Non-Abelian physics in light and sound

Non-Abelian phenomena arise when the sequence of operations on physical systems influences their behaviors. By possessing internal degrees of freedom such as polarization, light and sound can be subjected to various manipulations, including constituent materials, structured environments, and tailored source conditions. These manipulations enable the creation of a great variety of Hamiltonians, through which rich non-Abelian phenomena can be explored and observed. Recent developments have constituted a versatile testbed for exploring non-Abelian physics at the intersection of atomic, molecular, and optical physics; condensed matter physics; and mathematical physics. These fundamental endeavors could enable photonic and acoustic devices with multiplexing functionalities. Our review aims to provide a timely and comprehensive account of this emerging topic. Starting from the foundation of matrix-valued geometric phases, we address non-Abelian topological charges, non-Abelian gauge fields, non-Abelian braiding, non-Hermitian non-Abelian phenomena, and their realizations with photonics and acoustics and conclude with future prospects.

Photochemical Reaction Enabling the Engineering of Photonic Spin-Orbit Coupling in Organic-Crystal Optical Microcavities

The control and active manipulation of spin-orbit coupling (SOC) in photonic systems are fundamental in the development of modern spin optics and topological photonic devices. Here, we demonstrate the control of an artificial Rashba-Dresselhaus (RD) SOC mediated by photochemical reactions in a microcavity filled with an organic single crystal of photochromic phase-change character. Splitting of the circular polarization components of the optical modes induced by photonic RD SOC is observed experimentally in momentum space. By applying an ultraviolet light beam, we control the spatial molecular orientation through a photochemical reaction, and with that we control the energies of the photonic modes. This way, we realize a reversible conversion of spin splitting of the optical modes with different energies, leading to an optically controlled switching between circularly and linearly polarized optical modes in our device. Our strategy of in situ and reversible engineering of SOC induced by a light field provides a promising approach to actively design and manipulate synthetic gauge fields toward future on-chip integration in photonics and topological photonic devices.

Synthetic Non-Abelian Gauge Fields for Non-Hermitian Systems

Non-Abelian gauge fields are versatile tools for synthesizing topological phenomena, but have so far been mostly studied in Hermitian systems, where gauge flux has to be defined from a closed loop in order for vector potentials, whether Abelian or non-Abelian, to become physically meaningful. We show that this condition can be relaxed in non-Hermitian systems by proposing and studying a generalized Hatano-Nelson model with imbalanced non-Abelian hopping. Despite lacking gauge flux in one dimension, non-Abelian gauge fields create rich non-Hermitian topological consequences. With SU(2) gauge fields, the braiding degrees that can be achieved are twice the highest hopping order of a lattice model, indicating the utility of spinful freedom to attain high-order nontrivial braiding. At both ends of an open chain, non-Abelian gauge fields lead to the simultaneous presence of non-Hermitian skin modes, whose population can be effectively tuned near the exceptional points. Generalizing to two dimensions, the gauge invariance of Wilson loops can also break down in non-Hermitian lattices dressed with non-Abelian gauge fields. Toward realization, we present a concrete experimental proposal for non-Abelian gauge fields in non-Hermitian systems via the synthetic frequency dimension of a polarization-multiplexed fiber ring resonator.

Demonstrating Path-Independent Anyonic Braiding on a Modular Superconducting Quantum Processor

Anyons, exotic quasiparticles in two-dimensional space exhibiting nontrivial exchange statistics, play a crucial role in universal topological quantum computing. One notable proposal to manifest the fractional statistics of anyons is the toric code model; however, scaling up its size through quantum simulation poses a serious challenge because of its highly entangled ground state. In this Letter, we demonstrate that a modular superconducting quantum processor enables hardware-pragmatic implementation of the toric code model. Through in-parallel control across separate modules, we generate a 10-qubit toric code ground state in four steps and realize six distinct braiding paths to benchmark the performance of anyonic statistics. The path independence of the anyonic braiding statistics is verified by correlation measurements in an efficient and scalable fashion. Our modular approach, serving as a hardware embodiment of the toric code model, offers a promising avenue toward scalable simulation of topological phases, paving the way for quantum simulation in a distributed fashion.

