Unconventional Flatband Line States in Photonic Lieb Lattices

Quantum percolation on Lieb Lattices

We theoretically investigate the quantum percolation problem on Lieb lattices in two and three dimensions. We study the statistics of the energy levels through random matrix theory and determine the level spacing distributions, which, with the aid of finite-size scaling theory, allows us to obtain accurate estimates for site- and bond-percolation thresholds and critical exponents. Our numerical investigation supports a localized-delocalized transition at finite threshold, which decreases as the average coordination number increases. The precise determination of the localization length exponent enables us to claim that quantum site- and bond-percolation problems on Lieb lattices belong to the same universality class, with ν decreasing with lattice dimensionality d, similarly to the classical percolation problem. In addition, we verify that, in three dimensions, quantum percolation on Lieb lattices belongs to the same universality class as the Anderson impurity model.

Bulk-Hole Correspondence and Inner Robust Boundary Modes in Singular Flatband Lattices

Topological entities based on bulk-boundary correspondence are ubiquitous, from conventional to higher-order topological insulators, where the protected states are typically localized at the outer boundaries (edges or corners). A less explored scenario involves protected states that are localized at the inner boundaries, sharing the same energy as the bulk states. Here, we propose and demonstrate what we refer to as the "bulk-hole correspondence''-a relation between the inner robust boundary modes (RBMs) and the existence of multiple "holes" in singular flatband lattices, mediated by the immovable discontinuity of the bulk Bloch wave functions. We find that the number of independent flatband states always equals the sum of the number of independent compact localized states and the number of nontrivial inner RBMs, as captured by the Betti number that also counts the hole number from topological data analysis. This correspondence is universal for singular flatband lattices, regardless of the lattice shape and the hole shape. Using laser-written kagome lattices as a platform, we experimentally observe such inner RBMs, demonstrating their real-space topological nature and robustness. Our results may extend to other singular flatband systems beyond photonics, including non-Euclidean lattices, providing a new approach for understanding nontrivial flatband states and topology in hole-bearing lattice systems.

Topological orbital angular momentum extraction and twofold protection of vortex transport

Vortex phenomena are ubiquitous in nature. In optics, despite the availability of numerous techniques for vortex generation and detection, topological protection of vortex transport with desired orbital angular momentum (OAM) remains a challenge. Here, by use of topological disclination, we demonstrate a scheme to confine and guide vortices featuring arbitrary high-order charges. Such a scheme relies on twofold topological protection: a non-trivial winding in momentum space due to chiral symmetry, and a non-trivial winding in real space due to the complex coupling of OAM modes across the disclination structure. We unveil a vorticity-coordinated rotational symmetry, which sets up a universal relation between the vortex topological charge and the rotational symmetry order of the system. As an example, we construct photonic disclination lattices with a single core but different C n symmetries and achieve robust transport of an optical vortex with preserved OAM solely corresponding to one selected zero-energy vortex mode at the mid-gap. Furthermore, we show that such topological structures can be used for vortex filtering to extract a chosen OAM mode from mixed excitations. Our results illustrate the fundamental interplay of vorticity, disclination and higher-order topology, which may open a new pathway for the development of OAM-based photonic devices such as vortex guides, fibres and lasers.

Flat band fine-tuning and its photonic applications

Flat bands - single-particle energy bands - in tight-binding lattices, aka networks, have attracted attention due to the presence of macroscopic degeneracies and their sensitivity to perturbations. They support compact localized eigenstates protected by destructive interference. This makes them natural candidates for emerging exotic phases and unconventional orders. In this review we consider the recently proposed systematic ways to construct flat band networks based on symmetries or fine-tuning. We then discuss how the construction methods can be further extended, adapted or exploited in presence of perturbations, both single-particle and many-body. This strategy has lead to the discovery of non-perturbative metal-insulator transitions, fractal phases, nonlinear and quantum caging and many-body nonergodic quantum models. We discuss what implications these results may have for the design of fine-tuned nanophotonic systems including photonic crystals, nanocavities, and metasurfaces.

