Synthetic Gauge Research Articles

Phase-dependent topological interface state and spatial adiabatic passage in a generalized Su-Schrieffer-Heeger model

We extend the Su-Schrieffer-Heeger model to include both an additional real intercell coupling and a complex intracell coupling whose phase can be interpreted as the Peierls phase associated with a synthetic gauge field. Using different Peierls phases, we can realize a heterostructure supporting a topologically protected interface state, depending on the existence of one trivial and two nontrivial phases of distinct winding numbers. The spatial adiabatic passage of this localized state from the inner interface to either open boundary can be attained simply by modulating the corresponding Peierls phase, while its decay or growth rate is controlled by the on-site parity-time symmetric loss and gain terms. Our results provide a scheme to achieve dynamical control of topologically protected states while being of potential interest to topological lasers with an adjustable spatial profile.

Symmetry breaking and lattice kirigami: Finite temperature effects

Recent work has analysed how deformations due to the insertion of a defect in a flat hexagonal lattice affect the ground state structure of an interacting fermion field theory. Such modifications result in an increase of the order parameter in the vicinity of the defect and can be explained by a kirigami effect, that is the combined effect of the curvature, locally introduced by the deformation in the lattice tessellation, and of a synthetic gauge field induced by the boundary conditions along the cut, performed to introduce the defect. In this work, we extend the formalism and previous results to include finite temperature effects.

Linked and knotted synthetic magnetic fields

We show that the realization of synthetic magnetic fields via light-matter coupling in the $\mathrm{\ensuremath{\Lambda}}$ scheme implements a natural geometrical construction of magnetic fields, namely, as the pullback of the area element of the sphere to Euclidean space via certain maps. For suitable maps, this construction generates linked and knotted magnetic fields, and the synthetic realization amounts to the identification of the map with the ratio of two Rabi frequencies which represent the coupling of the internal energy levels of an ultracold atom. We consider examples of maps which can be physically realized in terms of Rabi frequencies and which lead to linked and knotted synthetic magnetic fields acting on the neutral atomic gas. We also show that the ground state of the Bose-Einstein condensate may inherit the topological properties of the synthetic gauge field, with linked and knotted vortex lines appearing in some cases.

En-gauging the light

Synthetic gauge fields that enable controllable confinement and guiding via bound states in the continuum are demonstrated, offering new ways to confine and manipulate localization of radiation in space and opportunities to new applications of artificial gauge fields in photonics.

Light guiding by artificial gauge fields

Artificial gauge fields enable uncharged particles to behave as if affected by external fields. Generated by geometry or modulation, artificial gauge fields are instrumental in realizing topological physics in photonics, cold atoms and acoustics. Here, we experimentally demonstrate waveguiding by artificial gauge fields. We construct artificial gauge fields by using waveguide arrays with non-trivial trajectories. Tilting the arrays results in gauge fields that are different in the core and cladding, shifting their dispersion curves, thereby confining the light to the core. In a more advanced setting, we demonstrate waveguiding in a medium with the same gauge and dispersion everywhere, where the only difference between the core and the cladding is a phase shift in the dynamics of the gauge fields, which facilitates waveguiding via bound states in the continuum. Waveguiding and bound states in the continuum via artificial gauge fields relate to a plethora of systems, ranging from photonics and microwaves to cold atoms and acoustics. Optical guiding by a synthetic gauge field is experimentally demonstrated through an array of evanescently coupled identical waveguides, opening the door to applications of artificial gauge fields in optical, microwave and acoustic systems and in cold atoms.

Observation of localized modes at effective gauge field interface in synthetic mesh lattice

We predict a generic mechanism of wave localization at an interface between uniform artificial gauge fields, arising due to propagation-dependent phase accumulation similar to Aharonov-Bohm phenomenon. We realize experimentally a synthetic mesh lattice with real-time control over the vector gauge field, and observe robust localization under a broad variation of gauge strength and direction, as well as structural lattice parameters. This suggests new possibilities for confining and guiding waves in diverse physical systems through the synthetic gauge fields.

Acoustic Landau quantization and quantum-Hall-like edge states

Many intriguing phenomena occur for electrons under strong magnetic fields. Recently, it was proposed that an appropriate strain texture in graphene can induce a synthetic gauge field, in which the electrons behave like in a real magnetic field. This opened the door to control quantum transport by mechanical means and to explore unprecedented physics in high-field regime. Such studies have been achieved in molecular and photonic lattices. Here we report the first experimental realization of giant uniform pseudomagnetic field in acoustics by introducing a simple uniaxial deformation to acoustic graphene. Benefited from the controllability of our macroscopic platform, we observe the acoustic Landau levels in frequency-resolved spectroscopy and their spatial localization in pressure-field distributions. We further visualize the quantum-Hall-like edge states (connected to the zeroth Landau level), which have been elusive before owing to the challenge in creating large-area uniform pseudomagnetic fields. These results, highly consistent with our full-wave simulations, establish a complete framework for artificial structures under constant pseudomagnetic fields. Our findings, conceptually novel in acoustics, may offer new opportunities to manipulate sound.