Evolution Operator Can Always Be Separated into the Product of Holonomy and Dynamic Operators

The geometric phase is a fundamental quantity characterizing the holonomic feature of quantum systems. It is well known that the evolution operator of a quantum system undergoing a cyclic evolution can be simply written as the product of holonomic and dynamical components for the three special cases concerning the Berry phase, adiabatic non-Abelian geometric phase, and nonadiabatic Abelian geometric phase. However, for the most general case concerning the nonadiabatic non-Abelian geometric phase, how to separate the evolution operator into holonomic and dynamical components is a long-standing open problem. In this Letter, we solve this open problem. We show that the evolution operator of a quantum system can always be separated into the product of holonomy and dynamic operators. Based on it, we further derive a matrix representation of this separation formula for cyclic evolution, and give a necessary and sufficient condition for a general evolution being purely holonomic. Our finding is not only of theoretical interest itself, but also of vital importance for the application of quantum holonomy. It unifies the representations of all four types of evolution concerning the adiabatic/nonadiabatic Abelian/non-Abelian geometric phase, and provides a general approach to realizing purely holonomic evolution.

A second wave of topological phenomena in photonics and acoustics

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

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.

Non-Abelian Inverse Anderson Transitions

Inverse Anderson transitions, where the flat-band localization is destroyed by disorder, have been wildly investigated in quantum and classical systems in the presence of Abelian gauge fields. Here, we report the first investigation on inverse Anderson transitions in the system with non-Abelian gauge fields. It is found that pseudospin-dependent localized and delocalized eigenstates coexist in the disordered non-Abelian Aharonov-Bohm cage, making inverse Anderson transitions depend on the relative phase of two internal pseudospins. Such an exotic phenomenon induced by the interplay between non-Abelian gauge fields and disorder has no Abelian analogy. Furthermore, we theoretically design and experimentally fabricate non-Abelian Aharonov-Bohm topolectrical circuits to observe the non-Abelian inverse Anderson transition. Through the direct measurements of frequency-dependent impedance responses and voltage dynamics, the pseudospin-dependent non-Abelian inverse Anderson transitions are observed. Our results establish the connection between inverse Anderson transitions and non-Abelian gauge fields, and thus comprise a new insight on the fundamental aspects of localization in disordered non-Abelian flat-band systems.

Non-Abelian effects in dissipative photonic topological lattices

Topology is central to phenomena that arise in a variety of fields, ranging from quantum field theory to quantum information science to condensed matter physics. Recently, the study of topology has been extended to open systems, leading to a plethora of intriguing effects such as topological lasing, exceptional surfaces, as well as non-Hermitian bulk-boundary correspondence. Here, we show that Bloch eigenstates associated with lattices with dissipatively coupled elements exhibit geometric properties that cannot be described via scalar Berry phases, in sharp contrast to conservative Hamiltonians with non-degenerate energy levels. This unusual behavior can be attributed to the significant population exchanges among the corresponding dissipation bands of such lattices. Using a one-dimensional example, we show both theoretically and experimentally that such population exchanges can manifest themselves via matrix-valued operators in the corresponding Bloch dynamics. In two-dimensional lattices, such matrix-valued operators can form non-commuting pairs and lead to non-Abelian dynamics, as confirmed by our numerical simulations. Our results point to new ways in which the combined effect of topology and engineered dissipation can lead to non-Abelian topological phenomena.

Artificial Non-Abelian Lattice Gauge Fields for Photons in the Synthetic Frequency Dimension

Non-Abelian gauge fields give rise to nontrivial topological physics. Here we develop a scheme to create an arbitrary SU(2) lattice gauge field for photons in the synthetic frequency dimension using an array of dynamically modulated ring resonators. The photon polarization is taken as the spin basis to implement the matrix-valued gauge fields. Using a non-Abelian generalization of the Harper-Hofstadter Hamiltonian as a specific example, we show that the measurement of the steady-state photon amplitudes inside the resonators can reveal the band structures of the Hamiltonian, which show signatures of the underlying non-Abelian gauge field. These results provide opportunities to explore novel topological phenomena associated with non-Abelian lattice gauge fields in photonic systems.

Experimental realization of non-Abelian permutations in a three-state non-Hermitian system

Eigenstates of a non-Hermitian system exist on complex Riemannian manifolds, with multiple sheets connecting at branch cuts and exceptional points (EPs). These eigenstates can evolve across different sheets-a process that naturally corresponds to state permutation. Here, we report the first experimental realization of non-Abelian permutations in a three-state non-Hermitian system. Our approach relies on the stroboscopic encircling of two different exceptional arcs (EAs), which are smooth trajectories of order-2 EPs appearing from the coalescence of two adjacent states. The non-Abelian characteristics are confirmed by encircling the EAs in opposite sequences. A total of five non-trivial permutations are experimentally realized, which together comprise a non-Abelian group. Our approach provides a reliable way of investigating non-Abelian state permutations and the related exotic winding effects in non-Hermitian systems.