Two-dimensional flat-band solitons in superhoneycomb lattices

Flat-band periodic materials are characterized by a linear spectrum containing at least one band where the propagation constant remains nearly constant irrespective of the Bloch momentum across the Brillouin zone. These materials provide a unique platform for investigating phenomena related to light localization. Meantime, the interaction between flat-band physics and nonlinearity in continuous systems remains largely unexplored, particularly in continuous systems where the band flatness deviates slightly from zero, in contrast to simplified discrete systems with exactly flat bands. Here, we use a continuous superhoneycomb lattice featuring a flat band in its spectrum to theoretically and numerically introduce a range of stable flat-band solitons. These solutions encompass fundamental, dipole, multi-peak, and even vortex solitons. Numerical analysis demonstrates that these solitons are stable in a broad range of powers. They do not bifurcate from the flat band and can be analyzed using Wannier function expansion leading to their designation as Wannier solitons. These solitons showcase novel possibilities for light localization and transmission within nonlinear flat-band systems.

Emergence of flat bands and ferromagnetic fluctuations via orbital-selective electron correlations in Mn-based kagome metal

Kagome lattice has been actively studied for the possible realization of frustration-induced two-dimensional flat bands and a number of correlation-induced phases. Currently, the search for kagome systems with a nearly dispersionless flat band close to the Fermi level is ongoing. Here, by combining theoretical and experimental tools, we present Sc3Mn3Al7Si5 as a novel realization of correlation-induced almost-flat bands in the kagome lattice in the vicinity of the Fermi level. Our magnetic susceptibility, 27Al nuclear magnetic resonance, transport, and optical conductivity measurements provide signatures of a correlated metallic phase with tantalizing ferromagnetic instability. Our dynamical mean-field calculations suggest that such ferromagnetic instability observed originates from the formation of nearly flat dispersions close to the Fermi level, where electron correlations induce strong orbital-selective renormalization and manifestation of the kagome-frustrated bands. In addition, a significant negative magnetoresistance signal is observed, which can be attributed to the suppression of flat-band-induced ferromagnetic fluctuation, which further supports the formation of flat bands in this compound. These findings broaden a new prospect to harness correlated topological phases via multiorbital correlations in 3d-based kagome systems.

Reconfigurable moiré nanolaser arrays with phase synchronization

Miniaturized lasers play a central role in the infrastructure of modern information society. The breakthrough in laser miniaturization beyond the wavelength scale has opened up new opportunities for a wide range of applications1-4, as well as for investigating light-matter interactions in extreme-optical-field localization and lasing-mode engineering5-19. An ultimate objective of microscale laser research is to develop reconfigurable coherent nanolaser arrays that can simultaneously enhance information capacity and functionality. However, the absence of a suitable physical mechanism for reconfiguring nanolaser cavities hinders the demonstration of nanolasers in either a single cavity or a fixed array. Here we propose and demonstrate moiré nanolaser arrays based on optical flatbands in twisted photonic graphene lattices, in which coherent nanolasing is realized from a single nanocavity to reconfigurable arrays of nanocavities. We observe synchronized nanolaser arrays exhibiting high spatial and spectral coherence, across a range of distinct patterns, including P, K and U shapes and the Chinese characters '' and '' ('China' in Chinese). Moreover, we obtain nanolaser arrays that emit with spatially varying relative phases, allowing us to manipulate emission directions. Our work lays the foundation for the development of reconfigurable active devices that have potential applications in communication, LiDAR (light detection and ranging), optical computing and imaging.

