Localization of Chiral Edge States by the Non-Hermitian Skin Effect.

Quantum Hall systems host chiral edge states extending along the one-dimensional boundary of any two-dimensional sample. In solid state materials, the edge states serve as perfectly robust transport channels that produce a quantized Hall conductance; due to their chirality, and the topological protection by the Chern number of the bulk band structure, they cannot be spatially localized by defects or disorder. Here, we show experimentally that the chiral edge states of a lossy quantum Hall system can be localized. In a gyromagnetic photonic crystal exhibiting the quantum Hall topological phase, an appropriately structured loss configuration imparts the edge states' complex energy spectrum with a feature known as point-gap winding. This intrinsically non-Hermitian topological invariant is distinct from the Chern number invariant of the bulk (which remains intact) and induces mode localization via the "non-Hermitian skin effect." The interplay of the two topological phenomena-the Chern number and point-gap winding-gives rise to a non-Hermitian generalization of the paradigmatic Chern-type bulk-boundary correspondence principle. Compared to previous realizations of the non-Hermitian skin effect, the skin modes in this system have superior robustness against local defects and disorders.

Three-dimensional flat Landau levels in an inhomogeneous acoustic crystal.

When electrons moving in two dimensions (2D) are subjected to a strong uniform magnetic field, they form flat bands called Landau levels (LLs). LLs can also arise from pseudomagnetic fields (PMFs) induced by lattice distortions. In three-dimensional (3D) systems, there has been no experimental demonstration of LLs  as a type of flat band thus far. Here, we report the experimental realization of a flat 3D LL in an acoustic crystal. Starting from a lattice whose bandstructure exhibits a nodal ring, we design an inhomogeneous distortion corresponding to a specific pseudomagnetic vector potential (PVP). This distortion causes the nodal ring states to break up into LLs, including a zeroth LL that is flat along all three directions. These findings suggest the possibility of using nodal ring materials to generate 3D flat bands, allowing access to strong interactions and other attractive physical regimes in 3D.

Photonic Majorana quantum cascade laser with polarization-winding emission.

Topological cavities, whose modes are protected against perturbations, are promising candidates for novel semiconductor laser devices. To date, there have been several demonstrations of topological lasers (TLs) exhibiting robust lasing modes. The possibility of achieving nontrivial beam profiles in TLs has recently been explored in the form of vortex wavefront emissions enabled by a structured optical pump or strong magnetic field, which are inconvenient for device applications. Electrically pumped TLs, by contrast, have attracted attention for their compact footprint and easy on-chip integration with photonic circuits. Here, we experimentally demonstrate an electrically pumped TL based on photonic analogue of a Majorana zero mode (MZM), implemented monolithically on a quantum cascade chip. We show that the MZM emits a cylindrical vector (CV) beam, with a topologically nontrivial polarization profile from a terahertz (THz) semiconductor laser.

Self-adaptive deep reinforcement learning for THz beamforming with silicon metasurfaces in 6G communications.

Exponential growth in data rate demands has driven efforts to develop novel beamforming techniques for realizing massive multiple-input and multiple-output (MIMO) systems in sixth-generation (6G) terabits per second wireless communications. Existing beamforming techniques rely on conventional optimization algorithms that are too computationally expensive for real-time applications and require complex digital processing yet to be achieved for phased array antennas at terahertz frequencies. Here, we develop an intelligent and self-adaptive beamforming scheme enabled by deep reinforcement learning, which can predict the spatial phase profiles required to produce arbitrary desired radiation patterns in real-time. Our deep learning model adaptively trains an artificial neural network in real-time by comparing the input and predicted intensity patterns via automatic differentiation of the phase-to-intensity function. As a proof of concept, we experimentally demonstrate two-dimensional beamforming by spatially modulating broadband terahertz waves using silicon metasurfaces designed with the aid of the deep learning model. Our work offers an efficient and robust deep learning model for real-time self-adaptive beamforming to enable multi-user massive MIMO systems for 6G terahertz wireless communications, as well as intelligent metasurfaces for other terahertz applications in imaging and sensing.

Topological Chern vectors in three-dimensional photonic crystals.

