Direct observation of geometric-phase interference in dynamics around a conical intersection

Conical intersections are ubiquitous in chemistry and physics, often governing processes such as light harvesting, vision, photocatalysis and chemical reactivity. They act as funnels between electronic states of molecules, allowing rapid and efficient relaxation during chemical dynamics. In addition, when a reaction path encircles a conical intersection, the molecular wavefunction experiences a geometric phase, which can affect the outcome of the reaction through quantum-mechanical interference. Past experiments have measured indirect signatures of geometric phases in scattering patterns and spectroscopic observables, but there has been no direct observation of the underlying wavepacket interference. Here we experimentally observe geometric-phase interference in the dynamics of a wavepacket travelling around an engineered conical intersection in a programmable trapped-ion quantum simulator. To achieve this, we develop a technique to reconstruct the two-dimensional wavepacket densities of a trapped ion. Experiments agree with the theoretical model, demonstrating the ability of analogue quantum simulators-such as those realized using trapped ions-to accurately describe nuclear quantum effects.

Berry Curvature and Bulk-Boundary Correspondence from Transport Measurement for Photonic Chern Bands

Berry curvature is a fundamental element to characterize topological quantum physics, while a full measurement of Berry curvature in momentum space was not reported for topological states. Here we achieve two-dimensional Berry curvature reconstruction in a photonic quantum anomalous Hall system via Hall transport measurement of a momentum-resolved wave packet. Integrating measured Berry curvature over the two-dimensional Brillouin zone, we obtain Chern numbers corresponding to -1 and 0. Further, we identify bulk-boundary correspondence by measuring topology-linked chiral edge states at the boundary. The full topological characterization of photonic Chern bands from Berry curvature, Chern number, and edge transport measurements enables our photonic system to serve as a versatile platform for further in-depth study of novel topological physics.

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.

Experimental realization of a high precision tunable hexagonal optical lattice

Hexagonal optical lattices offer a tunable platform to study exotic orbital physics in solid state materials. Here, we present a versatile high-precision scheme to implement a hexagonal optical lattice potential, which is engineered by overlapping two independent triangular optical sublattices generated by laser beams with slightly different wavelengths around 1064 nm. This enables us to precisely control the detailed structure of the hexagonal lattice by adjusting the relative position and the relative lattice depth of the two triangular optical sublattices. Taking advantage of the sensitive dependence of the second Bloch band on small lattice deformations, we propose a strategy to optimize the optical lattice geometry with an extremely high precision. This method can also be extended to other lattice configurations involving more than two sublattices. Our work provides the experimental requirements in the search for novel orbital physics of ultracold atoms, for example, in the flat p-band of the hexagonal optical lattice.

Synthetic Topological Vacua of Yang-Mills Fields in Bose-Einstein Condensates

Topological vacua are a family of degenerate ground states of Yang-Mills fields with zero field strength but nontrivial topological structures. They play a fundamental role in particle physics and quantum field theory, but have not yet been experimentally observed. Here we report the first theoretical proposal and experimental realization of synthetic topological vacua with a cloud of atomic Bose-Einstein condensates. Our setup provides a promising platform to demonstrate the fundamental concept that a vacuum, rather than being empty, has rich spatial structures. The Hamiltonian for the vacuum of topological number n=1 is synthesized and the related Hopf index is measured. The vacuum of topological number n=2 is also realized, and we find that vacua with different topological numbers have distinctive spin textures and Hopf links. Our Letter opens up opportunities for exploring topological vacua and related long-sought-after instantons in tabletop experiments.

Wave Packet Dynamics in Synthetic Non-Abelian Gauge Fields

It is generally admitted that in quantum mechanics, the electromagnetic potentials have physical interpretations otherwise absent in classical physics as illustrated by the Aharonov-Bohm effect. In 1984, Berry interpreted this effect as a geometrical phase factor. The same year, Wilczek and Zee generalized the concept of Berry phases to degenerate levels and showed that a non-Abelian gauge field arises in these systems. In sharp contrast with the Abelian case, spatially uniform non-Abelian gauge fields can induce particle noninertial motion. We explore this intriguing phenomenon with a degenerated Fermionic atomic gas subject to a two-dimensional synthetic SU(2) non-Abelian gauge field. We reveal the spin Hall nature of the noninertial dynamic as well as its anisotropy in amplitude and frequency due to the spin texture of the system. We finally draw the similarities and differences of the observed wave packet dynamic and the celebrated Zitterbewegung effect of the relativistic Dirac equation.

