On account of the ultrafast switching speed and precisely spatial controllability of the optical Stark effect, two applications, i.e., ports ratio tunable polariton splitter and programmable polariton router, were demonstrated, showing designable and rewritable functionality of all-optically controllable polariton topological insulators. This study paves the way to robustly and intelligently control/form polaritonic and spintronic devices for future classical and quantum information processing and application.

Floquet topological insulators with hybrid edges

Topological edge states form at the edges of periodic materials with specific degeneracies in their modal spectra, such as Dirac points, under the action of effects breaking certain symmetries of the system. In particular, in Floquet topological insulators unidirectional edge states appear upon breakup of the effective time-reversal symmetry due to dynamical modulations of the underlying lattice potential. However, such states are usually reported for certain simple lattice terminations, for example, at zigzag or bearded edges in honeycomb lattices. Here we show that unconventional topological edge states may exist in Floquet insulators based on arrays of helical waveguides with hybrid edges involving alternating zigzag and armchair segments, even if the latter are long. Such edge states appear in the largest part of the first Brillouin zone and show topological protection upon passage through the defects. Topological states at hybrid edges persist in the presence of focusing nonlinearity of the material. Our results can be extended to other lattice types and physical systems, they lift some of the constraints connected with lattice terminations that may not support edge states in the absence of effects breaking time-reversal symmetry of the system and expand the variety of geometrical shapes in which topological insulators can be constructed.

Quasidisorder-induced topology

We study the effects of quasidisorder and Anderson disorder on a two-dimensional topological superconductor with an applied external magnetic field. The cases of a $p$-wave superconductor and a noncentrosymmetric superconductor with mixed $p$- and $s$-wave pairings and the Rashba spin-orbit coupling are studied. We show that, for a perpendicular magnetic field, the introduction of quasidisorder leads to the appearance of topological phases in new regions, characterized by an integer value of the Chern number. For a parallel magnetic field, we identify regimes with the appearance of new Majorana flat bands and also new unidirectional Majorana edge states, as quasidisorder is introduced. We show that the Majorana flat bands have a quantized Berry phase of $\ensuremath{\pi}$ and identify it as a topological invariant. Two topological transitions are identified and the values of the critical exponents $z$ and $\ensuremath{\nu}$ are obtained. The fractal nature of the eigenstates is discussed both for Anderson and Aubry-Andr\'e disorders.

Mechanically‐Reconfigurable Edge States in an Ultrathin Valley‐Hall Topological Metamaterial

AbstractBroadband topological metamaterials hold the key for designing the next generation of integrated photonic platforms and microwave devices given their protected back‐scattering‐free and unidirectional edge states, among other exotic properties. However, synthesizing such metamaterial has proven challenging. Here, a broadband bandgap (relative bandwidth of more than 43%) Valley‐Hall topological metamaterial with deep subwavelength thickness is proposed. The present topological metamaterial is composed of three layers printed circuit boards whose total thickness is 1.524 mm ≈ λ/100. The topological phase transition is achieved by introducing an asymmetry parameter δr. Three mechanically reconfigurable edge states can be obtained by varying interlayer displacement. Their robust transmission is demonstrated through two kinds of waveguide domain walls with cavities and disorders. Exploiting the proposed topological metamaterial, a six‐way power divider is constructed and measured as a proof‐of‐concept of the potential of the proposed technology for future electromagnetic devices.