The paradigmatic example of a topological phase of matter, the two-dimensional Chern insulator1-5, is characterized by a topological invariant consisting of a single integer, the scalar Chern number. Extending the Chern insulator phase from two to three dimensions requires generalization of the Chern number to a three-vector6,7, similar to the three-dimensional (3D) quantum Hall effect8-13. Such Chern vectors for 3D Chern insulators have never been explored experimentally. Here we use magnetically tunable 3D photonic crystals to achieve the experimental demonstration of Chern vectors and their topological surface states. We demonstrate Chern vector magnitudes of up to six, higher than all scalar Chern numbers previously realized in topological materials. The isofrequency contours formed by the topological surface states in the surface Brillouin zone form torus knots or links, whose characteristic integers are determined by the Chern vectors. We demonstrate a sample with surface states forming a (2, 2) torus link or Hopf link in the surface Brillouin zone, which is topologically distinct from the surface states of other 3D topological phases. These results establish the Chern vector as an intrinsic bulk topological invariant in 3D topological materials, with surface states possessing unique topological characteristics.

Active Ultrahigh-Q (0.2 × 106 ) THz Topological Cavities on a Chip.

Rapid scaling of semiconductor devices has led to an increase in the number of processor cores and integrated functionalities onto a single chip to support the growing demands of high-speed and large-volume consumer electronics. To meet this burgeoning demand, an improved interconnect capacity in terms of bandwidth density and active tunability is required for enhanced throughput and energy efficiency. Low-loss terahertz silicon interconnects with larger bandwidth offer a solution for the existing inter-/intrachip bandwidth density and energy-efficiency bottleneck. Here, a low-loss terahertz topological interconnect-cavity system is presented that can actively route signals through sharp bends, by critically coupling to a topological cavity with an ultrahigh-quality (Q) factor of 0.2 × 106 . The topologically protected large Q factor cavity enables energy-efficient optical control showing 60 dB modulation. Dynamic control is further demonstrated of the critical coupling between the topological interconnect-cavity for on-chip active tailoring of the cavity resonance linewidth, frequency, and modulation through complete suppression of the back reflection. The silicon topological cavity is complementary metal-oxide-semiconductor (CMOS)-compatible and highly desirable for hybrid electronic-photonic technologies for sixth (6G) generation terahertz communication devices. Ultrahigh-Q cavity also paves the path for designing ultrasensitive topological sensors, terahertz topological integrated circuits, and nonlinear topological photonic devices.

Time-periodic corner states from Floquet higher-order topology.

The recent discoveries of higher-order topological insulators (HOTIs) have shifted the paradigm of topological materials, previously limited to topological states at boundaries of materials, to include topological states at boundaries of boundaries, such as corners. So far, all HOTI realisations have been based on static systems described by time-invariant Hamiltonians, without considering the time-variant situation. There is growing interest in Floquet systems, in which time-periodic driving can induce unconventional phenomena such as Floquet topological phases and time crystals. Recent theories have attempted to combine Floquet engineering and HOTIs, but there has been no experimental realisation so far. Here we report on the experimental demonstration of a two-dimensional (2D) Floquet HOTI in a three-dimensional (3D) acoustic lattice, with modulation along a spatial axis serving as an effective time-dependent drive. Acoustic measurements reveal Floquet corner states with double the period of the underlying drive; these oscillations are robust, like time crystal modes, except that the robustness arises from topological protection. This shows that space-time dynamics can induce anomalous higher-order topological phases unique to Floquet systems.

Demonstration of negative refraction induced by synthetic gauge fields.

Negative refraction is a counterintuitive wave phenomenon that has inspired the development of metamaterials and metasurfaces with negative refractive indices and surface phase discontinuities, respectively. Recent theories have proposed an alternative mechanism for negative refraction: Synthetic gauge fields, induced by either dynamical modulation or motion, can shift a material's dispersion in momentum space, forcing a positive refractive index medium to exhibit negative refraction above a certain threshold. However, this phenomenon has not previously been observed. Here, we report on the experimental demonstration of gauge field­induced negative refraction in a twisted bilayer acoustic metamaterial. The synthetic gauge fields arise in a projected two-dimensional geometry and can be continuously tuned by varying the wave number along the third dimension. Gauge field­induced waveguiding with backward-propagating modes is also demonstrated in a trilayer configuration. These results introduce a mechanism for performing wave manipulation in artificially engineered materials.

Ideal type-II Weyl points in topological circuits.

Weyl points (WPs), nodal degenerate points in three-dimensional (3D) momentum space, are said to be 'ideal' if they are symmetry-related and well-separated, and reside at the same energy and far from nontopological bands. Although type-II WPs have unique spectral characteristics compared with type-I counterparts, ideal type-II WPs have not yet been reported because of a lack of an experimental platform with enough flexibility to produce strongly tilted dispersion bands. Here, we experimentally realize a topological circuit that hosts only topological bands with a minimal number of four ideal type-II WPs. By stacking two-dimensional (2D) layers of inductor-capacitor (LC) resonator dimers with the broken parity inversion symmetry (P), we achieve a strongly tilted band structure with two group velocities in the same direction, and topological surface states in an incomplete bandgap. Our results establish an ideal system for the further study of Weyl physics and other exotic topological phenomena.