Direct geometric probe of singularities in band structure

A quantum system’s energy landscape may have points where multiple energy surfaces are degenerate and that exhibit singular geometry of the wave function manifold, with major consequences for the system’s properties. Ultracold atoms in optical lattices have been used to indirectly characterize such points in the band structure. We measured the non-Abelian transformation produced by transport directly through the singularities. We accelerated atoms along a quasi-momentum trajectory that enters, turns, and then exits the singularities at linear and quadratic band-touching points of a honeycomb lattice. Measurements after transport identified the topological winding numbers of these singularities to be 1 and 2, respectively. Our work introduces a distinct method for probing singularities that enables the study of non-Dirac singularities in ultracold-atom quantum simulators.

Large-S and Tensor-Network Methods for Strongly-Interacting Topological Insulators

The study of correlation effects in topological phases of matter can benefit from a multidisciplinary approach that combines techniques drawn from condensed matter, high-energy physics and quantum information science. In this work, we exploit these connections to study the strongly-interacting limit of certain lattice Hubbard models of topological insulators, which map onto four-Fermi quantum field theories with a Wilson-type discretisation and have been recently shown to be at reach of cold-atom quantum simulators based on synthetic spin-orbit coupling. We combine large-S and tensor-network techniques to explore the possible spontaneous symmetry-breaking phases that appear when the interactions of the topological insulators are sufficiently large. In particular, we show that varying the Wilson parameter r of the lattice discretisations leads to a novel Heisenberg–Ising compass model with critical lines that flow with the value of r.

Nonlinear Topological Effects in Optical Coupled Hexagonal Lattice

Topological physics in optical lattices have attracted much attention in recent years. The nonlinear effects on such optical systems remain well-explored and a large amount of progress has been achieved. In this paper, under the mean-field approximation for a nonlinearly optical coupled boson-hexagonal lattice system, we calculate the nonlinear Dirac cone and discuss its dependence on the parameters of the system. Due to the special structure of the cone, the Berry phase (two-dimensional Zak phase) acquired around these Dirac cones is quantized, and the critical value can be modulated by interactions between different lattices sites. We numerically calculate the overall Aharonov-Bohm (AB) phase and find that it is also quantized, which provides a possible topological number by which we can characterize the quantum phases. Furthermore, we find that topological phase transition occurs when the band gap closes at the nonlinear Dirac points. This is different from linear systems, in which the transition happens when the band gap closes and reopens at the Dirac points.

Measurement of Spin Chern Numbers in Quantum Simulated Topological Insulators

The topology of quantum systems has become a topic of great interest since the discovery of topological insulators. However, as a hallmark of the topological insulators, the spin Chern number has not yet been experimentally detected. The challenge to directly measure this topological invariant lies in the fact that this spin Chern number is defined based on artificially constructed wave functions. Here we experimentally mimic the celebrated Bernevig-Hughes-Zhang model with cold atoms, and then measure the spin Chern number with the linear response theory. We observe that, although the Chern number for each spin component is ill defined, the spin Chern number measured by their difference is still well defined when both energy and spin gaps are nonvanished.

Unsupervised machine learning of topological phase transitions from experimental data

Identifying phase transitions is one of the key challenges in quantum many-body physics. Recently, machine learning methods have been shown to be an alternative way of localising phase boundaries from noisy and imperfect data without the knowledge of the order parameter. Here, we apply different unsupervised machine learning techniques, including anomaly detection and influence functions, to experimental data from ultracold atoms. In this way, we obtain the topological phase diagram of the Haldane model in a completely unbiased fashion. We show that these methods can successfully be applied to experimental data at finite temperatures and to the data of Floquet systems when post-processing the data to a single micromotion phase. Our work provides a benchmark for the unsupervised detection of new exotic phases in complex many-body systems.

Measuring Topological Invariants in a Polaritonic Analog of Graphene

Topological materials rely on engineering global properties of their bulk energy bands called topological invariants. These invariants, usually defined over the entire Brillouin zone, are related to the existence of protected edge states. However, for an important class of Hamiltonians corresponding to 2D lattices with time-reversal and chiral symmetry (e.g., graphene), the existence of edge states is linked to invariants that are not defined over the full 2D Brillouin zone, but on reduced 1D subspaces. Here, we demonstrate a novel scheme based on a combined real- and momentum-space measurement to directly access these 1D topological invariants in lattices of semiconductor microcavities confining exciton polaritons. We extract these invariants in arrays emulating the physics of regular and critically compressed graphene where Dirac cones have merged. Our scheme provides a direct evidence of the bulk-edge correspondence in these systems and opens the door to the exploration of more complex topological effects, e.g., involving disorder and interactions.