Multiple flatbands and localized states in photonic super-Kagome lattices

We demonstrate multiple flatbands and compact localized states (CLSs) in a photonic super-Kagome lattice (SKL) that exhibits coexistence of singular and nonsingular flatbands within its unique band structure. Specifically, we find that the upper two flatbands of an SKL are singular-characterized by singularities due to band touching with their neighboring dispersive bands at the Brillouin zone center. Conversely, the lower three degenerate flatbands are nonsingular and remain spectrally isolated from other dispersive bands. The existence of such two distinct types of flatbands is experimentally demonstrated by observing stable evolution of the CLSs with various geometrical shapes in a laser-written SKL. We also discuss the classification of the flatbands in momentum space, using band-touching singularities of the Bloch wave functions. Furthermore, we validate this classification in real space based on unit cell occupancy of the CLSs in a single SKL plaquette. These results may provide insights for the study of flatband transport, dynamics, and nontrivial topological phenomena in other relevant systems.

Compact localized states in magnonic Lieb lattices

Lieb lattice is one of the simplest bipartite lattices, where compact localized states (CLS) are observed. This type of localization is induced by the peculiar topology of the unit cell, where the modes are localized only on selected sublattices due to the destructive interference of partial waves. We demonstrate the possibility of magnonic Lieb lattice realization, where flat bands and CLS can be observed in the planar structure of sub-micron in-plane sizes. Using forward volume configuration, the Ga-doped YIG layer with cylindrical inclusions (without Ga content) arranged in a Lieb lattice with 250 nm period was investigated numerically (finite-element method). The structure was tailored to observe, for a lowest magnonic bands, the oscillatory and evanescent spin waves in inclusions and matrix, respectively. Such a design reproduces the Lieb lattice of nodes (inclusions) coupled to each other by the matrix with the CLS in flat bands.

Noncontractible loop states from a partially flat band in a photonic borophene lattice

Flat band systems are commonly associated with compact localized states (CLSs) that arise from the macroscopic degeneracy of eigenstates at the flat band energy. However, in the case of singular flat bands, conventional localized flat band states are incomplete, leading to the existence of noncontractible loop states (NLSs) with nontrivial real-space topology. In this study, we experimentally and analytically demonstrate the existence of NLSs in a 2D photonic borophene lattice without a CLS counterpart, owing to a band that is flat only along high-symmetry lines and dispersive along others. Our findings challenge the conventional notion that NLSs are necessarily linked to robust boundary modes due to a bulk-boundary correspondence. Protected by the band flatness that originates from band touching, NLSs play a significant role in investigating the fundamental physics of flat band systems.

Photonic Realization of a Generic Type of Graphene Edge States Exhibiting Topological Flat Band

Cutting a honeycomb lattice (HCL) ends up with three types of edges (zigzag, bearded, and armchair), as is well known in the study of graphene edge states. Here, we propose and demonstrate a distinctive twig-shaped edge, thereby observing new edge states using a photonic platform. Our main findings are (i) the twig edge is a generic type of HCL edge complementary to the armchair edge, formed by choosing the right primitive cell rather than simple lattice cutting or Klein edge modification; (ii) the twig edge states form a complete flat band across the Brillouin zone with zero-energy degeneracy, characterized by nontrivial topological winding of the lattice Hamiltonian; (iii) the twig edge states can be elongated or compactly localized at the boundary, manifesting both flat band and topological features. Although realized here in a photonic graphene, such twig edge states should exist in other synthetic HCL structures. Moreover, our results may broaden the understanding of graphene edge states, as well as new avenues for realization of robust edge localization and nontrivial topological phases based on Dirac-like materials.

