Floquet Topological Insulators Research Articles

All‐Bands‐Flat Floquet Topological Photonic Insulators with Microring Lattices

Coupled microring lattices are versatile photonic systems that can be used to realize various topological phases of matter. In two‐dimensional (2D) microring lattices, the periodic and unidirectional circulation of light in each microring gives rise to a time‐like dimension, so that the lattice emulates a (2 + 1)D system with much richer topological behaviors than static 2D lattices. Accurate treatment of these systems requires a departure from the static tight‐binding model of coupled resonators and take into account the periodic coupling sequence of light in the lattice network. This article provides an overview of the theory and design of (2 + 1)D microring lattices for realizing Floquet topological photonic insulators (TPIs). Particular focus is placed on the microring Lieb lattice with perfect couplings, which emulates an anomalous Floquet insulator with all flat bands. Such a system exhibits some unique properties, including wide edge mode continuum exceeding a Floquet–Brillouin zone, super‐robustness to lattice disorder, Aharonov–Bohm (AB) caging and compact localized flat‐band states that can be used to realize high‐quality factor topological resonators. All‐bands‐flat Floquet–Lieb microring lattices provide a versatile platform for investigating topological physics as well as potential applications in realizing topologically‐protected photonic devices.

Dynamical chiral symmetry and symmetry-class conversion in Floquet topological insulators

Dynamical chiral symmetry and symmetry-class conversion in Floquet topological insulators

Four-dimensional Floquet topological insulator with an emergent second Chern number

Four-dimensional Floquet topological insulator with an emergent second Chern number

Floquet topological insulators with spin-orbit coupling

Floquet topological insulators with spin-orbit coupling

Observation of a Photonic Orbital Gauge Field.

Gauge field is widely studied in natural and artificial materials. With an effective magnetic field for uncharged particles, many intriguing phenomena are observed in several systems like photonic Floquet topological insulator. However, previous researches about the gauge field mostly focus on limited dimensions such as the Dirac spinor in graphene materials. Here, an orbital gauge field based on photonic triangular lattices is first proposed and experimentally observed. Disclination defects with Frank angle Ω created on such lattices breaks the original lattice symmetry and generates purely geometric gauge field operating on orbital basis functions. Interestingly, it is found that bound states near zero energy with the orbital angular momentum (OAM) l = 2 are intensively confined at the disclination as gradually expanding Ω. Moreover, the introduction of a vector potential field breaks the time-reversal symmetry of the orbital gauge field, experimentally manifested by the chiral transmission of light on helical waveguides. The orbital gauge field further suggests fantastic applications of manipulating the vortex light in photonic integrateddevices.

Floquet engineering of selective magnon–magnon coupling in synthetic antiferromagnets

Floquet engineering is a widely applied method for temporally periodic driving in various quantum systems, capable of inducing emergent phenomena, such as Floquet states, Floquet topological insulators, and temporal crystals. In this work, we study the selective magnon–magnon coupling between sidebands by introducing Floquet engineering into magnonic system: coupling between Floquet sidebands occurs only when there is an odd order difference. In addition to the coherent coupling between the optical and acoustic modes in synthetic antiferromagnet, a Floquet coupling is observed as the secondary splitting of the two hybridized magnon modes. The splitting width of Floquet coupling exhibits different dependence with driving amplitude, i.e., linearly increasing for weak driving regime and nonlinearly varying for strong driving regime. Specifically, the nonlinear dependence arises from the coupling between hybridized sidebands that exhibit odd order differences in their components. Our findings could promote the further development of Floquet magnonics and its application for quantum information.

Fractal photonic anomalous Floquet topological insulators to generate multiple quantum chiral edge states

Anomalous Floquet topological insulators with vanishing Chern numbers but supporting chiral edge modes are attracting more and more attention. Since the existing anomalous Floquet topological insulators usually support only one kind of chiral edge mode even at a large lattice size, they are unscalable and unapplicable for multistate topological quantum systems. Recently, fractal topological insulators with self-similarity have been explored to support more nontrivial modes. Here, we demonstrate the first experimental realization of fractal photonic anomalous Floquet topological insulators based on dual Sierpinski carpet consisting of directional couplers using the femtosecond laser direct writing. The fabricated lattices support much more kinds of chiral edge states with fewer waveguides and enable perfect hopping of quantum states with near unit transfer efficiency. Instead of zero-dimensional bound modes for quantum state transport in previous laser direct-written topological insulators, we generate multiple propagating single-photon chiral edge states in the fractal lattice and observe high-visibility quantum interferences. These suggest the successful realization of highly indistinguishable single-photon chiral edge states, which can be applied in various quantum operations. This work provides the potential for enhancing the multi-fold manipulation of quantum states, enlarging the encodable quantum information capacity in a single lattice via high-dimensional encoding and many other fractal applications.