Topological features without a lattice in Rashba spin-orbit coupled atoms

Topological order can be found in a wide range of physical systems, from crystalline solids, photonic meta-materials and even atmospheric waves to optomechanic, acoustic and atomic systems. Topological systems are a robust foundation for creating quantized channels for transporting electrical current, light, and atmospheric disturbances. These topological effects are quantified in terms of integer-valued 'invariants', such as the Chern number, applicable to the quantum Hall effect, or the [Formula: see text] invariant suitable for topological insulators. Here, we report the engineering of Rashba spin-orbit coupling for a cold atomic gas giving non-trivial topology, without the underlying crystalline structure that conventionally yields integer Chern numbers. We validated our procedure by spectroscopically measuring both branches of the Rashba dispersion relation which touch at a single Dirac point. We then measured the quantum geometry underlying the dispersion relation using matter-wave interferometry to implement a form of quantum state tomography, giving a Berry's phase with magnitude π. This implies that opening a gap at the Dirac point would give two dispersions (bands) each with half-integer Chern number, potentially implying new forms of topological transport.

Non-Abelian Bloch oscillations in higher-order topological insulators

Bloch oscillations (BOs) are a fundamental phenomenon by which a wave packet undergoes a periodic motion in a lattice when subjected to a force. Observed in a wide range of synthetic systems, BOs are intrinsically related to geometric and topological properties of the underlying band structure. This has established BOs as a prominent tool for the detection of Berry-phase effects, including those described by non-Abelian gauge fields. In this work, we unveil a unique topological effect that manifests in the BOs of higher-order topological insulators through the interplay of non-Abelian Berry curvature and quantized Wilson loops. It is characterized by an oscillating Hall drift synchronized with a topologically-protected inter-band beating and a multiplied Bloch period. We elucidate that the origin of this synchronization mechanism relies on the periodic quantum dynamics of Wannier centers. Our work paves the way to the experimental detection of non-Abelian topological properties through the measurement of Berry phases and center-of-mass displacements.

Floquet Higher-Order Topological Insulators with Anomalous Dynamical Polarization

Higher-order topological insulators (HOTIs) have emerged as a new class of phases, whose robust in-gap "corner" modes arise from the bulk higher-order multipoles beyond the dipoles in conventional topological insulators. Here, we incorporate Floquet driving into HOTIs, and report for the first time a dynamical polarization theory with anomalous nonequilibrium multipoles. Further, a proposal to detect not only corner states but also their dynamical origin in cold atoms is demonstrated, with the latter one never achieved before. Experimental determination of anomalous Floquet corner modes is also proposed.

Universal momentum-to-real-space mapping of topological singularities

Topological properties of materials are typically presented in momentum space. Here, we demonstrate a universal mapping of topological singularities from momentum to real space. By exciting Dirac-like cones in photonic honeycomb (pseudospin-1/2) and Lieb (pseudospin-1) lattices with vortex beams of topological charge l, optimally aligned with a given pseudospin state s, we directly observe topological charge conversion that follows the rule l → l + 2s. Although the mapping is observed in photonic lattices where pseudospin-orbit interaction takes place, we generalize the theory to show it is the nontrivial Berry phase winding that accounts for the conversion which persists even in systems where angular momentum is not conserved, unveiling its topological origin. Our results have direct impact on other branches of physics and material sciences beyond the 2D photonic platform: equivalent mapping occurs for 3D topological singularities such as Dirac-Weyl synthetic monopoles, achievable in mechanical, acoustic, or ultracold atomic systems, and even with electron beams.

Topological properties of materials are typically presented in momentum space. Here, the authors show a universal mapping of topological singularities from momentum to real space, potentially applicable to a wide range of systems.

Tunable Nonreciprocal Quantum Transport through a Dissipative Aharonov-Bohm Ring in Ultracold Atoms

We report the experimental observation of tunable, nonreciprocal quantum transport of a Bose-Einstein condensate in a momentum lattice. By implementing a dissipative Aharonov-Bohm (AB) ring in momentum space and sending atoms through it, we demonstrate a directional atom flow by measuring the momentum distribution of the condensate at different times. While the dissipative AB ring is characterized by the synthetic magnetic flux through the ring and the laser-induced loss on it, both the propagation direction and transport rate of the atom flow sensitively depend on these highly tunable parameters. We demonstrate that the nonreciprocity originates from the interplay of the synthetic magnetic flux and the laser-induced loss, which simultaneously breaks the inversion and the time-reversal symmetries. Our results open up the avenue for investigating nonreciprocal dynamics in cold atoms, and highlight the dissipative AB ring as a flexible building element for applications in quantum simulation and quantum information.

