Quantum Simulation
Topological properties of physical systems are reflected in so-called Chern numbers: A nonzero Chern number typically means that a system is topologically nontrivial. Sugawa et al. engineered a cold atom system with a nonzero second Chern number, in contrast to condensed matter physics, where only the first Chern number is usually invoked. The exotic topology relates to the emergence of a type of magnetic monopole called the Yang monopole (known from theoretical high-energy physics) in a five-dimensional space of internal degrees of freedom in a rubidium Bose-Einstein condensate. The results illustrate the potential of cold atoms physics to simulate high-energy phenomena.
Science , this issue p. [1429][1]
[1]: /lookup/doi/10.1126/science.aam9031
Quantum Simulation
Topological properties of physical systems are reflected in so-called Chern numbers: A nonzero Chern number typically means that a system is topologically nontrivial. Sugawa et al. engineered a cold atom system with a nonzero second Chern number, in contrast to condensed matter physics, where only the first Chern number is usually invoked. The exotic topology relates to the emergence of a type of magnetic monopole called the Yang monopole (known from theoretical high-energy physics) in a five-dimensional space of internal degrees of freedom in a rubidium Bose-Einstein condensate. The results illustrate the potential of cold atoms physics to simulate high-energy phenomena.
Topological order is often quantified in terms of Chern numbers, each of which classifies a topological singularity. Here, inspired by concepts from high-energy physics, we use quantum simulation based on the spin degrees of freedom of atomic Bose-Einstein condensates to characterize a singularity present in five-dimensional non-Abelian gauge theories-a Yang monopole. We quantify the monopole in terms of Chern numbers measured on enclosing manifolds: Whereas the well-known first Chern number vanishes, the second Chern number does not. By displacing the manifold, we induce and observe a topological transition, where the topology of the manifold changes to a trivial state.
Bose–Einstein condensation is a quantum statistical phase transition that occurs in a system consisting of bosons when a single-particle quantum state becomes macroscopically occupied. This peculiar state of matter was first predicted in 1925 and finally realized seventy years later in vapours of weakly-interacting alkali-metal atoms. Since then, Bose–Einstein condensates have been one of the most fascinating research fields in modern physics. \n \nThe gaseous condensates offer a robust platform to accurately study interacting many-particle systems from the first principles. Experimentally, the possibility to precisely control a condensate with external fields and directly image its order parameter provides unforeseen opportunities to obtain deep insight into phenomena across different subfields of physics. In particular, gaseous condensates can emulate complicated models that arise in atomic, condensed-matter, and even particle physics, allowing realizations of exotic phenomena that are elusive in their original contexts. An outstanding example is the existence of various topological defects in ultacold quantum gases with internal degrees of freedom. \n \nIn this thesis, we investigate the creation, stability, and dynamical properties of various topological defects in spinor Bose–Einstein condensates. The majority of the theoretical results is obtained by numerically solving the dynamics using Gross–Pitaevskii equations for spin-1 condensates. Many theoretical predictions are confirmed by the very good agreement with experiments. \n \nThe first experimental observations of a topological point defect in an order parameter describing the quantum gas are presented. Such a point defect is reminiscent to the magnetic monopole particle appearing in grand unified theories. Therefore, the discovery of monopoles in quantum gases further encourages the quest for finding magnetic monopoles in natural electromagnetic fields, a search largely initiated by Paul Dirac almost a century ago. \n \nSubsequently, the fine structure and decay dynamics of the point defect are studied numerically and verified in experiments. The created monopole is gradually destroyed during the polar-to-ferromagnetic quantum phase transition, which results in the spontaneous emergence of a Dirac monopole in synthetic magnetic field. In addition to the singular point defect, the first experimental realization of a knot soliton in the context of quantum field is reported. This thesis lays the foundation for studies of the dynamics and stability of three-dimensional topological structures in quantum systems.
Different topological phases of quantum systems have become areas of increased focus in recent decades. In particular, the question of how to realize and manipulate systems with a nontrivial first Chern number is pursued both experimentally and theoretically. Here we go beyond typical spin-1/2 systems and consider both single and coupled spin-1 systems as a means of realizing higher first Chern numbers and studying the emergence of different topological phases. We show that rich topological phase diagrams can be realized by coupling two spin-1 systems using both numerical and analytical methods. Furthermore, we consider a concrete realization of spin-1 systems using a superconducting circuit. This realization includes nonstandard spin-spin interaction terms that may endanger the topological properties. We argue, however, that the realistic circuit Hamiltonian including all terms should be expected to show the rich phase structure as well. This puts our theoretical predictions within reach of state-of-the-art experimental setups.
