Topological phases associated with non-Abelian charges can exhibit a distinguished bulk-edge correspondence compared with Abelian phases, although elucidating this relationship remains challenging in traditional solid-state systems. In this paper, we propose a theoretical framework for synthesizing non-Abelian quaternion charges in ultracold atomic gases. By designing artificial spin-orbit coupling patterns, the topological edge modes demonstrate a clear correspondence with the band topology determined by various quaternion charges. This paves the way for observing the interface modes whose existence is attributed to the nonconservation multiplication relation, which is fundamental to non-Abelian charges. This scheme can be readily implemented using current ultracold atom techniques, offering a promising approach to explore the intriguing non-Abelian characteristics of the system.
Topological phases associated with non-Abelian charges can exhibit a distinguished bulk-edge correspondence compared with Abelian phases, although elucidating this relationship remains challenging in traditional solid-state systems. In this paper, we propose a theoretical framework for synthesizing non-Abelian quaternion charges in ultracold atomic gases. By designing artificial spin-orbit coupling patterns, the topological edge modes demonstrate a clear correspondence with the band topology determined by various quaternion charges. This paves the way for observing the interface modes whose existence is attributed to the nonconservation multiplication relation, which is fundamental to non-Abelian charges. This scheme can be readily implemented using current ultracold atom techniques, offering a promising approach to explore the intriguing non-Abelian characteristics of the system.
Non-Abelian topological charges (NATCs), characterized by their noncommutative algebra, offer a framework for describing multigap topological phases beyond conventional Abelian invariants. While higher-order topological phases (HOTPs) host boundary states at corners or hinges, their characterization has largely relied on Abelian invariants such as winding and Chern numbers. Here, we propose a coupled-wire scheme of constructing non-Abelian HOTPs and analyze a non-Abelian second-order topological insulator as its minimal model. The resulting Hamiltonian supports hybridized corner modes, protected by parity-time-reversal plus sublattice symmetries and described by a topological vector that unites a non-Abelian quaternion charge with an Abelian winding number. Corner states emerge only when both invariants are nontrivial, whereas weak topological edge states of non-Abelian origins arise when the quaternion charge is nontrivial, enriching the bulk-edge-corner correspondence. The system further exhibits both non-Abelian and Abelian topological phase transitions, providing a unified platform that bridges these two distinct topological classes. Our work extends the understanding of HOTPs into non-Abelian regimes and suggests feasible experimental realizations in synthetic quantum systems, such as photonic or acoustic metamaterials.
Cold atoms with laser-induced spin-orbit (SO) interactions provide promising platforms to explore novel quantum physics, in particular the exotic topological phases, beyond natural conditions of solids. The past several years have witnessed important progresses in both theory and experiment in the study of SO coupling and novel quantum states for ultracold atoms. Here we review the physics of the SO coupled quantum gases, focusing on the latest theoretical and experimental progresses of realizing SO couplings beyond one-dimension (1D), and the further investigation of novel topological quantum phases in such systems, including the topological insulating phases and topological superfluids. A pedagogical introduction to the SO coupling for ultracold atoms and topological quantum phases is presented. We show that the so-called optical Raman lattice schemes, which combine the creation of the conventional optical lattice and Raman lattice with topological stability, can provide minimal methods with high experimental feasibility to realize 1D to 3D SO couplings. The optical Raman lattices exhibit novel intrinsic symmetries, which enable the natural realization of topological phases belonging to different symmetry classes, with the topology being detectable through minimal measurement strategies. We introduce how the non-Abelian Majorana modes emerge in the SO coupled superfluid phases which can be topologically nontrivial or trivial, for which a few fundamental theorems are presented and discussed. The experimental schemes for achieving non-Abelian superfluid phases are given. Finally, we point out the future important issues in this rapidly growing research field.
The realization and detection of topological phases with ultracold atomic gases is at the frontier of current theoretical and experimental research. Here, we identify cold atoms in optical ladders subjected to synthetic magnetic fields as readily realizable bridges between one-dimensional spin-orbit (time-reversal) topological insulators and two-dimensional quantum Hall insulators. We reveal three instances of their promising potential: (i) they realize spin-orbit coupling, with the left-right leg degree of freedom playing the role of an effective spin, (ii) their energy bands and eigenstates exactly reproduce the topological chiral edge modes of two-dimensional Chern insulators, and (iii) they can be tailored to realize a topological phase transition from a trivial to a topological insulating phase. We propose realistic schemes to observe the chiral and topological properties of ladder systems with current optical lattice-based experiments. Our findings open a door to the exploration of the physics of the edges of quantum Hall insulators and to the realization of spin-orbit coupling and topological superfluid phases with ultracold atomic gases.