Experimental measurement of the quantum geometric tensor using coupled qubits in diamond

Geometry and topology are fundamental concepts, which underlie a wide range of fascinating physical phenomena such as topological states of matter and topological defects. In quantum mechanics, the geometry of quantum states is fully captured by the quantum geometric tensor. Using a qubit formed by an NV center in diamond, we perform the first experimental measurement of the complete quantum geometric tensor. Our approach builds on a strong connection between coherent Rabi oscillations upon parametric modulations and the quantum geometry of the underlying states. We then apply our method to a system of two interacting qubits, by exploiting the coupling between the NV center spin and a neighboring ¹³C nuclear spin. Our results establish coherent dynamical responses as a versatile probe for quantum geometry, and they pave the way for the detection of novel topological phenomena in solid state.

Nonlinear Hall Acceleration and the Quantum Rectification Sum Rule

Electrons moving in a Bloch band are known to acquire an anomalous Hall velocity proportional to the Berry curvature of the band which is responsible for the intrinsic linear Hall effect in materials with broken time-reversal symmetry. Here, we demonstrate that there is also an anomalous correction to the electron acceleration which is proportional to the Berry curvature dipole and is responsible for the nonlinear Hall effect recently discovered in materials with broken inversion symmetry. This allows us to uncover a deeper meaning of the Berry curvature dipole as a nonlinear version of the Drude weight that serves as a measurable order parameter for broken inversion symmetry in metals. We also derive a quantum rectification sum rule in time reversal invariant materials by showing that the integral over frequency of the rectification conductivity depends solely on the Berry connection and not on the band energies. The intraband spectral weight of this sum rule is exhausted by the Berry curvature dipole Drude-like peak, and the interband weight is also entirely controlled by the Berry connection. This sum rule opens a door to search for alternative photovoltaic technologies based on the Berry geometry of bands. We also describe the rectification properties of Weyl semimetals which are a promising platform to investigate these effects.

How to Directly Measure Floquet Topological Invariants in Optical Lattices

The classification of topological Floquet systems with time-periodic Hamiltonians transcends that of static systems. For example, spinless fermions in periodically driven two-dimensional lattices are not completely characterized by the Chern numbers of the quasienergy bands, but rather by a set of winding numbers associated with the gaps. We propose a feasible scheme for measuring these winding numbers in a periodically driven optical lattice efficiently and directly. It is based on the construction of a one-parameter family of drives, continuously connecting the Floquet system of interest to a trivial reference system. The winding numbers are then determined by the identification and the tomography of the band-touching singularities occurring on the way. As a by-product, we also propose a method for probing spectral properties of time evolution operators via a time analog of crystallography.

Experimental Measurement of the Quantum Metric Tensor and Related Topological Phase Transition with a Superconducting Qubit

A Berry curvature is an imaginary component of the quantum geometric tensor (QGT) and is well studied in many branches of modern physics; however, the quantum metric as a real component of the QGT is less explored. Here, by using tunable superconducting circuits, we experimentally demonstrate two methods to directly measure the quantum metric tensor for characterizing the geometry and topology of underlying quantum states in parameter space. The first method is to probe the transition probability after a sudden quench, and the second one is to detect the excitation rate under weak periodic driving. Furthermore, based on quantum metric and Berry-curvature measurements, we explore a topological phase transition in a simulated time-reversal-symmetric system. The work opens up a unique approach to explore the topology of quantum states with the QGT.

Valley Vortex States and Degeneracy Lifting via Photonic Higher-Band Excitation

We demonstrate valley-dependent vortex generation in photonic graphene. Without breaking inversion symmetry, the excitation of two valleys leads to the formation of an optical vortex upon Bragg reflection to the third equivalent valley, with its chirality determined by the valley degree of freedom. Vortex-antivortex pairs with valley-dependent topological charge flipping are also observed and corroborated by numerical simulations. Furthermore, we develop a three-band effective Hamiltonian model to describe the dynamics of the coupled valleys and find that the commonly used two-band model is not sufficient to explain the observed vortex degeneracy lifting. Such valley-polarized vortex states arise from high-band excitation without a synthetic-field-induced gap opening. Our results from a photonic setting may provide insight for the study of valley contrasting and Berry-phase-mediated topological phenomena in other systems.

Topological bands for ultracold atoms

There have been significant recent advances in realizing band structures with geometrical and topological features in experiments on cold atomic gases. This review summarizes these developments, beginning with a summary of the key concepts of geometry and topology for Bloch bands. Descriptions are given of the different methods that have been used to generate these novel band structures for cold atoms and of the physical observables that have allowed their characterization. The focus is on the physical principles that underlie the different experimental approaches, providing a conceptual framework within which to view these developments. Also described is how specific experimental implementations can influence physical properties. Moving beyond single-particle effects, descriptions are given of the forms of interparticle interactions that emerge when atoms are subjected to these energy bands and of some of the many-body phases that may be sought in future experiments.