Topological Edge States on Different Domain Walls of Two Opposed Helical Waveguide Arrays

Floquet topological insulators (FTIs) have richer topological properties than static systems. In this work, we designed different domain wall (DW) structures consisting of a Floquet photonic lattice with opposite helical directions. We investigated the existence and types of edge states in three shared coupling structures and the impact of these shared coupling structures on edge states. When two opposite helical lattices share a straight waveguide array coupling, the edge states are localized on the straight waveguide. When two opposite helical lattices share a clockwise (or anticlockwise) helical waveguide array coupling, the DWs consist of zigzag and bearded edges, but the positions of the zigzag and bearded edges of the shared clockwise waveguide array are different from those of the shared anticlockwise waveguide array. The slope and transmission rate of the edge states both vary with the degree of coupling between the shared waveguides. The characteristics of these edge states, such as transmission speed and band gap width, are also affected by the incidence angle, modulation phase factor, and helical radii, and the methods for controlling the edge states in different shared coupling structures are provided. This will help deepen our understanding of how topological structures influence the electronic and photonic properties of materials. This could also lead to combining topology with metasurface-based structured light, which would highlight many novel properties with great application potential for various fields, such as imaging, metrology, communication, quantum information processing, and light–matter interaction.

Floquet non-Abelian topological insulator and multifold bulk-edge correspondence

Topological phases characterized by non-Abelian charges are beyond the scope of the paradigmatic tenfold way and have gained increasing attention recently. Here we investigate topological insulators with multiple tangled gaps in Floquet settings and identify uncharted Floquet non-Abelian topological insulators without any static or Abelian analog. We demonstrate that the bulk-edge correspondence is multifold and follows the multiplication rule of the quaternion group Q8. The same quaternion charge corresponds to several distinct edge-state configurations that are fully determined by phase-band singularities of the time evolution. In the anomalous non-Abelian phase, edge states appear in all bandgaps despite trivial quaternion charge. Furthermore, we uncover an exotic swap effect—the emergence of interface modes with swapped driving, which is a signature of the non-Abelian dynamics and absent in Floquet Abelian systems. Our work, for the first time, presents Floquet topological insulators characterized by non-Abelian charges and opens up exciting possibilities for exploring the rich and uncharted territory of non-equilibrium topological phases.

Non-Hermitian Floquet Topological Matter-A Review.

The past few years have witnessed a surge of interest in non-Hermitian Floquet topological matter due to its exotic properties resulting from the interplay between driving fields and non-Hermiticity. The present review sums up our studies on non-Hermitian Floquet topological matter in one and two spatial dimensions. We first give a bird's-eye view of the literature for clarifying the physical significance of non-Hermitian Floquet systems. We then introduce, in a pedagogical manner, a number of useful tools tailored for the study of non-Hermitian Floquet systems and their topological properties. With the aid of these tools, we present typical examples of non-Hermitian Floquet topological insulators, superconductors, and quasicrystals, with a focus on their topological invariants, bulk-edge correspondences, non-Hermitian skin effects, dynamical properties, and localization transitions. We conclude this review by summarizing our main findings and presenting our vision of future directions.

Electric circuit simulation of Floquet topological insulators in Fourier space

We present a method for simulating any non-interacting and time-periodic tight-binding Hamiltonian in Fourier space using electric circuits made of inductors and capacitors. We first map the time-periodic Hamiltonian to a Floquet Hamiltonian, which converts the time dimension into a Floquet dimension. In electric circuits, this Floquet dimension is simulated as an extraspatial dimension without any time dependency in the electrical elements. The number of replicas needed in the Floquet Hamiltonian depends on the frequency and strength of the drive. We also demonstrate that we can detect the topological edge states (including the anomalous edge states in the dynamical gap) in an electric circuit by measuring the two-point impedance between the nodes. Our method paves a simple and promising way to explore and control Floquet topological phases in electric circuits.

Sound non-reciprocity based on synthetic magnetism

Synthetic magnetism has been recently realized using spatiotemporal modulation patterns, producing non-reciprocal steering of charge-neutral particles such as photons and phonons. Here, we design and experimentally demonstrate a non-reciprocal acoustic system composed of three compact cavities interlinked with both dynamic and static couplings, in which phase-correlated modulations induce a synthetic magnetic flux that breaks time-reversal symmetry. Within the rotating wave approximation, the transport properties of the system are controlled to efficiently realize large non-reciprocal acoustic transport. By optimizing the coupling strengths and modulation phases, we achieve frequency-preserved unidirectional transport with 45-dB isolation ratio and 0.85 forward transmission. Our results open to the realization of acoustic non-reciprocal technologies with high efficiency and large isolation, and offer a route towards Floquet topological insulators for sound.