Collective modes of vortex lattices in two-component Bose–Einstein condensates under synthetic gauge fields

We study collective modes of vortex lattices in two-component Bose–Einstein condensates subject to synthetic magnetic fields in mutually parallel or antiparallel directions. By means of the Bogoliubov theory with the lowest-Landau-level approximation, we numerically calculate the excitation spectra for a rich variety of vortex lattices that appear commonly for parallel and antiparallel synthetic fields. We find that in all of these cases there appear two distinct modes with linear and quadratic dispersion relations at low energies, which exhibit anisotropy reflecting the symmetry of each lattice structure. Remarkably, the low-energy spectra for the two types of fields are found to be related to each other by simple rescaling when vortices in different components overlap owing to an intercomponent attraction. These results are consistent with an effective field theory analysis. However, the rescaling relations break down for interlaced vortex lattices appearing with an intercomponent repulsion, indicating a nontrivial effect of an intercomponent vortex displacement beyond the effective field theory. We also find that high-energy parts of the excitation bands exhibit line or point nodes as a consequence of a fractional translation symmetry present in some of the lattice structures.

Rotating Atomic Quantum Gases with Light-Induced Azimuthal Gauge Potentials and the Observation of the Hess-Fairbank Effect.

We demonstrate synthetic azimuthal gauge potentials for Bose-Einstein condensates from engineering atom-light couplings. The gauge potential is created by adiabatically loading the condensate into the lowest energy Raman-dressed state, achieving a coreless vortex state. The azimuthal gauge potentials act as effective rotations and are tunable by the Raman coupling and detuning. We characterize the spin textures of the dressed states, in agreements with the theory. The lowest energy dressed state is stable with a 4.5-s half-atom-number-fraction lifetime. In addition, we exploit the azimuthal gauge potential to demonstrate the Hess-Fairbank effect, the analogue of Meissner effect in superconductors. The atoms in the absolute ground state has a zero quasiangular momentum and transits into a polar-core vortex when the synthetic magnetic flux is tuned to exceed a critical value. Our demonstration serves as a paradigm to create topological excitations by tailoring atom-light interactions where both types of SO(3) vortices in the |⟨F[over →]⟩|=1 manifold, coreless vortices and polar-core vortices, are created in our experiment. The gauge field in the stationary Hamiltonian opens a path to investigating rotation properties of atomic superfluids under thermal equilibrium.

Quantization of the Hall conductivity in the Harper-Hofstadter model

We study the robustness of the quantization of the Hall conductivity in the Harper-Hofstadter model towards the details of the protocol with which a longitudinal uniform driving force $F_x(t)$ is turned on. In the vector potential gauge, through Peierls substitution, this involves the switching-on of complex time-dependent hopping amplitudes $\mathrm{e}^{-\frac{i}{\hbar}\mathcal{A}_x(t)}$ in the $\hat{\mathbf{x}}$-direction such that $\partial_t \mathcal{A}_x(t)=F_x(t)$. The switching-on can be sudden, $F_x(t)=\theta(t) F$, where $F$ is the steady driving force, or more generally smooth $F_x(t)=f(t/t_{0}) F$, where $f(t/t_{0})$ is such that $f(0)=0$ and $f(1)=1$. We investigate how the time-averaged (steady-state) particle current density $j_y$ in the $\hat{\mathbf{y}}$-direction deviates from the quantized value $j_y \, h/F = n$ due to the finite value of $F$ and the details of the switching-on protocol. Exploiting the time-periodicity of the Hamiltonian $\hat{H}(t)$, we use Floquet techniques to study this problem. In this picture the (Kubo) linear response $F\to 0$ regime corresponds to the adiabatic limit for $\hat{H}(t)$. In the case of a sudden quench $j_y \, h/F$ shows $F^2$ corrections to the perfectly quantized limit. When the switching-on is smooth, the result depends on the switch-on time $t_{0}$: for a fixed $t_{0}$ we observe a crossover force $F^*$ between a quadratic regime for $F<F^*$ and a {\em non-analytic} exponential $\mathrm{e}^{-\gamma/|F|}$ for $F>F^*$. The crossover $F^*$ decreases as $t_{0}$ increases, eventually recovering the topological robustness. These effects are in principle amenable to experimental tests in optical lattice cold atomic systems with synthetic gauge fields.