Image reconstruction through a multimode fiber with a simple neural network architecture.

Multimode fibers (MMFs) have the potential to carry complex images for endoscopy and related applications, but decoding the complex speckle patterns produced by mode-mixing and modal dispersion in MMFs is a serious challenge. Several groups have recently shown that convolutional neural networks (CNNs) can be trained to perform high-fidelity MMF image reconstruction. We find that a considerably simpler neural network architecture, the single hidden layer dense neural network, performs at least as well as previously-used CNNs in terms of image reconstruction fidelity, and is superior in terms of training time and computing resources required. The trained networks can accurately reconstruct MMF images collected over a week after the cessation of the training set, with the dense network performing as well as the CNN over the entire period.

Ideal Unconventional Weyl Point in a Chiral Photonic Metamaterial.

Unconventional Weyl points (WPs), carrying topological charge 2 or higher, possess interesting properties different from ordinary charge-1 WPs, including multiple Fermi arcs that stretch over a large portion of the Brillouin zone. Thus far, such WPs have been observed in chiral materials and acoustic metamaterials, but there has been no clean demonstration in photonics in which the unconventional photonic WPs are separated from trivial bands. We experimentally realize an ideal symmetry-protected photonic charge-2 WP in a three-dimensional topological chiral microwave metamaterial. We use field mapping to directly observe the projected bulk dispersion, as well as the two long surface arcs that form a noncontractible loop wrapping around the surface Brillouin zone. The surface states span a record-wide frequency window of around 22.7% relative bandwidth. We demonstrate that the surface states exhibit a novel topological self-collimation property and are robust against disorder. This work provides an ideal photonic platform for exploring fundamental physics and applications of unconventional WPs.

Topological Anderson Insulator in Disordered Photonic Crystals.

Recent studies have revealed the counterintuitive possibility that increasing disorder can turn a topologically trivial insulator into a nontrivial insulator, called a topological Anderson insulator (TAI). Here, we propose and experimentally demonstrate a photonic TAI in a two-dimensional disordered gyromagnetic photonic crystal in the microwave regime. We directly observe the disorder-induced topological phase transition from a trivial insulator to a TAI with robust chiral edge states. We also demonstrate topological heterostructures that host edge states at interfaces between domains with different disorder parameters.

Photonic amorphous topological insulator.

The current understanding of topological insulators and their classical wave analogs, such as photonic topological insulators, is mainly based on topological band theory. However, standard band theory does not apply to amorphous phases of matter, which are formed by non-crystalline lattices with no long-range positional order but only short-range order, exhibiting unique phenomena such as the glass-to-liquid transition. Here, we experimentally investigate amorphous variants of a Chern number-based photonic topological insulator. By tuning the disorder strength in the lattice, we demonstrate that photonic topological edge states can persist into the amorphous regime prior to the glass-to-liquid transition. After the transition to a liquid-like lattice configuration, the signatures of topological edge states disappear. This interplay between topology and short-range order in amorphous lattices paves the way for new classes of non-crystalline topological photonic bandgap materials.

Observation of an acoustic octupole topological insulator.

Berry phase associated with energy bands in crystals can lead to quantised observables like quantised dipole polarizations in one-dimensional topological insulators. Recent theories have generalised the concept of quantised dipoles to multipoles, resulting in the discovery of multipole topological insulators which exhibit a hierarchy of multipole topology: a quantised octupole moment in a three-dimensional bulk induces quantised quadrupole moments on its two-dimensional surfaces, which in turn induce quantised dipole moments on one-dimensional hinges. Here, we report on the realisation of an octupole topological insulator in a three-dimensional acoustic metamaterial. We observe zero-dimensional topological corner states, one-dimensional gapped hinge states, two-dimensional gapped surface states, and three-dimensional gapped bulk states, representing the hierarchy of octupole, quadrupole and dipole moments. Conditions for forming a nontrivial octupole moment are demonstrated by comparisons with two different lattice configurations having trivial octupole moments. Our work establishes the multipole topology and its full hierarchy in three-dimensional geometries.

Observation of an unpaired photonic Dirac point.