Quantized charge transport in disordered Floquet topological insulators in the absence of Anderson localization

Quantized charge transport in disordered Floquet topological insulators in the absence of Anderson localization

Optical conductivity and orbital magnetization of Floquet vortex states

Motivated by recent experimental demonstrations of Floquet topological insulators, there have been several theoretical proposals for using structured light, either spatial or spectral, to create other properties such as flat bands and vortex states. In particular, the generation of vortex states in a massive Dirac fermion insulator irradiated by light carrying nonzero orbital angular momentum (OAM) has been proposed. Here, we evaluate the orbital magnetization and optical conductivity as physical observables for such a system. We show that the OAM of light induces nonzero orbital magnetization and current density. The orbital magnetization density increases linearly as a function of the OAM degree. In certain regimes, we find that orbital magnetization density is independent of the system size, width, and Rabi frequency of light. It is shown that the orbital magnetization arising from our Floquet theory is large and can be probed by magnetometry measurements. Furthermore, we study the optical conductivity for various types of electron transitions between different states such as vortex, edge, and bulk that are present in the system. Based on the peaks in conductance, a scheme for the detection of vortex states is proposed.

Magneto-optical conductivity and giant Faraday-Kerr rotation in Floquet topological insulators

Magneto-optical conductivity and giant Faraday-Kerr rotation in Floquet topological insulators

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.

Steering edge currents through a Floquet topological insulator

Periodic driving may cause topologically protected, chiral transport along edges of a two-dimensional (2D) lattice that, without driving, would be topologically trivial. We study what happens if one adds a different on-site potential along the diagonal of such a 2D grid. In addition to the usual bulk and edge states, the system then also exhibits doublon states, analogous to two interacting particles in one dimension. A particle initially located at an edge propagates along the system's boundary. Its wave function splits when it hits the diagonal and continues propagating simultaneously along the edge and the diagonal. The strength of the diagonal potential determines the ratio between both parts. We show that for specific values of the diagonal potential, hopping onto the diagonal is prohibited so that the system effectively separates into two triangular lattices. For other values of the diagonal potential, we find a temporal delay between the two contributions traveling around and through the system. This behavior could enable the steering of topologically protected transport of light along the edges and through the bulk of laser-inscribed photonic waveguide arrays.

Kitaev formula for periodic, quasicrystal, and fractal Floquet topological insulators

Topological insulators are attracting researchers to reveal related fundamental physics, whose fractal versions are expanding topological physics into the fractal regime. The fractal topological systems, however, show a challenge to the Kitaev formula, because it leads to a noninteger Chern number. The Chern number within the band gap oscillates more and more actively as the system size increases. We uncover that such a challenge should be attributed to the coupling, brought about by the lattice fractalization, between the internal and external edge states. To confirm this, examples of periodic, quasicrystal, and fractal Floquet topological insulators are compared with each other by analyzing the topological edge states and local Chern markers. The sum of the local Chern markers on all the edges of the fractal lattice presents a fractional power relation with the system size, reflecting the fractalization of the topological edge states. We reveal that once such a coupling is shut down, the Kitaev formula would once again result in integers.

Various topological phases and their abnormal effects of topological acoustic metamaterials

AbstractThe last 20 years have witnessed growing impacts of the topological concept on the branches of physics, including materials, electronics, photonics, and acoustics. Topology describes objects with some global invariant property under continuous deformation, which in mathematics could date back to the 17th century and mature in the 20th century. In physics, it successfully underpinned the physics of the Quantum Hall effect in 1984. To date, topology has been extensively applied to describe topological phases in acoustic metamaterials. As artificial structures, acoustic metamaterials could be well theoretically analyzed, on‐demand designed, and easily fabricated by modern techniques, such as three‐dimensional printing. Some new theoretical topological models were first discovered in acoustic metamaterials analogous to electronic counterparts, associated with novel effects for acoustics closer to applications. In this review, we focused on the concept of topology and its realization in airborne acoustic crystals, solid elastic phononic crystals, and surface acoustic wave systems. We also introduced emerging concepts of non‐Hermitian, higher‐order, and Floquet topological insulators in acoustics. It has been shown that the topology theory has such a powerful generality that among the disciplines from electron to photon and phonon, from electronic to photonics and acoustics, from acoustic topological theory to acoustic devices, could interact and be analogous to fertilize fantastic new ideas and prototype devices, which might find applications in acoustic engineering and noise‐vibration control engineering in the near future.

Anomalous broadband Floquet topological metasurface with pure site rings

Photonic and acoustic topological insulators exhibiting one-way transportation that is robust against defects and impurities are typically realized in coupled arrays of two-dimensional ring resonators. These systems have produced a series of applications, including optical isolators, delay lines, and lasers. However, the structures are complicated because an additional coupler ring between neighboring rings is needed to construct photonic pseudospin. A photonic anomalous Floquet topological insulator is proposed and experimentally demonstrated in the microwave regime. This improved design takes advantage of the efficient and backward coupling of negative-index media. The results contribute to the understanding of topological structures in metamaterials and point toward a unique direction for constructing useful topological photonic devices.