Topological circuits of inductors and capacitors

Alternating current (ac) circuits can have electromagnetic edge modes protected by symmetries, analogous to topological band insulators or semimetals. How to make such a topological circuit? This paper illustrates a particular design idea by analyzing a series of topological circuits consisting purely of inductors (L) and capacitors (C) connected to each other by wires to form periodic lattices. All the examples are treated using a unifying approach based on Lagrangians and the dynamical H-matrix. First, the building blocks and permutation wiring are introduced using simple circuits in one dimension, the SSH transmission line and a braided ladder analogous to the ice-tray model also known as the π-flux ladder. Then, more general building blocks (loops and stars) and wiring schemes (m-shifts) are introduced. The key concepts of emergent pseudo-spin degrees of freedom and synthetic gauge fields are discussed, and the connection to quantum lattice Hamiltonians is clarified. A diagrammatic notation is introduced to simplify the design and presentation of more complicated circuits. These building blocks are then used to construct topological circuits in higher dimensions. The examples include the circuit analog of Haldane’s Chern insulator in two dimensions and quantum Hall insulator in four dimensions featuring finite second Chern numbers. The topological invariants and symmetry protection of the edge modes are discussed based on the H-matrix.

Synthetic gauge fields for lattices with multi-orbital unit cells: routes towards a π-flux dice lattice with flat bands

We propose a general strategy for generating synthetic magnetic fields in complex lattices with non-trivial connectivity based on light–matter coupling in cold atomic gases. Our approach starts from an underlying optical flux lattice in which a synthetic magnetic field is generated by coupling several internal states. Starting from a high-symmetry optical flux lattice, we superpose a scalar potential with a super- or sublattice period in order to eliminate links between the original lattice sites. As an alternative to changing connectivity, the approach can also be used to create or remove lattice sites from the underlying parent lattice. To demonstrate our concept, we consider the dice lattice geometry as an explicit example, and construct a dice lattice with a flux density of half a flux quantum per plaquette, providing a pathway to flat bands with a large band gap. While the intuition for our proposal stems from the analysis of deep optical lattices, we demonstrate that the approach is robust even for shallow optical flux lattices far from the tight-binding limit. We also provide an alternative experimental proposal to realise a synthetic gauge field in a fully frustrated dice lattice based on laser-induced hoppings along individual bonds of the lattice, again involving a superlattice potential. In this approach, atoms with a long-lived excited state are trapped using an ‘anti-magic’ wavelength of light, allowing the desired complex hopping elements to be induced in a specific laser coupling scheme for the dice lattice geometry. We conclude by comparing the complexity of these alternative approaches, and advocate that complex optical flux lattices provide the more elegant and easily generalisable strategy.

Topological phase transition in a two-species fermion system: Effects of a rotating trap potential or a synthetic gauge field

We numerically investigate the quantum phases and phase transition in a system made of two species of fermionic atoms that interact with each other via $s$-wave Feshbach resonance, and are subject to rotation or a synthetic gauge field that puts the fermions at Landau level filling factor $\nu_f = 2$. We show that the system undergoes a continuous quantum phase transition from a $\nu_f = 2$ fermionic integer quantum Hall state formed by atoms, to a $\nu_b = 1/2$ bosonic fractional quantum Hall state formed by bosonic diatomic molecules. In the disk geometry we use, these two different topological phases are distinguished by their different gapless edge excitation spectra, and the quantum phase transition between them is signaled by the closing of the energy gap in the bulk. Comparisons will be made with field theoretical predictions, and the case of $p$-wave pairing.

Condensates in double-well potential with synthetic gauge potentials and vortex seeding

We demonstrate an enhancement in the vortex generation when artificial gauge potential is introduced to condensates confined in a double well potential. This is due to the lower energy required to create a vortex in the low condensate density region within the barrier. Furthermore, we study the transport of vortices between the two wells, and show that the traverse time for vortices is longer for the lower height of the well. We also show that the critical value of synthetic magnetic field to inject vortices into the bulk of the condensate is lower in the double-well potential compared to the harmonic confining potential.

Non‐Hermitian Gauged Topological Laser Arrays

AbstractStable and phase‐locked emission in an extended topological supermode of coupled laser arrays, based on concepts of non‐Hermitian and topological photonics, is theoretically suggested. A non‐Hermitian Su–Schrieffer–Heeger chain of coupled microring resonators is considered, and it is shown that application of a synthetic imaginary gauge field via auxiliary passive microrings leads to all supermodes of the chain, except one, to become edge states. The only extended supermode, that retains some topological protection, can stably oscillate suppressing all other non‐topological edge supermodes. Numerical simulations based on a rate equation model of the semiconductor laser arrays confirm stable anti‐phase laser emission in the extended topological supermode and the role of the synthetic gauge field to enhance laser stability.