Doping-induced magnetism and magnetoelectric coupling in one-dimensional NbOCl3 and NbOBr3

Low-dimensional multiferroic systems with magnetoelectric coupling have attracted considerable attention due to their important applications in high-density low-power storage. Based on the first-principles calculations, we demonstrated that the recently proposed one-dimensional (1D) ferroelectric materials NbOCl₃ and NbOBr₃ have good stabilities, and found that they can be easily separated from the bulk phase. Due to the flat band near the Fermi level, the itinerant ferromagnetism can be induced over a wide range of electron-doping concentrations, and it leads to the coexistence of ferroelectricity and ferromagnetism in 1D NbOX₃ (X = Cl, Br) and finite-length nanochains. More interestingly, there is strong magnetoelectric coupling on finite-length nanochains, which is caused by the spontaneous electrical polarization and redistribution of magnetic carriers. In addition, magnetism also can be introduced by oxygen vacancies. We also analyzed the effects of doping concentration, strain, and length on ferroelectric polarization and magnetism. Our findings provide a way to design and search low-dimensional multiferroics.

Experimental realization of a reconfigurable Lieb photonic lattice in a coherent atomic medium

We have experimentally demonstrated the realization of an instantaneously reconfigurable Lieb photonic lattice with a flatband in a three-level Λ-type rubidium atomic configuration. Such a coherently controllable Lieb photonic lattice is optically induced by a coupling field possessing a spatially periodic intensity distribution (generated via a spatial light modulator) under the condition of electromagnetically induced transparency. The incident weak Gaussian probe field can experience discrete diffraction and the observed probe beam at the output surface of the medium exhibits the same Lieb pattern, verifying the formation of the refractive index with a Lieb profile inside the atomic vapor cell. The potential wells and the band structure of the Lieb photonic lattice can be effectively manipulated by easily tuning the frequency of the involved laser beams. The current work can promisingly pave the way for exploring the exotic dynamics as well as tunable photonic devices in Lieb photonic lattices.

Compact localized boundary states in a quasi-1D electronic diamond-necklace chain

Zero-energy modes localized at the ends of one-dimensional (1D) wires hold great potential as qubits for fault-tolerant quantum computing. However, all the candidates known to date exhibit a wave function that decays exponentially into the bulk and hybridizes with other nearby zero-modes, thus hampering their use for braiding operations. Here, we show that a quasi-1D diamond-necklace chain exhibits an unforeseen type of robust boundary state, namely compact localized zero-energy modes that do not decay into the bulk. We find that this state emerges due to the presence of a latent symmetry in the system. We experimentally realize the diamond-necklace chain in an electronic quantum simulator setup.

Photonic flatband resonances for free-electron radiation

Flatbands have become a cornerstone of contemporary condensed-matter physics and photonics. In electronics, flatbands entail comparable energy bandwidth and Coulomb interaction, leading to correlated phenomena such as the fractional quantum Hall effect and recently those in magic-angle systems. In photonics, they enable properties including slow light1 and lasing2. Notably, flatbands support supercollimation-diffractionless wavepacket propagation-in both systems3,4. Despite these intense parallel efforts, flatbands have never been shown to affect the core interaction between free electrons and photons. Their interaction, pivotal for free-electron lasers5, microscopy and spectroscopy6,7, and particle accelerators8,9, is, in fact, limited by a dimensionality mismatch between localized electrons and extended photons. Here we reveal theoretically that photonic flatbands can overcome this mismatch and thus remarkably boost their interaction. We design flatband resonances in a silicon-on-insulator photonic crystal slab to control and enhance the associated free-electron radiation by tuning their trajectory and velocity. We observe signatures of flatband enhancement, recording a two-order increase from the conventional diffraction-enabled Smith-Purcell radiation. The enhancement enables polarization shaping of free-electron radiation and characterization of photonic bands through electron-beam measurements. Our results support the use of flatbands as test beds for strong light-electron interaction, particularly relevant for efficient and compact free-electron light sources and accelerators.