Electrically tunable Berry curvature and strong light-matter coupling in liquid crystal microcavities with 2D perovskite

The field of spinoptronics is underpinned by good control over photonic spin-orbit coupling in devices that have strong optical nonlinearities. Such devices might hold the key to a new era of optoelectronics where momentum and polarization degrees of freedom of light are interwoven and interfaced with electronics. However, manipulating photons through electrical means is a daunting task given their charge neutrality. In this work, we present electrically tunable microcavity exciton-polariton resonances in a Rashba-Dresselhaus spin-orbit coupling field. We show that different spin-orbit coupling fields and the reduced cavity symmetry lead to tunable formation of the Berry curvature, the hallmark of quantum geometrical effects. For this, we have implemented an architecture of a photonic structure with a two-dimensional perovskite layer incorporated into a microcavity filled with nematic liquid crystal. Our work interfaces spinoptronic devices with electronics by combining electrical control over both the strong light-matter coupling conditions and artificial gauge fields.

Observation of Non-Abelian Thouless Pump

Thouless pump provides robust ways to realize quantized transport of waves and particles, and it casts the static 2D quantum Hall effect onto 1D dynamic systems where one of the momentum dimensions is replaced by the evolution time or path parameter. In the past few decades, various types of Abelian Thouless pump have been achieved theoretically and experimentally. However, the study of non-Abelian Thouless pump is scarce, which tells us that the order of two evolution loops with the same base point cannot be changed, and there has been no experimental observation of non-Abelian Thouless pump. Here we report the observation of a non-Abelian Thouless pump in coupled acoustic waveguide array. The non-Abelian property originates from the noncommutative combination of two different ℤ_{3} pump cycles that traverse across multiple band degeneracies in the parameter space in a three-band system. Moreover, we can pump a specific initial state to any state on any lattice site by applying these two ℤ_{3} pump cycles multiple times in a well-designed sequence. Our study paves the way for exploring and utilizing non-Abelian dynamical effects in classical wave systems and may offer different recipes for quantum walking, quantum optics, and quantum computation.

Polarization Singularities of Photonic Quasicrystals in Momentum Space

We report the observation of polarization singularities in momentum space of 2D photonic quasicrystal slabs. Supercell approximation and band-unfolding approach are applied to obtain approximate photonic dispersions and the far-field polarization states defined on them. We discuss the relations between the topological charges of the polarization vortex singularities at Γ points and the symmetries of photonic quasicrystal slabs. With a perspective of multipolar expansions for the supercell, we confirm that the singularities are protected by the point-group symmetry of the photonic quasicrystal slab. We further uncover that the polarization singularities of photonic quasicrystal slab correspond to quasibound states in the continuum with exceptionally high-quality factors. Polarization singularities of different topological charges are also experimentally verified. Our Letter introduces core concepts of optical singularities into quasiperiodic systems, providing new platforms for explorations merging topological and singular optics.

Low-Loss Broadband Transverse Electric Pass Hybrid Plasmonic Fiber Polarizers Using Metallic Nanomaterials

Metals were for decades perceived as devoid of interesting optical properties that could be harnessed for optical components and devices. However, with the development of accurate nanofabrication techniques and precise control over architectural parameters, metals can be structured and characterized on the nanoscale. Metallic plasmonic nanomaterials exhibit a number of unique structural and optical properties, which offer the potential for developing new types of plasmonic devices. Here, we demonstrate a low-loss broadband polarizer based on a hybrid plasmonic fiber structure using metals as polarization-selective absorption materials. The polarization mechanism, design, fabrication, and characteristics of the plasmonic polarizers are investigated theoretically, numerically, and experimentally. The theoretical analysis predicts that the polarization-selective absorption with insensitivity to wavelength enables hybrid plasmonic fibers to function as broadband polarizers. Numerical simulations give the comparison of the polarization-selective absorption of various metallic nanomaterials (Ag, Au, In, Al, Cr) and show that aluminum is regarded as the optimum absorption material for the plasmonic polarizer. Experimental results show that through precise control over geometrical parameters, this device is capable of offering a high polarization extinction ratio (PER) of over 40 dB and a low insertion loss (IL) of less than 1.3 dB in the wavelength region of 810.1-870.0 nm. Compared with commercial birefringent-crystal-fiber polarizers, the plasmonic fiber polarizer has a better PER and IL bandwidth. These merits, combined with a compact and robust configuration, enable the plasmonic polarizer to have great potential in a broad range of applications.

