Higher-dimensional Hilbert spaces in quantum simulation, as in all quantum science, expand the range of accessible phenomena. In this work, we experimentally realize a synthetic monopole using an ultracold spin-1 ensemble, where the monopole charge is quantified by the topologically invariant first Chern number and sources a synthetic magnetic field quantified by the Berry curvature. By using a three-level system with tunable spin-tensor coupling, we introduce anisotropy to the field, directly measure the Chern number, and observe a topological phase transition. We verify the robustness of the monopole's topological charge under deformation, and observe signatures of the topological phases using spin-texture and Majorana-star measurements. This work demonstrates spin-tensor coupling as a tuning parameter for engineering both geometric anisotropy and a rich topological phase space.
A magnetic monopole is a hypothetical elementary particle with an isolated magnetic pole. Its existence would directly lead to the quantization of electric charges. In recent years, analogs of magnetic monopoles, represented by topological defects in parameter spaces, have been studied in a wide range of physical systems. These works mainly focused on Abelian Dirac monopoles in spin-1/2 or non-Abelian Yang monopoles in spin-3/2 systems. Here we propose to realize three types of spin-1 topological monopoles and study their geometric properties using the parameter space formed by three hyperfine states of ultracold atoms coupled by radio-frequency fields. These spin-1 monopoles, characterized by different monopole charges, possess distinct Berry curvature fields and spin textures, which are directly measurable in experiments. The topological phase transitions between different monopoles are accompanied by the emergence of a spin ``vortex'' and can be intuitively visualized using Majorana's stellar representation. We show how to determine the Berry curvature, hence the geometric phase and monopole charge from dynamical effects. Our scheme provides a simple and highly tunable platform for observing and manipulating spin-1 topological monopoles, paving the way for exploring new topological quantum matter.
Monopoles play a center role in gauge theories and topological matter. There are two fundamental types of monopoles in physics: vector monopoles and tensor monopoles. Examples of vector monopoles include the Dirac monopole in three dimensions and Yang monopole in five dimensions, which have been extensively studied and observed in condensed matter or artificial systems. However, tensor monopoles are less studied, and their observation has not been reported. Here we experimentally construct a tunable spin-1 Hamiltonian to generate a tensor monopole and then measure its unique features with superconducting quantum circuits. The energy structure of a 4D Weyl-like Hamiltonian with threefold degenerate points acting as tensor monopoles is imaged. Through quantum-metric measurements, we report the first experiment that measures the Dixmier-Douady invariant, the topological charge of the tensor monopole. Moreover, we observe topological phase transitions characterized by the topological Dixmier-Douady invariant, rather than the Chern numbers as used for conventional monopoles in odd-dimensional spaces.
Here, we elaborate on and develop the geometrical approach introduced in K. Le Hur, Physics Reports 1104 1-42 (2025) between the magnetic monopole created from a radial field, quantum physics and topological lattice models through quantum phase transitions. We introduce an effective magnetic moment for a monopole when applying an additional source field along z-direction which also mediates the quantum phase transition. We present its relation with the transverse pumped quantum Hall current. The magnetic susceptibility can be introduced as a measure of the topological invariant i.e. it remains quantized within the topological phase until the transition. We show the relation with two-dimensional topological lattice models such as a honeycomb Haldane model in real space. We develop the theory and present a numerical analysis between local invariants in momentum space introduced from Dirac points, correlation functions and the responses to circularly polarized light. We develop the formalism for coupled-planes materials including the possibility of quantum spin Hall effect and address a relation between the Ramanujan infinite alternating series and an interface in real space with a topological number one-half.
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.
We propose a theory to characterize equilibrium topological phase with nonequilibrium quantum dynamics by introducing the concept of high-order topological charges, with novel phenomena being predicted. Through a dimension reduction approach, we can characterize a
d-dimensional (
dD) integer-invariant topological phase with lower dimensional topological number quantified by high-order topological charges, of which the
sth-order topological charges denote the monopoles confined on the $(s\ensuremath{-}1)\mathrm{th}$-order band inversion surfaces (BISs) that are $(d\ensuremath{-}s+1)\mathrm{D}$ momentum subspaces. The bulk topology is determined by the
sth-order topological charges enclosed by the
sth-order BISs. By quenching the system from trivial phase to topological regime, we show that the bulk topology of postquench Hamiltonian can be detected through a high-order dynamical bulk-surface correspondence, in which both the high-order topological charges and high-order BISs are identified from quench dynamics. This characterization theory has essential advantages in two aspects. First, the highest (
dth) order topological charges are characterized by only discrete signs of spin polarization in zero dimension (i.e., the
0th Chern numbers), whose measurement is much easier than the first-order topological charges that are characterized by the continuous charge-related spin texture in higher dimensional space. Second, a more striking result is that a first-order high integer valued topological charge always reduces to multiple highest order topological charges with unit charge value, and the latter can be readily detected in experiments. The two fundamental features greatly simplify the characterization and detection of the topological charges and also topological phases, which shall advance the experimental studies in the near future.
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.