Topological Flat Bands in Self-Complementary Plasmonic Metasurfaces

Photonics can be confined in real space with dispersion vanishing in the momentum space due to destructive interference. In this Letter, we report the experimental realization of flat bands with nontrivial topology in a self-complementary plasmonic metasurface. The band diagram and compact localized states are measured. In these nontrivial band gaps, we observe the topological edge states by near-field measurements. Furthermore, we propose a digitalized metasurface by loading controllable diodes with C_{3} symmetry in every unit cell. By pumping a digital signal into the metasurface, we investigate the interaction between incident waves and the dynamic metasurface. Experimental results indicate that compact localized states in the nontrivial flat band could enhance the wave-matter interactions to convert more incident waves to time-modulated harmonic photonics. Although our experiments are conducted in the microwave regime, extending the related concepts into the optical plasmonic systems is feasible. Our findings pave an avenue toward planar integrated photonic devices with nontrivial flat bands and exotic transmission phenomena.

Electronic Quantum Materials Simulated with Artificial Model Lattices

The band structure and electronic properties of a material are defined by the sort of elements, the atomic registry in the crystal, the dimensions, the presence of spin–orbit coupling, and the electronic interactions. In natural crystals, the interplay of these factors is difficult to unravel, since it is usually not possible to vary one of these factors in an independent way, keeping the others constant. In other words, a complete understanding of complex electronic materials remains challenging to date. The geometry of two- and one-dimensional crystals can be mimicked in artificial lattices. Moreover, geometries that do not exist in nature can be created for the sake of further insight. Such engineered artificial lattices can be better controlled and fine-tuned than natural crystals. This makes it easier to vary the lattice geometry, dimensions, spin–orbit coupling, and interactions independently from each other. Thus, engineering and characterization of artificial lattices can provide unique insights. In this Review, we focus on artificial lattices that are built atom-by-atom on atomically flat metals, using atomic manipulation in a scanning tunneling microscope. Cryogenic scanning tunneling microscopy allows for consecutive creation, microscopic characterization, and band-structure analysis by tunneling spectroscopy, amounting in the analogue quantum simulation of a given lattice type. We first review the physical elements of this method. We then discuss the creation and characterization of artificial atoms and molecules. For the lattices, we review works on honeycomb and Lieb lattices and lattices that result in crystalline topological insulators, such as the Kekulé and “breathing” kagome lattice. Geometric but nonperiodic structures such as electronic quasi-crystals and fractals are discussed as well. Finally, we consider the option to transfer the knowledge gained back to real materials, engineered by geometric patterning of semiconductor quantum wells.

Bimorphic Floquet topological insulators

Topological theories have established a unique set of rules that govern the transport properties in a wide variety of wave-mechanical settings. In a marked departure from the established approaches that induce Floquet topological phases by specifically tailored discrete coupling protocols or helical lattice motions, we introduce a class of bimorphic Floquet topological insulators that leverage connective chains with periodically modulated on-site potentials to reveal rich topological features in the system. In exploring a 'chain-driven' generalization of the archetypical Floquet honeycomb lattice, we identify a rich phase structure that can host multiple non-trivial topological phases associated simultaneously with both Chern-type and anomalous chiral states. Experiments carried out in photonic waveguide lattices reveal a strongly confined helical edge state that, owing to its origin in bulk flat bands, can be set into motion in a topologically protected fashion, or halted at will, without compromising its adherence to individual lattice sites.

Engineering of flat bands and Dirac bands in two-dimensional covalent organic frameworks (COFs): relationships among molecular orbital symmetry, lattice symmetry, and electronic-structure characteristics

Two-dimensional covalent organic frameworks (2D-COFs), also referred to as 2D polymer networks, display unusual electronic-structure characteristics, which can significantly enrich and broaden the fields of electronics and spintronics. In this Focus article, our objective is to lay the groundwork for the conceptual description of the fundamental relationships among the COF electronic structures, the symmetries of their 2D lattices, and the frontier molecular orbitals (MOs) of their core and linker components. We focus on monolayers of hexagonal COFs and use tight-binding model analyses to highlight the critical role of the frontier-MO symmetry, in addition to lattice symmetry, in determining the nature of the electronic bands near the Fermi level. We rationalize the intriguing feature that, when the core unit has degenerate highest occupied MOs [or lowest unoccupied MOs], the COF highest valence band [or lowest conduction band] is flat but degenerate with a dispersive band at a high-symmetry point of the Brillouin zone; the consequences of having such band characteristics are briefly described. Multi-layer and bulk 2D COFs are found to maintain the salient features of the monolayer electronic structures albeit with a reduced bandgap due to the interlayer coupling. This Focus article is thus meant to provide an effective framework for the engineering of flat and Dirac bands in 2D polymer networks.