At photonic Dirac points, electromagnetic waves are governed by the same equations as two-component massless relativistic fermions. However, photonic Dirac points are known to occur in pairs in "photonic graphene" and other similar photonic crystals, which necessitates special precautions to excite only one valley state. Systems hosting unpaired photonic Dirac points are significantly harder to realize, as they require broken time-reversal symmetry. Here, we report on the observation of an unpaired Dirac point in a planar two-dimensional photonic crystal. The structure incorporates gyromagnetic materials, which break time-reversal symmetry; the unpaired Dirac point occurs when a parity-breaking parameter is fine-tuned to a topological transition between a photonic Chern insulator and a conventional photonic insulator phase. Evidence for the unpaired Dirac point is provided by transmission and field-mapping experiments, including a demonstration of strongly non-reciprocal reflection. This unpaired Dirac point may have applications in valley filters and angular selective photonic devices.

Electrically pumped topological laser with valley edge modes.

Quantum cascade lasers are compact, electrically pumped light sources in the technologically important mid-infrared and terahertz region of the electromagnetic spectrum1,2. Recently, the concept of topology3 has been expanded from condensed matter physics into photonics4, giving rise to a new type of lasing5-8 using topologically protected photonic modes that can efficiently bypass corners and defects4. Previous demonstrations of topological lasers have required an external laser source for optical pumping and have operated in the conventional optical frequency regime5-8. Here we demonstrate an electrically pumped terahertz quantum cascade laser based on topologically protected valley edge states9-11. Unlike topological lasers that rely on large-scale features to impart topological protection, our compact design makes use of the valley degree of freedom in photonic crystals10,11, analogous to two-dimensional gapped valleytronic materials12. Lasing with regularly spaced emission peaks occurs in a sharp-cornered triangular cavity, even if perturbations are introduced into the underlying structure, owing to the existence of topologically protected valley edge states that circulate around the cavity without experiencing localization. We probe the properties of the topological lasing modes by adding different outcouplers to the topological cavity. The laser based on valley edge states may open routes to the practical use of topological protection in electrically driven laser sources.

Observation of a topological nodal surface and its surface-state arcs in an artificial acoustic crystal.

Three-dimensional (3D) gapless topological phases can be classified by the dimensionality of the band degeneracies, including zero-dimensional (0D) nodal points, one-dimensional (1D) nodal lines, and two-dimensional (2D) nodal surfaces. Both nodal points and nodal lines have been realized recently in photonics and acoustics. However, a nodal surface has never been observed in any classical-wave system. Here, we report on the experimental observation of a twofold symmetry-enforced nodal surface in a 3D chiral acoustic crystal. In particular, the demonstrated nodal surface carries a topological charge of 2, constituting the first realization of a higher-dimensional topologically-charged band degeneracy. Using direct acoustic field measurements, we observe the projected nodal surface and its Fermi-arc-like surface states and demonstrate topologically-induced robustness of the surface states against disorders. This discovery of a higher-dimensional topologically-charged band degeneracy paves the way toward further explorations of the physics and applications of new topological semimetal phases.

Realization of an Acoustic Third-Order Topological Insulator.

The recent discovery of higher-order topological insulators (TIs) has opened new possibilities in the search for novel topological materials and metamaterials. Second-order TIs have been implemented in two-dimensional (2D) systems exhibiting topological "corner states," as well as three-dimensional (3D) systems having one-dimensional (1D) topological "hinge states." Third-order TIs, which have topological states three dimensions lower than the bulk (which must thus be 3D or higher), have not yet been reported. Here, we describe the realization of a third-order TI in an anisotropic diamond-lattice acoustic metamaterial. The bulk acoustic band structure has nontrivial topology characterized by quantized Wannier centers. By direct acoustic measurement, we observe corner states at two corners of a rhombohedronlike structure, as predicted by the quantized Wannier centers. This work extends topological corner states from 2D to 3D, and may find applications in novel acoustic devices.

Acoustic higher-order topological insulator on a kagome lattice.

Higher-order topological insulators1-5 are a family of recently predicted topological phases of matter that obey an extended topological bulk-boundary correspondence principle. For example, a two-dimensional (2D) second-order topological insulator does not exhibit gapless one-dimensional (1D) topological edge states, like a standard 2D topological insulator, but instead has topologically protected zero-dimensional (0D) corner states. The first prediction of a second-order topological insulator1, based on quantized quadrupole polarization, was demonstrated in classical mechanical6 and electromagnetic7,8 metamaterials. Here we experimentally realize a second-order topological insulator in an acoustic metamaterial, based on a 'breathing' kagome lattice9 that has zero quadrupole polarization but a non-trivial bulk topology characterized by quantized Wannier centres2,9,10. Unlike previous higher-order topological insulator realizations, the corner states depend not only on the bulk topology but also on the corner shape; we show experimentally that they exist at acute-angled corners of the kagome lattice, but not at obtuse-angled corners. This shape dependence allows corner states to act as topologically protected but reconfigurable local resonances.