Observation of Unidirectional Solitonlike Edge States in Nonlinear Floquet Topological Insulators

A salient feature of solid-state quantum-Hall-type topological materials in two dimensions is the presence of conducting electronic edge states that are insensitive to scattering by disorder. Such unidirectional edge states have been predicted and observed in many other experimental settings, including photonics, mechanical, and acoustic structures. It is of great interest to understand how topological states behave in the presence of interparticle interactions and nonlinearity. Here, we experimentally demonstrate backscatter-immune unidirectional solitonlike nonlinear states on the edge of photonic topological insulators consisting of laser-written waveguides. As a result of the optical Kerr nonlinearity of the ambient glass, the solitonlike wave packet forms a long-lived quasilocalized coherent structure that slowly radiates power into the bulk and along the edge. The realization of solitonlike edge states paves the way to an understanding of nonlinear and interacting topological systems.2 MoreReceived 4 May 2021Revised 18 October 2021Accepted 21 October 2021DOI:https://doi.org/10.1103/PhysRevX.11.041057Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasKerr effectOptical solitonsPhotonicsTopological effects in photonic systemsTopological materialsAtomic, Molecular & OpticalCondensed Matter, Materials & Applied PhysicsNonlinear Dynamics

Dual-polarization topological phases and phase transition in magnetic photonic crystalline insulator

Two-dimensional topological photonic crystals have rapidly emerged as a recent and fascinating branch of photonic research. However, most of them were limited to a specific type of polarization, TE or TM polarization. Here, we explored the dual-polarization topological phases in two-dimensional magnetic photonic crystal (PC) which are composed of ferrite rod clusters in the plasma background. Under the perturbations of the bias magnetic field and/or the cluster distortion in the unit cell, the PC exhibited dual-polarization topological phases, including the quantum Hall (QH) phase, the higher-order quantum spin Hall (HO-QSH) phase and the conventional insulator (CI) phase. We studied the topological nature of these phases by the Wilson loop, Chern number, and unidirectional edge states. Intriguingly, we showed that the HO-QSH phases could present in PC of C 3v symmetry instead of being restricted to C 6v symmetry. The lower symmetry enlarges the gap in the edge states, which helps for the emergence of corner states. By continuously deforming the unit cell configuration, we demonstrated the phase transition in the system was dual-polarization. Our results extend the topological phases in the PCs and pave the way for the dual-polarization topological devices and their applications.

Berry bands and pseudo-spin of topological photonic phases

Realising photonic analogues of the robust, unidirectional edge states of electronic topological insulators would improve our control of light on the nanoscale and revolutionise the performance of photonic devices. Here we show that new symmetry protected topological phases can be detected by reformulating energy eigenproblems as Berry curvature eigenproblems. The "Berry bands" span the same eigenspace as the original valence energy bands, but separate into pseudo-spinful and pseudo-spinless subspaces in $\mathrm{C}_2\mathcal{T}$-symmetric crystals. We demonstrate the method on the well-known case of Wu & Hu [Phys. Rev. Lett. 114, 223901 (2015)] and a recently discovered fragilely topological crystal, and show that both crystals belong to the same $\mathrm{C}_2\mathcal{T}$-protected $\mathbb{Z}_2$ topological phase. This work helps unite theory and numerics, and is useful in defining and identifying new symmetry-protected phases in photonics and electronics.

Far-field detection of two-dimensional terahertz topological photonic crystals

Originating from the investigation of condensed matter states, we have discovered a new material with phase transition no-trivial properties——topological insulators. With the progress in artificial metamaterials, the photonic crystals which can give rise to topological edge states that propagate around sharp corners immune to scattering have captured much attention. In this paper, according to the spatial and time reversal symmetry and quantum spin Hall effect theory, we presented a topological photonic crystal made of metallic cylinders at terahertz frequencies. By slightly expanding and shrinking the unit cell, the spatial reversal symmetry was broken and therefore, the band degeneracy was broken, realizing the topological band gap. Unidirectional topological edge states were proven to exist on a domain wall connecting two photonic crystals with opposite topological properties without back-scattering. Then we investigated the reflection spectra of the designed topological photonic crystals in the terahertz regime, which provided a new means to verify topological photonic crystals.