Extended SSH Model: Non-Local Couplings and Non-Monotonous Edge States

We construct a generalized system by introducing an additional long-range hopping to the well-known Su-Schrieffer-Heeger (SSH) model. This system exhibits richer topological properties including non-trivial topological phases and associated localized edge states. We study the zero-energy edge states in detail and derive the edge-state wave functions using two different methods. Furthermore, we propose a possible setup using octupole moments optically excited on an array of dielectric particles for the realization of the system, and by adjusting the coupling strengths between neighboring particles we can control the hotspots (near-field enhancement) in such structures.

Photonics meets topology

The topological phases in materials have been studied in recent decades for their unique boundary states and transport properties. Photonic systems with band structures embrace the topological phases closely, where they not only provide platforms to testify the topological band theory, but also shed light on designing novel optical devices. In this review, we present exciting developments, supported by brief descriptions of prominent milestones of topological phases in photonic systems in recent years. These studies may sustain further developments of optical devices and offer novel methods for light manipulations.

Experimental Observation of Aharonov-Bohm Cages in Photonic Lattices

We report on the experimental realization of a uniform synthetic magnetic flux and the observation of Aharonov-Bohm cages in photonic lattices. Considering a rhombic array of optical waveguides, we engineer modulation-assisted tunneling processes that effectively produce nonzero magnetic flux per plaquette. This synthetic magnetic field for light can be tuned at will by varying the phase of the modulation. In the regime where half a flux quantum is realized in each plaquette, all the energy bands dramatically collapse into nondispersive (flat) bands and all eigenstates are completely localized. We demonstrate this Aharonov-Bohm caging by studying the propagation of light in the bulk of the photonic lattice. Besides, we explore the dynamics on the edge of the lattice and discuss how the corresponding edge states can be continuously connected to the topological edge states of the Creutz ladder. Our photonic lattice constitutes an appealing platform where the interplay between engineered gauge fields, frustration, localization, and topological properties can be finely studied.

Observation of Valley Landau-Zener-Bloch Oscillations and Pseudospin Imbalance in Photonic Graphene

We demonstrate intervalley Bloch oscillation (BO) and Landau-Zener tunneling (LZT) in an optically induced honeycomb lattice with a refractive-index gradient. Unlike previously observed BO in a gapped square lattice, we show nonadiabatic beam dynamics that are highly sensitive to the direction of the index gradient and the choice of the Dirac cones. In particular, a symmetry-preserving potential leads to nearly perfect LZT and coherent BO between the inequivalent valleys, whereas a symmetry-breaking potential generates asymmetric scattering, imperfect LZT, and valley-sensitive generation of vortices mediated by a pseudospin imbalance. This clearly indicates that, near the Dirac points, the transverse gradient does not always act as a simple scalar force, as commonly assumed, and the LZT probability is strongly affected by the sublattice symmetry as analyzed from an effective Landau-Zener Hamiltonian. Our results illustrate the anisotropic response of an otherwise isotropic Dirac platform to real-space potentials acting as strong driving fields, which may be useful for manipulation of pseudospin and valley degrees of freedom in graphenelike systems.

Chiral Orbital Magnetism of p-Orbital Bosons in Optical Lattices

Chiral magnetism is a fascinating quantum phenomena that has been found in low-dimensional magnetic materials. It is not only interesting for understanding the concept of chirality, but also important for potential applications in spintronics. Past studies show that chiral magnets require both a lack of inversion symmetry and spin-orbit coupling to induce the Dzyaloshinskii-Moriya interaction. Here we report that the combination of inversion symmetry breaking and quantum degeneracy of orbital degrees of freedom will provide a new paradigm to achieve chiral orbital magnetism. By means of density matrix renormalization group calculation, we demonstrate that chiral orbital magnetism can be found when considering bosonic atoms loaded in the p band of an optical lattice in the Mott regime. The high tunability of our scheme is also illustrated through simply manipulating the inversion symmetry of the system for cold atom experimental conditions.

Yang Monopoles and Emergent Three-Dimensional Topological Defects in Interacting Bosons

The Yang monopole as a zero-dimensional topological defect has been well established in multiple fields in physics. However, it remains an intriguing question to understand the interaction effects on Yang monopoles. Here, we show that the collective motion of many interacting bosons gives rise to exotic topological defects that are distinct from Yang monopoles seen by a single particle. Whereas interactions may distribute Yang monopoles in the parameter space or glue them to a single giant one of multiple charges, three-dimensional topological defects also arise from continuous manifolds of degenerate many-body eigenstates. Their projections in lower dimensions lead to knotted nodal lines and nodal rings. Our results suggest that ultracold bosonic atoms can be used to create emergent topological defects and directly measure topological invariants that are not easy to access in solids.