Synthetic Gauge Fields in a Single Optomechanical Resonator

Synthetic gauge fields have recently emerged, arising in the context of quantum simulations, topological matter, and the protected transportation of excitations against defects. For example, an ultracold atom experiences a light-induced effective magnetic field when tunneling in an optical lattice, and offering a platform to simulate the quantum Hall effect and topological insulators. Similarly, the magnetic field associated with photon transport between sites has been demonstrated in a coupled resonator array. Here, we report the first experimental demonstration of a synthetic gauge field in the virtual lattices of bosonic modes in a single optomechanical resonator. By employing degenerate clockwise and counterclockwise optical modes and a mechanical mode, a controllable synthetic gauge field is realized by tuning the phase of the driving lasers. The nonreciprocal conversion between the three modes is realized for different synthetic magnetic fluxes. As a proof-of-principle demonstration, we also show the dynamics of the system under a fast-varying synthetic gauge field, and demonstrate synthetic electric field. Our demonstration not only provides a versatile and controllable platform for studying synthetic gauge fields in high dimensions but also enables an exploration of ultrafast gauge field tuning with a large dynamic range, which is restricted for a magnetic field.

Nontrivial band geometry in an optically active system

Optical activity, also called circular birefringence, is known for two hundred years, but its applications for topological photonics remain unexplored. Unlike the Faraday effect, the optical activity provokes rotation of the linear polarization of light without magnetic effects, thus preserving the time-reversal symmetry. In this work, we report a direct measurement of the Berry curvature and quantum metric of the photonic modes of a planar cavity, containing a birefringent organic microcrystal (perylene) and exhibiting emergent optical activity. This experiment, performed at room temperature and at visible wavelength, establishes the potential of organic materials for implementing non-magnetic and low-cost topological photonic devices.

Layer Pseudospin Dynamics and Genuine Non-Abelian Berry Phase in Inhomogeneously Strained Moiré Pattern

Periodicity of long wavelength moiré patterns is very often destroyed by the inhomogeneous strain introduced in fabrications of van der Waals layered structures. We present a framework to describe massive Dirac fermions in such distorted moiré pattern of transition metal dichalcogenides homobilayers, accounting for the dynamics of layer pseudospin. In decoupled bilayers, we show two causes of in-plane layer pseudospin precession: By the coupling of layer antisymmetric strain to valley magnetic moment; and by the Aharonov-Bohm effect in the SU(2) gauge potential for the case of R-type bilayer under antisymmetric strain and H-type under symmetric strain. With interlayer coupling in the moiré, its interplay with strain manifests as a non-Abelian gauge field. We show a genuine non-Abelian Aharonov-Bohm effect in such field, where the evolution operators for different loops are noncommutative. This provides an exciting platform to explore non-Abelian gauge field effects on electron, with remarkable tunability of the field by strain and interlayer bias.

Non-Abelian generalizations of the Hofstadter model: spin-orbit-coupled butterfly pairs

The Hofstadter model, well known for its fractal butterfly spectrum, describes two-dimensional electrons under a perpendicular magnetic field, which gives rise to the integer quantum Hall effect. Inspired by the real-space building blocks of non-Abelian gauge fields from a recent experiment, we introduce and theoretically study two non-Abelian generalizations of the Hofstadter model. Each model describes two pairs of Hofstadter butterflies that are spin-orbit coupled. In contrast to the original Hofstadter model that can be equivalently studied in the Landau and symmetric gauges, the corresponding non-Abelian generalizations exhibit distinct spectra due to the non-commutativity of the gauge fields. We derive the genuine (necessary and sufficient) non-Abelian condition for the two models from the commutativity of their arbitrary loop operators. At zero energy, the models are gapless and host Weyl and Dirac points protected by internal and crystalline symmetries. Double (8-fold), triple (12-fold), and quadrupole (16-fold) Dirac points also emerge, especially under equal hopping phases of the non-Abelian potentials. At other fillings, the gapped phases of the models give rise to topological insulators. We conclude by discussing possible schemes for experimental realization of the models on photonic platforms.

Nonreciprocal Optomechanical Entanglement against Backscattering Losses

We propose how to achieve nonreciprocal quantum entanglement of light and motion and reveal its counterintuitive robustness against random losses. We find that by splitting the counterpropagating lights of a spinning resonator via the Sagnac effect, photons and phonons can be entangled strongly in a chosen direction but fully uncorrelated in the other. This makes it possible both to realize quantum nonreciprocity even in the absence of any classical nonreciprocity and also to achieve significant entanglement revival against backscattering losses in practical devices. Our work provides a way to protect and engineer quantum resources by utilizing diverse nonreciprocal devices, for building noise-tolerant quantum processors, realizing chiral networks, and backaction-immune quantum sensors.