Band evolution and Landau-Zener Bloch oscillations in strained photonic rhombic lattices

We investigate band evolution of chiral and non-chiral symmetric flatband photonic rhombic lattices by applying a strain along the diagonal direction, and thereby demonstrating Landau-Zener Bloch (LZB) oscillations in the presence of a refractive index gradient. The chiral and non-chiral symmetric rhombic lattices are obtained by adding a detuning to uniform lattices. For the chiral symmetric lattices, the middle flatband is perturbed due to the chiral symmetry breaking while a nearly flatband appears as the bottom band with the increase of strain-induced next-nearest-neighbor hopping. Consequently, LZB oscillations exhibit intriguing characteristics such as asymmetric energy transitions and almost complete suppression of the oscillations. Nevertheless, for the non-chiral symmetric lattices, flatband persists owing to the retained particle-hole symmetry and evolves into the bottom band. Remarkably, the band gap can be readily tuned, which allows controlling of the amplitude of Landau-Zener tunneling (LZT) rate and may lead to thorough LZT. Our analysis provides an alternative perspective on the generation of tunable flatband and may also bring insight to study the symmetry and topological characterization of the flatband.

Strain induced localization to delocalization transition on a Lieb photonic ribbon lattice

Ribbon lattices are kind of transition systems in between one and two dimensions, and their study is crucial to understand the origin of different emerging properties. In this work, we study a Lieb ribbon lattice and the localization-delocalization transition occurring due to a reduction of lattice distances (compression) and the corresponding flat band deformation. We observe how above a critical compression ratio the energy spreads out and propagates freely across the lattice, therefore transforming the system from being a kind of insulator into a conductor. We implement an experiment on a photonic platform and show an excellent agreement with the predicted phenomenology. Our findings suggest and prove experimentally the use of compression or mechanical deformation of lattices to switch the transport properties of a given system.

Generation of gradient photonic moiré lattice fields

We designed and generated gradient photonic moiré lattice fields comprising three varying periodic moiré wavefields. Because of the common twisted angles between periodic triangular and hexagonal moiré wavefields, gradient patterns can be easily obtained through coherent superposition of hexagonal-triangular-hexagonal photonic moiré lattice fields. In addition, two specific twisted angles of Δα|_C=3 and Δα|_C=5 are proposed, which not only guarantee the periodicity of moiré fields but also provide an additional degree of freedom to control the structural arrangement of the gradient photonic moiré lattice fields. Further study reveals the non-diffracting character of the gradient photonic moiré lattice field generated using the holographic method. This study proposes an easy way to generate and control the structures of gradient moiré lattice fields that can be used to fabricate photonic lattices in optical storage media for light modulation.

Nonlinear control of photonic higher-order topological bound states in the continuum

Higher-order topological insulators (HOTIs) are recently discovered topological phases, possessing symmetry-protected corner states with fractional charges. An unexpected connection between these states and the seemingly unrelated phenomenon of bound states in the continuum (BICs) was recently unveiled. When nonlinearity is added to the HOTI system, a number of fundamentally important questions arise. For example, how does nonlinearity couple higher-order topological BICs with the rest of the system, including continuum states? In fact, thus far BICs in nonlinear HOTIs have remained unexplored. Here we unveil the interplay of nonlinearity, higher-order topology, and BICs in a photonic platform. We observe topological corner states that are also BICs in a laser-written second-order topological lattice and further demonstrate their nonlinear coupling with edge (but not bulk) modes under the proper action of both self-focusing and defocusing nonlinearities. Theoretically, we calculate the eigenvalue spectrum and analog of the Zak phase in the nonlinear regime, illustrating that a topological BIC can be actively tuned by nonlinearity in such a photonic HOTI. Our studies are applicable to other nonlinear HOTI systems, with promising applications in emerging topology-driven devices.