Measurement and significance of Wilson loops in synthetic gauge fields

We study Wilson loops as a necessary tool for unambiguous identification of non-Abelian synthetic gauge fields, with attention to certain crucial but often overlooked features, such as the requirement of at least three distinct loops. We devise a method to determine the complete Wilson loop matrix from the time evolved amplitudes of the internal atomic states of laser-coupled ultracold atoms that does not require lattice confinement. The analysis is done in the context of a new cyclic model that can realize both Abelian and non-Abelian structures within a single configuration with continuous variation possible between U(1) and U(2) gauge groups by varying the detuning of the laser fields.

Optical circulation in a multimode optomechanical resonator

Breaking the symmetry of electromagnetic wave propagation enables important technological functionality. In particular, circulators are nonreciprocal components that can route photons directionally in classical or quantum photonic circuits and offer prospects for fundamental research on electromagnetic transport. Developing highly efficient circulators thus presents an important challenge, especially to realise compact reconfigurable implementations that do not rely on magnetic fields to break reciprocity. We demonstrate optical circulation utilising radiation pressure interactions in an on-chip multimode optomechanical system. Mechanically mediated optical mode conversion in a silica microtoroid provides a synthetic gauge bias for light, enabling four-port circulation that exploits tailored interference between appropriate light paths. We identify two sideband conditions under which ideal circulation is approached. This allows to experimentally demonstrate ~10 dB isolation and <3 dB insertion loss in all relevant channels. We show the possibility of actively controlling the circulator properties, enabling ideal opportunities for reconfigurable integrated nanophotonic circuits.

Phase diagrams and Hofstadter butterflies in the strongly correlated bosonic systems on the lattices with Dirac points

Gauge potentials with different configurations have been recently realized in the optical lattice experiments. It is remarkable that one of the simplest gauge potential can generate particle energy spectrum with the self-similar structure known as a Hofstadter butterfly. We investigate theoretically the impact of strong on-site interaction on such a spectrum in the bosonic Mott insulator within Bose–Hubbard model. In particular, it is shown that the fractal structure is encoded in the quasi-particle and hole bosonic branches for different lattice backgrounds. For example a square lattice and other structures (brick-wall and staggered magnetic flux lattice) which contain Dirac points in energy dispersions are considered. This shows that single-particle physics is still present even in the strong interaction limit for whole Hofstadter spectrum. Additionally we observe, that although in brick-wall and staggered flux lattices the quasi-particle densities of states look qualitatively similar, the corresponding Hofstadter butterfly assumes different forms. In particular, we use a superposition of two different synthetic gauge fields which appears to be a generator of non-trivial phenomena in the optical lattice systems. We also discuss the consequences of these phenomena on the phase diagrams between bosonic Mott insulator and superfluid phase. The analysis is carried out within the strong coupling expansion method on the finite size lattices and also at finite temperatures which are relevant for the currently made experiments.

Advances in synthetic gauge fields for light through dynamic modulation.

Photons are weak particles that do not directly couple to magnetic fields. However, it is possible to generate a photonic gauge field by breaking reciprocity such that the phase of light depends on its direction of propagation. This non-reciprocal phase indicates the presence of an effective magnetic field for the light itself. By suitable tailoring of this phase, it is possible to demonstrate quantum effects typically associated with electrons, and, as has been recently shown, non-trivial topological properties of light. This paper reviews dynamic modulation as a process for breaking the time-reversal symmetry of light and generating a synthetic gauge field, and discusses its role in topological photonics, as well as recent developments in exploring topological photonics in higher dimensions.

Competing orders in the Hofstadter t–J model

The Hofstadter model describes non-interacting fermions on a lattice in the presence of an external magnetic field. Motivated by the plethora of solid-state phases emerging from electron interactions, we consider an interacting version of the Hofstadter model including a Hubbard repulsion U. We investigate this model in the large-U limit corresponding to a t-J Hamiltonian with an external (orbital) magnetic field. By using renormalized mean field theory supplemented by exact diagonalization calculations of small clusters, we find evidence for competing symmetry-breaking phases, exhibiting (possibly co-existing) charge, bond and superconducting orders. Topological properties of the states are also investigated and some of our results are compared to related experiments involving ultra-cold atoms loaded on optical lattices in the presence of a synthetic gauge field.