Viscous Maxwell-Chern-Simons theory for topological electromagnetic phases of matter

We present the fundamental model of a topological electromagnetic phase of matter: viscous Maxwell-Chern-Simons theory. Our model applies to a quantum Hall fluids with viscosity. We solve both continuum and lattice regularized systems to demonstrate that this is the minimal (exactly solvable) gauge theory with a nontrivial photonic Chern number ($C\neq 0$) for electromagnetic waves coupled to a quantum Hall fluid. The interplay of symmetry and topology is also captured by the spin-1 representations of a photonic skyrmion at high-symmetry points in the Brillouin zone. To rigorously analyze the topological physics, we introduce the viscous Maxwell-Chern-Simons Lagrangian and derive the equations of motion, as well as the boundary conditions, from the principle of least action. We discover topologically-protected chiral (unidirectional) edge states which minimize the surface variation and correspond to massless photonic excitations costing an infinitesimal amount of energy. Physically, our predicted electromagnetic phases are connected to a dynamical photonic mass in the integer quantum Hall fluid. This arises from viscous (nonlocal) Hall conductivity and we identify the nonlocal Chern-Simons coupling with the Hall viscosity. The electromagnetic phase is topologically nontrivial $C\neq 0$ when the Hall viscosity inhibits the total bulk Hall response. Our work bridges the gap between electromagnetic and condensed matter topological physics while also demonstrating the central role of spin-1 quantization in nontrivial photonic phases.

Floquet topological photonic crystals with temporally modulated media.

We show that Floquet topological insulating states can exist in two-dimensional photonic crystals made of time-variant optical materials. By arranging the modulating phases, it facilitates effective gauge fields that give rise to topological effects. The band structures demonstrate the existence of topologically non-trivial bandgaps, thereby leading to back-scattering immune unidirectional edge states owing to bulk-edge correspondence. With these first-principle numerical results, we then verify the topological order for every Floquet band via Wilson loop approach. In the final paragraph, the possible experimental implementation for Floquet topological photonics is also discussed.

A semiconductor topological photonic ring resonator

Unidirectional photonic edge states arise at the interface between two topologically distinct photonic crystals. Here, we demonstrate a micrometer-scale GaAs photonic ring resonator, created using a spin Hall-type topological photonic crystal waveguide. Embedded InGaAs quantum dots are used to probe the mode structure of the device. We map the spatial profile of the resonator modes and demonstrate the control of the mode confinement through tuning of the photonic crystal lattice parameters. The intrinsic chirality of the edge states makes them of interest for applications in integrated quantum photonics, and the resonator represents an important building block toward the development of such devices with embedded quantum emitters.

Unidirectional helical edge states for all-dielectric topological photonic crystals

The study of one-way edge states of electromagnetic wave based on quantum Hall effect in photonic crystals (PCs) has recently made a great progress. In particular, the pseudo-spin states in topological PCs made of dielectric materials can be realized by breaking pseudo-time-reversal symmetry. We perform a research on the influence of parameters, such as lattice constant, dielectric permittivity of cylinders, radius of the dielectric cylinder and scaling factor of PCs, on the photonic bandgaps of topological PCs. Based on the analysis results, we present a 2D topological PC configuration that supports robust unidirectional helical edge states along zigzag interface between the topologically trivial and nontrivial PCs. With the structure, an operating bandwidth as larger as of $$0.0173\;(2\pi c/a)$$, robust helical edge states with energy-condensed and great nonreciprocity can be achieved by exciting a circularly polarized source. Utilizing this kind of architecture, we construct Z-shaped channel to demonstrate the robustness of helical edge states transport in different cases. The research offers a valid way for realizing unidirectional edge states with robust transport in practicable quantum communication applications.

Gapless unidirectional photonic transport using all-dielectric kagome lattices

Photonic topological insulators are a promising photonic platform due to the possibility of unidirectional edge states with insensitivity to bending, fabrication imperfections or environmental fluctuation. Here we demonstrate highly efficient unidirectional photonic edge mode propagation facilitated by an optical analogue of the quantum valley Hall effect. With an all-dielectric kagome lattice design, we demonstrate broadband suppressed reflection in the presence of sharp corners and further show negligible vertical losses in a semiconductor-based device at telecommunication wavelengths.