The present chapter discusses methods by which topological Bloch bands can be prepared in cold-atom setups. Focusing on the case of Chern bands for two-dimensional systems, we describe how topological properties can be triggered by driving atomic gases, either by dressing internal levels with light or through time-periodic modulations. We illustrate these methods with concrete examples, and we discuss recent experiments where geometrical and topological band properties have been identified. Introduction Ultracold atoms constitute a promising physical platform for the preparation and exploration of novel states of matter [1, 2, 3]. In particular, the engineering of topological band structures with cold-atom systems, together with the capability of tuning interactions between the particles, opens an interesting route toward the realization of intriguing strongly correlated states with topological features, such as fractional topological insulators and quantum Hall liquids [4]. This chapter is dedicated to the preparation and the detection of topological band structures characterized by nonzero Chern numbers [5]. Such Chern bands , which constitute the building blocks for realizing (fractional) Chern insulators [6, 7], arise in two-dimensional (2D) systems presenting time-reversal-symmetry (TRS) breaking effects. For instance, nontrivial Chern bands naturally appear in the Harper-Hofstadter model [8], a lattice penetrated by a uniform flux, where they generalize the (nondispersive) Landau levels to the lattice framework. Additionally, Chern bands also appear in staggered flux configurations, such as in Haldane's honeycomb-lattice model [9] or in lattice systems combining Rashba spin-orbit coupling and Zeeman (exchange) fields. The atoms being charge neutral, “magnetic” fluxes cannot be simply produced by subjecting optical lattices to “real” magnetic fields. It is the aim of this chapter to review several schemes that have been recently implemented in laboratories with the goal of realizing synthetic magnetic fields leading to Chern bands for cold atoms. Our presentation is complementary to that of Chapter 15, as we focus on synthetic magnetic fields in lattice-based systems for which the flux density can be made very large (with magnetic length comparable to interparticle spacing). The chapter is structured as follows: Section 14.2 describes how the Chern number is related to physical observables defined in a lattice framework. In particular, it clarifies the link between recent Chern-number measurements performed in cold bosonic gases and the more conventional (electronic) quantum Hall effect.
Two-dimensional Euler insulators are novel kinds of systems that host multigap topological phases, quantified by a quantized first Euler number in their bulk. Recently, these phases have been experimentally realized in suitable two-dimensional synthetic matter setups. Here, we introduce the second Euler invariant, a familiar invariant in both differential topology (Chern-Gauss-Bonnet theorem) and in four-dimensional Euclidean gravity, whose existence still needs to be explored in condensed matter systems. Specifically, we first define two specific models in four dimensions that support a nonzero second Euler number in the bulk together with peculiar gapless boundary states. Second, we discuss its robustness in general spacetime-inversion invariant phases and its role in the classification of topological degenerate real bands through real Grassmannians. Our results naturally generalize the second Chern and spin Chern numbers to the case of four-dimensional phases that are characterized by real Hamiltonians and open doors for implementing such higher-dimensional phases in artificial engineered systems, ranging from ultracold atoms to photonics and electric circuits.
Mathematical tools and concepts initially developed to study topological properties of geometrical objects have allowed for better understanding and interpretation of a number of phenomena in condensed matter physics, from the quantum Hall effect to topological insulators. It would be interesting to realize analogs of these phenomena with light scattered by cold atoms because atomic systems are simpler and allow for better control than complex solid-state compounds in which electrons interact with each other and with phonons, and where disorder is omnipresent. At the same time, mastering topological photonics of cold atoms may lead to interesting applications in quantum information science, for example. We study the opening of topological band gaps in two-dimensional honeycomb lattices of two-level atoms [1], the impact of disorder on their properties [2], as well as the constructive role that disorder can play by inducing a topologically nontrivial phase known as “topological Anderson insulator” [2,3]. Modifying the structure of the atomic lattice in an appropriate way allows for constructing a synthetic spin degree of freedom and thus for exploring the physics of spinful particles, such as the quantum spin Hall effect, for example. The analogy between topological phenomena in solid-state physics and in optics of cold atoms is not complete: atomic systems exhibit a number of new features due to the vector nature of light and to the differences in the impact of disorder on their properties.1. P. Wulles and S.E. Skipetrov, Topological photonic band gaps in honeycomb atomic arrays, arXiv:2308.134232. S.E. Skipetrov and P. Wulles, Topological transitions and Anderson localization of light in disordered atomic arrays, Phys. Rev. A 105, 043514 (2022)3. S.E. Skipetrov and P. Wulles, Photonic topological Anderson insulator in a two-dimensional atomic lattice, Comptes Rendus. Physique 24(S3), 39–54 (2023)