This is an introductory review of the physics of topological quantum matter with cold atoms. Topological quantum phases, originally discovered and investigated in condensed matter physics, have recently been explored in a range of different systems, which produced both fascinating physics findings and exciting opportunities for applications. Among the physical systems that have been considered to realize and probe these intriguing phases, ultracold atoms become promising platforms due to their high flexibility and controllability. Quantum simulation of topological phases with cold atomic gases is a rapidly evolving field, and recent theoretical and experimental developments reveal that some toy models originally proposed in condensed matter physics have been realized with this artificial quantum system. The purpose of this article is to introduce these developments. The article begins with a tutorial review of topological invariants and the methods to control parameters in the Hamiltonians of neutral atoms. Next, topological quantum phases in optical lattices are introduced in some detail, especially several celebrated models, such as the Su–Schrieffer–Heeger model, the Hofstadter–Harper model, the Haldane model and the Kane–Mele model. The theoretical proposals and experimental implementations of these models are discussed. Notably, many of these models cannot be directly realized in conventional solid-state experiments. The newly developed methods for probing the intrinsic properties of the topological phases in cold-atom systems are also reviewed. Finally, some topological phases with cold atoms in the continuum and in the presence of interactions are discussed, and an outlook on future work is given.This is an introductory review of the physics of topological quantum matter with cold atoms. Topological quantum phases, originally discovered and investigated in condensed matter physics, have recently been explored in a range of different systems, which produced both fascinating physics findings and exciting opportunities for applications. Among the physical systems that have been considered to realize and probe these intriguing phases, ultracold atoms become promising platforms due to their high flexibility and controllability. Quantum simulation of topological phases with cold atomic gases is a rapidly evolving field, and recent theoretical and experimental developments reveal that some toy models originally proposed in condensed matter physics have been realized with this artificial quantum system. The purpose of this article is to introduce these developments. The article begins with a tutorial review of topological invariants and the methods to control parameters in the Hamiltonians of neutral atoms. Next, topological quantum phases in optical lattices are introduced in some detail, especially several celebrated models, such as the Su–Schrieffer–Heeger model, the Hofstadter–Harper model, the Haldane model and the Kane–Mele model. The theoretical proposals and experimental implementations of these models are discussed. Notably, many of these models cannot be directly realized in conventional solid-state experiments. The newly developed methods for probing the intrinsic properties of the topological phases in cold-atom systems are also reviewed. Finally, some topological phases with cold atoms in the continuum and in the presence of interactions are discussed, and an outlook on future work is given.

arXiv:1810.09228Advances In PhysicsOct 2, 2018317 citations

Detecting complex-energy braiding topology in a dissipative atomic simulator with transformer-based geometric tomography

Yang Yue, Nan Li, Xin Zhang +8 more

Machine learning (ML) is shaping our exploration of topological matter, whose existence is inherently tied to the geometry of quantum states or energy spectra. In non-Hermitian systems, distinctive spectral geometry can lead to topological braiding of complex-energy bands, yet directly probing this topology-geometry interplay remains challenging. Here, we introduce a Transformer-based ML framework to capture this interplay and experimentally demonstrate it in a dissipative cold-atom simulator. Using a Bose-Einstein condensate, we engineer tunable dissipative two-level systems whose complex eigenenergies form braids. Owing to the density-dependent dissipation, the instantaneous energy braids exhibit topologically distinct structures at short and long times. The Transformer not only accurately predicts topological invariants for diverse energy braids but also, through its self-attention mechanism, autonomously highlights band crossings as the governing underlying geometric feature. Our work paves the way for ML-guided exploration of non-Hermitian topological phases in cold atoms and beyond. Despite the increasing applications of machine learning in the quantum sciences, applications in the context of topological matter are still limited. Here, the authors demonstrate topological braiding of complex energy bands in a dissipative ultracold atom platform based on a transformer approach, advancing machine-learning-aided search of topological phases.Machine learning (ML) is shaping our exploration of topological matter, whose existence is inherently tied to the geometry of quantum states or energy spectra. In non-Hermitian systems, distinctive spectral geometry can lead to topological braiding of complex-energy bands, yet directly probing this topology-geometry interplay remains challenging. Here, we introduce a Transformer-based ML framework to capture this interplay and experimentally demonstrate it in a dissipative cold-atom simulator. Using a Bose-Einstein condensate, we engineer tunable dissipative two-level systems whose complex eigenenergies form braids. Owing to the density-dependent dissipation, the instantaneous energy braids exhibit topologically distinct structures at short and long times. The Transformer not only accurately predicts topological invariants for diverse energy braids but also, through its self-attention mechanism, autonomously highlights band crossings as the governing underlying geometric feature. Our work paves the way for ML-guided exploration of non-Hermitian topological phases in cold atoms and beyond. Despite the increasing applications of machine learning in the quantum sciences, applications in the context of topological matter are still limited. Here, the authors demonstrate topological braiding of complex energy bands in a dissipative ultracold atom platform based on a transformer approach, advancing machine-learning-aided search of topological phases.

Nature CommunicationsApr 20, 20260 citations

Some topological states in one-dimensional cold atomic systems

Feng Mei, Dan-Wei Zhang, Shi-Liang Zhu

Annals of PhysicsJan 9, 20155 citations

Pairing Symmetry, Phase diagram and Edge Modes in Topological Fulde-Ferrell-Larkin-Ovchinnikov Phase

Chan Chun Fai, Ming Gong

The realizations of spin-orbit coupling in cold atoms lead to a burst of research activities in the searching of topological matters in ultracold atom systems. The very recent theoretical predictions show that topological Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superfluids can be realized with proper spin-orbit coupling and Zeeman fields. In this work, a comprehensive understanding of the pairing symmetry, phase diagram and the edge modes in this new topological matter are presented. The momentum of the Cooper pairs plays the role of renormalizing the in-plane Zeeman field and chemical potential. The in-plane Zeeman field and finite momentum pairing induce asymmetry to the effective $p$ -wave pairing, apart from a small fraction of higher orbital components. The phase diagram is composed by different phases, which are determined by the topology and band gap nature of the superfluids. Especially, the gapped and gapless topological FFLO phase have totally different finite size effect. These novel features show that the spin-orbit coupled cold atoms provides an important platform in realizing topological matters which may not be materialized with solids.The realizations of spin-orbit coupling in cold atoms lead to a burst of research activities in the searching of topological matters in ultracold atom systems. The very recent theoretical predictions show that topological Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superfluids can be realized with proper spin-orbit coupling and Zeeman fields. In this work, a comprehensive understanding of the pairing symmetry, phase diagram and the edge modes in this new topological matter are presented. The momentum of the Cooper pairs plays the role of renormalizing the in-plane Zeeman field and chemical potential. The in-plane Zeeman field and finite momentum pairing induce asymmetry to the effective $p$-wave pairing, apart from a small fraction of higher orbital components. The phase diagram is composed by different phases, which are determined by the topology and band gap nature of the superfluids. Especially, the gapped and gapless topological FFLO phase have totally different finite size effect. These novel features show that the spin-orbit coupled cold atoms provides an important platform in realizing topological matters which may not be materialized with solids.

arXiv:1312.3000Dec 10, 20131 citation

p

-wave pairing, apart from a small fraction of higher orbital components. The phase diagram is composed by different phases, which are determined by the topology and band gap nature of the superfluids. Especially, the gapped and gapless topological FFLO phase have totally different finite size effect. These novel features show that the spin-orbit coupled cold atoms provides an important platform in realizing topological matters which may not be materialized with solids.