Realizing the topological bands of helical states poses a challenge in studying ultracold atomic gases. Motivated by the recent experimental success in realizing chiral optical ladders, here we present a scheme for synthesizing topological quantum matter, especially the quantum spin Hall phase, in the chiral optical ladders. More precisely, we first establish the synthetic pseudo-spin-orbit coupling and Zeeman splitting in the chiral ladders. After analyzing the band structure of the ladders exposed to the bichromatic optical potentials, we report the existence of quantum spin Hall phase. We further identify a rich phase diagram of the bichromatic chiral ladders, illustrating that our proposal features a large space of system parameters exhibiting quantum phase transitions. Our scheme is within reach of the existing ladder optical lattices and hence provides a new method to engineer the elaborate topological bands for cold atomic gases.
Abstract Non-Hermitian skin effect(NHSE) describes a unique non-Hermitian phenomenon that all eigen-modes are localized near the boundary, and has profound impact on a wide range of bulk properties. In particular, topological systems with NHSE have stimulated extensive research interests recently, given the fresh theoretical and experimental challenges therein. Here we propose a readily implementable scheme for achieving NHSE with band topology in ultracold gases. Specifically, the scheme realizes the one-dimensional optical Raman lattice with two types of spin-orbit coupling (SOC) and an additional laser-induced dissipation. By tuning the dissipation and the SOC strengths, NHSE and band topology can be individually controlled such that they can coexist in a considerable parameter regime. To identify the topological phase in the presence of NHSE, we have restored the bulk-boundary correspondence by invoking the non-Bloch band theory, and discussed the dynamic signals for detection. Our work serves as a guideline for engineering topological lattices with NHSE in the highly tunable environment of cold atoms, paving the way for future studies of exotic non-Hermitian physics in a genuine quantum many-body setting.
Topological phenomena arise in a wide range of systems, with fascinating physical consequences. There is great interest in finding new ways to measure such consequences in ultracold atomic gas experiments. These experiments have significant advantages over the solid-state as ultracold atoms are controllable, tuneable and clean. They can also be used to investigate properties which are inaccessible in other quantum systems. We explore some of the novel features of topological energy bands and topological solitons in ultracold gases. Topological energy bands have important geometrical properties described by the Berry curvature. Bands with nonzero Berry curvature arise in key areas of current research, such as optical lattices with more than one band; strong artificial magnetic fields and 2D spin-orbit coupling. Topological solitons are also relevant to cutting-edge experiments as they can be created and studied with high temporal and spatial resolution. In this thesis, we investigate the consequences of Berry curvature for the semiclassical dynamics of a wavepacket and the collective modes of an ultracold gas. We also study theoretically the dynamics of skyrmion-antiskyrmion pairs in a Bose Einstein condensate. Firstly, we propose a general method by which experiments can map the Berry curvature across the Brillouin zone, and thereby determine the topological properties of the energy bands of optical lattices. The Berry curvature modifies the semiclassical dynamics and hence the trajectory of a wavepacket undergoing Bloch oscillations. Our general protocol allows a clean measurement from the semiclassical dynamics of the Berry curvature over the Brillouin zone. We discuss how this protocol may be implemented and explore the semiclassical dynamics for three relevant systems. We discuss general experimental considerations for observing Berry curvature effects before reviewing some of the progress in the field since the publication of our work. Secondly, we show that the Berry curvature changes the hydrodynamic equations of motion for a trapped Bose-Einstein condensate, and causes significant modifications to the collective mode frequencies. We illustrate our results for the case of two-dimensional Rashba spin-orbit coupling in a Zeeman field, where we also apply both a sum rule and an operator approach to the dipole mode. Extending the operator method, we derive the effects of Berry curvature on the dipole mode in very general settings. We show that the sizes of these effects can be large and readily detected in experiment. Collective modes therefore provide a sensitive way to measure geometrical properties of topological energy bands. Lastly, we study theoretically the dynamics of two-component Bose-Einstein condensates in two dimensions, which admit topological excitations related to the skyrmions of nuclear physics. We explore a branch of uniformly propagating solitary waves, which, at high momentum, can be viewed as skyrmion-antiskyrmion pairs. We study these solitary waves for a range of interaction regimes and show that, for experimentally relevant cases, there is a transition to spatially extended spin-wave states at low momentum. We explain how this can be understood by analogy to the two-dimensional ferromagnet and discuss how such solitary waves can be generated and studied in experiment.