Polarization-resolved edge states in terahertz topological photonic crystal.

The discovery of topological photonic states has revolutionized our understanding of electromagnetic propagation and scattering. With the introduction of topology, some attractive properties such as unidirectional propagation and robustness against defects and impurities will be endowed to photonic edge modes. In this study, two-dimensionally confined topological edge states were achieved at terahertz (THz) frequency based on an all-dielectric photonic crystal structure. Trivial and nontrivial bandgaps of two deforming honeycomb lattices as well as unidirectional topological edge states were observed. Because the topological edge states with opposite helicity propagated in opposite directions at the interface, a polarization-resolved characteristic was demonstrated here, and thus a continuously tunable power splitter was achieved. This study provides some insights for further THz topological studies and possibilities for THz integrated platforms.

Vortex index identification and unidirectional propagation in Kagome photonic crystals

Abstract Vortex field in photonic metamaterials has been revealed in recent years, which provides an extra degree of freedom to control the behavior of electromagnetic waves, such as the unidirectional propagation in topological metamaterials and orbital angular momentum multiplexing in optical fibers. Achieving topologically protected unidirectionality needs an external source either with a vortex index of m = ±1 (i.e. 2πm phase winding around phase singularity) or with a spin index of ±1 (i.e. right- or left-handed circular polarization). However, it is still an open question on the relationship between higher-order (|m | > 1) vortex field and topological metamaterials. Here, we study the excitation of valley bulk states by a source with a higher-order vortex index in Kagome photonic crystals. The selection rule for the excitation of valley bulk states is derived and can be applied to predict the selective excitation of valley bulk states by a source with a higher-order vortex index. Interestingly, we find that the Fourier spectrum and the averaged Fourier spectrum of the Ez component enable us to identify the vortex index of the source. Besides, it is shown that the unidirectional edge states can be excited by a source with a vortex index of 1 or 2 in a straight or 120° detour waveguide. Our work paves the way for the demonstration of the selection rule and may have potential applications in optical mode identification.

Topological plasmonic edge states in a planar array of metallic nanoparticles

Abstract Photonic topological insulators (PTIs) are electromagnetic structures with highly robust unidirectional edge states, originating from their nontrivial bulk band topology. Here, we propose a plasmonic PTI that supports highly confined one-way edge states capable of transporting deep subwavelength optical frequency plasmons through arbitrary paths without back-scattering. The structure consists of a simple planar array of coupled plasmonic nanoparticles arranged in a perturbed honeycomb lattice that exhibits nontrivial band topology. The operation frequency of the emergent edge states is shown to be independent of the lattice constant, allowing for the miniaturization of the structure. As a high-frequency PTI with a simple and planar design, this structure is compatible with well-established integrated nanofabrication technologies and may find application in planar, compact, and topologically robust integrated nanophotonic devices.

Proof of the Bulk-Edge Correspondence through a Link between Topological Photonics and Fluctuation-Electrodynamics

The bulk-edge correspondence links the Chern-topological numbers with the net number of unidirectional states supported at an interface of the relevant materials. This fundamental principle is perhaps the most consequential result of topological photonics, as it determines the precise physical manifestations of nontrivial topological features. Even though the bulk-edge correspondence has been extensively discussed and used in the literature, it seems that in the general photonic case with dispersive materials it has no solid mathematical foundation and is essentially a conjecture. Here, I present a rigorous demonstration of this fundamental principle by showing that the thermal fluctuation-induced light-angular momentum spectral density in a closed cavity can be expressed in terms of the photonic gap Chern number, as well as in terms of the net number of unidirectional edge states. In particular, I highlight the rather fundamental connections between topological numbers in Chern-type photonic insulators and the fluctuation-induced light-momentum