Observation and Uses of Position-Space Bloch Oscillations in an Ultracold Gas

We report the observation and characterization of position-space Bloch oscillations using cold atoms in a tilted optical lattice. While momentum-space Bloch oscillations are a common feature of optical lattice experiments, the real-space center-of-mass dynamics are typically unresolvable. In a regime of rapid tunneling and low force, we observe real-space Bloch oscillation amplitudes of hundreds of lattice sites, in both ground and excited bands. We demonstrate two unique capabilities enabled by tracking of Bloch dynamics in position space: measurement of the full position-momentum phase-space evolution during a Bloch cycle, and direct imaging of the lattice band structure. These techniques, along with the ability to exert long-distance coherent control of quantum gases without modulation, may open up new possibilities for quantum control and metrology.

Topological Maxwell Metal Bands in a Superconducting Qutrit

We experimentally explore the topological Maxwell metal bands by mapping the momentum space of condensed-matter models to the tunable parameter space of superconducting quantum circuits. An exotic band structure that is effectively described by the spin-1 Maxwell equations is imaged. Threefold degenerate points dubbed Maxwell points are observed in the Maxwell metal bands. Moreover, we engineer and observe the topological phase transition from the topological Maxwell metal to a trivial insulator, and report the first experiment to measure the Chern numbers that are higher than one.

Quantum anomalous Hall phase in a one-dimensional optical lattice

We propose to simulate and detect quantum anomalous Hall phase with ultracold atoms in a one-dimensional optical lattice, with the other synthetic dimension being realized by modulating spin-orbit coupling. We show that the system manifests a topologically nontrivial phase with two chiral edge states which can be readily detected in this synthetic two-dimensional system. Moreover, it is interesting that at the phase transition point there is a flat energy band and this system can also be in a topologically nontrivial phase with two Fermi zero modes existing at the boundaries by considering the synthetic dimension as a modulated parameter. We also show how to measure these topological phases experimentally in ultracold atoms. Another model with a random Rashba and Dresselhaus spin-orbit coupling strength is also found to exhibit topological nontrivial phase, and the impact of the disorder to the system is revealed.

Quantum simulations with ultracold atoms in optical lattices

Quantum simulation, a subdiscipline of quantum computation, can provide valuable insight into difficult quantum problems in physics or chemistry. Ultracold atoms in optical lattices represent an ideal platform for simulations of quantum many-body problems. Within this setting, quantum gas microscopes enable single atom observation and manipulation in large samples. Ultracold atom-based quantum simulators have already been used to probe quantum magnetism, to realize and detect topological quantum matter, and to study quantum systems with controlled long-range interactions. Experiments on many-body systems out of equilibrium have also provided results in regimes unavailable to the most advanced supercomputers. We review recent experimental progress in this field and comment on future directions.

Measurement of the Berry curvature of solids using high-harmonic spectroscopy

Berry phase and Berry curvature have become ubiquitous concepts in physics, relevant to a variety of phenomena, such as polarization, various Hall effects, etc. Studies of these phenomena call for characterization of Berry phase or curvature which is largely limited to theory, and a few measurements in optical lattices. In this work, we report polarimetry of high-harmonic emission from solids and exploit this novel capability to directly retrieve the Berry curvature of α-quartz. We show that the two manifestations of broken inversion symmetry in solids lead to perpendicular or parallel polarization of even harmonics with respect to the driving field. Using semiclassical transport theory, we retrieve the Berry curvature from spectra measured in perpendicular polarization, the results being supported by ab initio calculation. Our work demonstrates an approach for the direct measurement of Berry curvature in solids, which could serve as a benchmark for theoretical studies.

Manipulating the quantum interference effect and magnetotransport of ZnO nanowires through interfacial doping

We carefully prepared interfacial Al-doped (IAD) and interfacial natively-doped (IND) ZnO nanowires (NWs) by introducing atomic-layer interfacial Δ-doping between the two steps of CVD growth. Variable-temperature electron transport as well as magnetotransport behaviours of these NWs were systematically investigated. By virtue of the unique architecture and the quality-guaranteed growth technique, a series of quantum interference effects were clearly observed in the IAD ZnO NWs, including weak localization, universal conductance fluctuation and Altshuler-Aronov-Spivak oscillations. The phase-coherence length (L_φ) of electrons exceeds 100 nm in the IAD ZnO NWs, much longer than those in the IND ones and most conventionally doped ZnO NWs. This ability to efficiently manipulate a variety of quantum interference effects in ZnO NWs is very desirable for applications in nano-optoelectronics, nano- & quantum-electronics and solid-state quantum computing.