Probing Rotated Weyl Physics on Nonlinear Lithium Niobate-on-Insulator Chips

Topological photonics, featured by stable topological edge states resistant to perturbations, has been utilized to design robust integrated devices. Here, we present a study exploring the intriguing topological rotated Weyl physics in a 3D parameter space based on quaternary waveguide arrays on lithium niobate-on-insulator (LNOI) chips. Unlike previous works that focus on the Fermi arc surface states of a single Weyl structure, we can experimentally construct arbitrary interfaces between two Weyl structures whose orientations can be freely rotated in the synthetic parameter space. This intriguing system was difficult to realize in usual 3D Weyl semimetals due to lattice mismatch. We found whether the interface can host gapless topological interface states or not is determined by the relative rotational directions of the two Weyl structures. In the experiment, we have probed the local characteristics of the TISs through linear optical transmission and nonlinear second harmonic generation. Our study introduces a novel path to explore topological photonics on LNOI chips and various applications in integrated nonlinear and quantum optics.

Nonlinear tuning of PT symmetry and non-Hermitian topological states

Topology, parity-time (PT) symmetry, and nonlinearity are at the origin of many fundamental phenomena in complex systems across the natural sciences, but their mutual interplay remains unexplored. We established a nonlinear non-Hermitian topological platform for active tuning of PT symmetry and topological states. We found that the loss in a topological defect potential in a non-Hermitian photonic lattice can be tuned solely by nonlinearity, enabling the transition between PT-symmetric and non-PT-symmetric regimes and the maneuvering of topological zero modes. The interaction between two apparently antagonistic effects is revealed: the sensitivity close to exceptional points and the robustness of non-Hermitian topological states. Our scheme using single-channel control of global PT symmetry and topology via local nonlinearity may provide opportunities for unconventional light manipulation and device applications.

Symmetry-Induced Error Filtering in a Photonic Lieb Lattice

Quantum computation promises intrinsically parallel information processing capacity by harnessing the superposition and entanglement of quantum states. However, it is still challenging to realize universal quantum computation due that the reliability and scalability are limited by unavoidable noises on qubits. Nontrivial topological properties like quantum Hall phases are found capable of offering protection, but require stringent conditions of topological band gaps and broken time-reversal symmetry. Here, we propose and experimentally demonstrate a symmetry-induced error filtering scheme, showing a more general role of geometry in protection mechanism and applications. We encode qubits in a superposition of two spatial modes on a photonic Lieb lattice. The geometric symmetry endows the system with topological properties featuring a flat band touching, leading to distinctive transmission behaviors of π-phase qubits and 0-phase qubits. The geometry exhibits a significant effect on filtering phase errors, which also enables it to monitor phase deviations in real time. The symmetry-induced error filtering can be a key element for encoding and protecting quantum states, suggesting an emerging field of symmetry-protected universal quantum computation and noisy intermediate-scale quantum technologies.

Flat-Band Localization in Creutz Superradiance Lattices

Flat bands play an important role in diffraction-free photonics and attract fundamental interest in many-body physics. Here we report the engineering of flat-band localization of collective excited states of atoms in Creutz superradiance lattices with tunable synthetic gauge fields. Magnitudes and phases of the lattice hopping coefficients can be independently tuned to control the state components of the flat band and the Aharonov-Bohm phases. We can selectively excite the flat band and control the flat-band localization with the synthetic gauge field. Our study provides a room-temperature platform for flat bands of atoms and holds promising applications in exploring correlated topological materials.