Detecting Topological Transitions in Two Dimensions by Hamiltonian Evolution

We show that the evolution of two-component particles governed by a two-dimensional spin-orbit lattice Hamiltonian can reveal transitions between topological phases. A kink in the mean width of the particle distribution signals the closing of the band gap, a prerequisite for a quantum phase transition between topological phases. Furthermore, for realistic and experimentally motivated Hamiltonians, the density profile in topologically nontrivial phases displays characteristic rings in the vicinity of the origin that are absent in trivial phases. The results are expected to have an immediate application to systems of ultracold atoms and photonic lattices.

Realizing and Detecting a Topological Insulator in the AIII Symmetry Class

Topological insulators in the AIII (chiral unitary) symmetry class lack experimental realization. Moreover, fractionalization in one-dimensional topological insulators has not been yet directly observed. Our work might open possibilities for both challenges. We propose a one-dimensional model realizing the AIII symmetry class which can be realized in current experiments with ultracold atomic gases. We further report on a distinctive property of topological edge modes in the AIII class: in contrast to those in the well-studied BDI (chiral orthogonal) class, they have nonzero momentum. Exploiting this feature we propose a path for the detection of fractionalization. A fermion added to an AIII system splits into two halves localized at opposite momenta, which can be detected by imaging the momentum distribution.

Probing topology by "heating": Quantized circular dichroism in ultracold atoms

We reveal an intriguing manifestation of topology, which appears in the depletion rate of topological states of matter in response to an external drive. This phenomenon is presented by analyzing the response of a generic two-dimensional (2D) Chern insulator subjected to a circular time-periodic perturbation. Because of the system's chiral nature, the depletion rate is shown to depend on the orientation of the circular shake; taking the difference between the rates obtained from two opposite orientations of the drive, and integrating over a proper drive-frequency range, provides a direct measure of the topological Chern number (ν) of the populated band: This "differential integrated rate" is directly related to the strength of the driving field through the quantized coefficient η0 = ν/ℏ2, where h = 2π ℏ is Planck's constant. Contrary to the integer quantum Hall effect, this quantized response is found to be nonlinear with respect to the strength of the driving field, and it explicitly involves interband transitions. We investigate the possibility of probing this phenomenon in ultracold gases and highlight the crucial role played by edge states in this effect. We extend our results to 3D lattices, establishing a link between depletion rates and the nonlinear photogalvanic effect predicted for Weyl semimetals. The quantized circular dichroism revealed in this work designates depletion rate measurements as a universal probe for topological order in quantum matter.

Observation of Topological Bloch-State Defects and Their Merging Transition

Topological defects in Bloch bands, such as Dirac points in graphene, and their resulting Berry phases play an important role for the electronic dynamics in solid state crystals. Such defects can arise in systems with a two-atomic basis due to the momentum-dependent coupling of the two sublattice states, which gives rise to a pseudospin texture. The topological defects appear as vortices in the azimuthal phase of this pseudospin texture. Here, we demonstrate a complete measurement of the azimuthal phase in a hexagonal optical lattice employing a versatile method based on time-of-flight imaging after off-resonant lattice modulation. Furthermore, we map out the merging transition of the two Dirac points induced by beam imbalance. Our work paves the way to accessing geometric properties in optical lattices also with spin-orbit coupling and interactions.

Vortex degeneracy lifting and Aharonov-Bohm-like interference in deformed photonic graphene

Photonic graphene, a honeycomb lattice of evanescently coupled waveguides, has provided a superior platform for investigating a host of fundamental phenomena such as unconventional edge states, synthetic magnetic fields, photonic Landau levels, Floquet topological insulators, and pseudospin effects. Here, we demonstrate both experimentally and numerically, topological vortex degeneracy lifting and Aharonov-Bohm-like interference from local deformation in a photonic honeycomb lattice. When a single valley is excited, lattice deformation leads to the generation of a vortex pair due to the lifting of degeneracy associated with pseudospin states. In the case of double-valley excitation, we observe the Aharonov-Bohm-like interference merely due to the deformation of the graphene lattice, which gives rise to an artificial gauge field. Our results may provide insight into the understanding of similar phenomena in other graphene-like materials and structures.

Synthetic Dimensions and Spin-Orbit Coupling with an Optical Clock Transition

We demonstrate a novel way of synthesizing spin-orbit interactions in ultracold quantum gases, based on a single-photon optical clock transition coupling two long-lived electronic states of two-electron ^{173}Yb atoms. By mapping the electronic states onto effective sites along a synthetic "electronic" dimension, we have engineered fermionic ladders with synthetic magnetic flux in an experimental configuration that has allowed us to achieve uniform fluxes on a lattice with minimal requirements and unprecedented tunability. We have detected the spin-orbit coupling with fiber-link-enhanced clock spectroscopy and directly measured the emergence of chiral edge currents, probing them as a function of the flux. These results open new directions for the investigation of topological states of matter with ultracold atomic gases.

Photo-electrons unveil topological transitions in graphene-like systems

The topological structure of the wavefunctions of particles in periodic potentials is characterized by the Berry curvature Ω_kn whose integral on the Brillouin zone is a topological invariant known as the Chern number. The bulk-boundary correspondence states that these numbers define the number of edge or surface topologically protected states. It is then of primary interest to find experimental techniques able to measure the Berry curvature. However, up to now, there are no spectroscopic experiments that proved to be capable to obtain information on Ω_kn to distinguish different topological structures of the bulk wavefunctions of semiconducting materials. Based on experimental results of the dipolar matrix elements for graphene, here we show that ARPES experiments with the appropriate x-ray energies and polarization can unambiguously detect changes of the Chern numbers in dynamically driven graphene and graphene-like materials opening new routes towards the experimental study of topological properties of condensed matter systems.

π-Flux Dirac Bosons and Topological Edge Excitations in a Bosonic Chiral p-Wave Superfluid

We study the topological properties of elementary excitations in a staggered p_{x}±ip_{y} Bose-Einstein condensate realized in recent orbital optical lattice experiments. The condensate wave function may be viewed as a configuration space variant of the famous p_{x}+ip_{y} momentum space order parameter of strontium ruthenate superconductors. We show that its elementary excitation spectrum possesses Dirac bosons with π Berry flux. Remarkably, if we induce a population imbalance between the p_{x}+ip_{y} and p_{x}-ip_{y} condensate components, a gap opens up in the excitation spectrum resulting in a nonzero Chern invariant and topologically protected edge excitation modes. We give a detailed description of how our proposal can be implemented with standard experimental technology.

Quantum spin dynamics with pairwise-tunable, long-range interactions

We present a platform for the simulation of quantum magnetism with full control of interactions between pairs of spins at arbitrary distances in 1D and 2D lattices. In our scheme, two internal atomic states represent a pseudospin for atoms trapped within a photonic crystal waveguide (PCW). With the atomic transition frequency aligned inside a band gap of the PCW, virtual photons mediate coherent spin-spin interactions between lattice sites. To obtain full control of interaction coefficients at arbitrary atom-atom separations, ground-state energy shifts are introduced as a function of distance across the PCW. In conjunction with auxiliary pump fields, spin-exchange versus atom-atom separation can be engineered with arbitrary magnitude and phase, and arranged to introduce nontrivial Berry phases in the spin lattice, thus opening new avenues for realizing topological spin models. We illustrate the broad applicability of our scheme by explicit construction for several well-known spin models.

Interferometric measurements of many-body topological invariants using mobile impurities

Topological quantum phases cannot be characterized by Ginzburg-Landau type order parameters, and are instead described by non-local topological invariants. Experimental platforms capable of realizing such exotic states now include synthetic many-body systems such as ultracold atoms or photons. Unique tools available in these systems enable a new characterization of strongly correlated many-body states. Here we propose a general scheme for detecting topological order using interferometric measurements of elementary excitations. The key ingredient is the use of mobile impurities that bind to quasiparticles of a host many-body system. Specifically, we show how fractional charges can be probed in the bulk of fractional quantum Hall systems. We demonstrate that combining Ramsey interference with Bloch oscillations can be used to measure Chern numbers characterizing the dispersion of individual quasiparticles, which gives a direct probe of their fractional charges. Possible extensions of our method to other many-body systems, such as spin liquids, are conceivable.

Geometrical Pumping with a Bose-Einstein Condensate

We realized a quantum geometric "charge" pump for a Bose-Einstein condensate (BEC) in the lowest Bloch band of a novel bipartite magnetic lattice. Topological charge pumps in filled bands yield quantized pumping set by the global-topological-properties of the bands. In contrast, our geometric charge pump for a BEC occupying just a single crystal momentum state exhibits nonquantized charge pumping set by local-geometrical-properties of the band structure. Like topological charge pumps, for each pump cycle we observed an overall displacement (here, not quantized) and a temporal modulation of the atomic wave packet's position in each unit cell, i.e., the polarization.

Scaling theory of topological phase transitions

Topologically ordered systems are characterized by topological invariants that are often calculated from the momentum space integration of a certain function that represents the curvature of the many-body state. The curvature function may be Berry curvature, Berry connection, or other quantities depending on the system. Akin to stretching a messy string to reveal the number of knots it contains, a scaling procedure is proposed for the curvature function in inversion symmetric systems, from which the topological phase transition can be identified from the flow of the driving energy parameters that control the topology (hopping, chemical potential, etc) under scaling. At an infinitesimal operation, one obtains the renormalization group (RG) equations for the driving energy parameters. A length scale defined from the curvature function near the gap-closing momentum is suggested to characterize the scale invariance at critical points and fixed points, and displays a universal critical behavior in a variety of systems examined.