Localized modes in a two-dimensional lattice with a pluslike geometry

We investigate analytically and numerically the existence and dynamical stability of different localized modes in a two-dimensional photonic lattice comprising a square plaquette inscribed in the dodecagon lattices. The eigenvalue spectrum of the underlying linear lattice is characterized by a net formed of one flat band and four dispersive bands. By tailoring the intersite coupling coefficient ratio, opening of gaps between two pairs of neighboring dispersive bands can be induced, while the fully degenerate flat band characterized by compact eigenmodes stays nested between two inner dispersive bands. The nonlinearity destabilizes the compact modes and gives rise to unique families of localized modes in the newly opened gaps, as well as in the semi-infinite gaps. The governing mechanism of mode localization in that case is the light energy self-trapping effect. We have shown the stability of a few families of nonlinear modes in gaps. The suggested lattice model may serve for probing various artificial flat-band systems such as ultracold atoms in optical lattices, periodic electronic networks, and polariton condensates.

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.

Direct Observation of Flatband Loop States Arising from Nontrivial Real-Space Topology

Topological properties of lattices are typically revealed in momentum space using concepts such as the Chern number. Here, we study unconventional loop states, namely, the noncontractible loop states (NLSs) and robust boundary modes, mediated by nontrivial topology in real space. While such states play a key role in understanding fundamental physics of flatband systems, their experimental observation has been hampered because of the challenge in realizing desired boundary conditions. Using a laser-writing technique, we optically establish photonic kagome lattices with both an open boundary by properly truncating the lattice, and a periodic boundary by shaping the lattice into a Corbino geometry. We thereby demonstrate the robust boundary modes winding around the entire edge of the open lattice and, more directly, the NLSs winding in a closed loop akin to that in a torus. We prove that the NLSs due to real-space topology persist in ideal Corbino-shaped kagome lattices of arbitrary size. Our results could be of great importance for our understanding of the singular flatbands and the intriguing physics phenomenon applicable for strongly interacting systems.

Experimental Realization of Two-Dimensional Buckled Lieb Lattice

Two-dimensional (2D) materials with a Lieb lattice host exotic electronic band structures. Such a system does not exist in nature, and it is also difficult to obtain in the laboratory due to its structural instability. Here, we experimentally realized a 2D system composed of a tin overlayer on an aluminum substrate by molecular beam epitaxy. The specific arrangement of Sn atoms on the Al(100) surface, which benefits from favorable interface interactions, forms a stabilized buckled Lieb lattice. Theoretical calculations indicate a partially broken nodal line loop and a topologically nontrivial insulating state with a spin-orbital coupling effect in the band structure of this Lieb lattice. The electronic structure of this system is experimentally characterized by angle-resolved photoemission spectroscopy, in which the hybridized states between topmost Al atoms and Sn atoms are revealed. Our work provides an appealing method for constructing 2D quantum materials based on the Lieb lattice.

Quantum Network Transfer and Storage with Compact Localized States Induced by Local Symmetries

We propose modulation protocols designed to generate, store, and transfer compact localized states in a quantum network. Induced by parameter tuning or local reflection symmetries, such states vanish outside selected domains of the complete system and are therefore ideal for information storage. Their creation and transfer is here achieved either via amplitude phase flips or via optimal temporal control of intersite couplings. We apply the concept to a decorated, locally symmetric Lieb lattice where one sublattice is dimerized, and also demonstrate it for more complex setups. The approach allows for a flexible storage and transfer of states along independent paths in lattices supporting flat energetic bands. We further demonstrate a method to equip any network featuring static perfect state transfer of single-site excitations with compact localized states, thus increasing the storage ability of these networks. We show that these compact localized states can likewise be perfectly transferred through the corresponding network by suitable, time-dependent modifications. The generic network and protocols proposed can be utilized in various physical setups such as atomic or molecular spin lattices, photonic waveguide arrays, and acoustic setups.