Session D18: Invited Session: Detecting Topological Order in Cold Atoms

Constructing and Deconstructing Non-Abelian Anyons

Non-Abelian anyons are profoundly unintuitive quasiparticles. When braiding them, the order of the braids matters, dramatically changing the properties of the underlying piece of quantum matter. They are predicted to occur as excitations of certain quantum Hall liquids and as Majorana fermions attached to vortices in special superconductors. But their experimental realization remains a major challenge, possibly because our theoretical understanding of non-Abelian matter is also far from complete. In this talk I will deconstruct non-Abelian anyons, revealing that they are made of clusters of simpler quasiparticles, Abelian anyons, which become indistinguishable. I will show that deconstruction into identical indistinguishable components is a useful framework for the theoretical understanding of non-Abelian anyons, providing an intuitive picture for the physical mechanism leading to their emergence. Moreover, deconstruction opens a route for the construction and characterization of non-Abelian physical models and for their experimental realization in nature. To illustrate the approach, I will construct and characterize a non-Abelian spin-1 lattice model, discussing directions to detect the emergent non-Abelian anyons in experiments with ultracold atoms.   

Novel ways of creating and detecting topological order with cold atoms and ions

In my talk I will focus on novel physics and novel quantum phases that are expected in lattice systems of ultra-cold atoms or ions in synthetic gauge fields, generated via lattice modulations and shaking. I will discuss fractal energy spectra and topological phases in long-range spin chains realized with trapped ions or atoms in nanofibers, and synthetic gauge fields in synthetic dimensions. I will spend large part of the talk discussing the ways to detect topological effects and order, via tomography of band insulators from quench dynamics, or via direct imaging of topological edge states.   

Effects of Berry Curvature in Ultracold Atomic Gases

Topological energy bands exhibit many fascinating physical phenomena. For instance, topological invariants underlie both the quantum Hall effect and more general topological insulators. There is currently great interest in exploring such physics in ultracold gases. Recent experiments have explored optical lattices with novel geometrical and topological features, and there is much ongoing activity to extend to other situations. Less widely appreciated is the fact that the energy bands of these new forms of optical lattice also have important geometrical properties. In particular, the Berry curvature is a geometrical property of the energy eigenstates, defined locally in the Brillouin zone. When integrated over the Brillouin zone of a two-dimensional band, it gives the Chern number, the topological invariant of the quantum Hall effect. The Berry curvature has many physical consequences in 2D and 3D systems, such as in the anomalous quantum Hall effect. I shall summarize how the Berry curvature can manifest itself in experimental measurements of transport and of collective modes in ultracold atomic gases.   

Realization of the topological Haldane model

A topologically non-trivial band structure appears in a hexagonal lattice if time-reversal symmetry is broken, as suggested by F. D. M. Haldane. He further pointed out that, in combination with broken inversion symmetry, this gives rise to a phase diagram containing topologically distinct phases, yet without the necessity of a magnetic field. Studying the band structure of a hexagonal lattice with broken time reversal symmetry induced by complex valued next-nearest neighbor couplings, he showed that the boundaries of the topologically different phases are gap opening-and-closing transitions at the Dirac points. Whilst a realization of this model in a material was hardly conceivable, it provided the conceptual basis for other topological insulators and the quantum spin Hall effect. Prospects to realize the model with cold atoms emerged by advances in generating effective magnetic fields for neutral atoms and the idea to employ time-dependent fields to break time-reversal symmetry in a hexagonal lattice. Here we report on the implementation of the Haldane model in a periodically driven honeycomb optical lattice and the characterization of the topological Bloch bands using non-interacting fermionic atoms. Modulating the position of the lattice sites along a circular trajectory generates complex next-nearest-neighbor tunneling and a gap opens at the Dirac points, which we measure using momentum-resolved inter-band transitions. In analogy to a Hall conductance we observe a characteristic displacements of the atomic cloud under a constant force. By additionally breaking the inversion-symmetry, we identify the closing of the gap at an individual Dirac point, associated with the transition between the topologically distinct phases, obtaining good agreement with the calculated phase diagram. Whilst the physics of the non-interacting system is determined by the single-particle band structure, as studied in this work, the cold atom systems is also suited to explore the interplay between topology and interactions.   

An Aharonov-Bohom interferometer for determining Bloch band topology

The geometric structure of an energy band in a solid is fundamental for a wide range of many-body phenomena in condensed matter and is uniquely characterized by the distribution of Berry curvature over the Brillouin zone. In analogy to an Aharonov-Bohm interferometer that measures the magnetic flux penetrating a given area in real space, we realize an atomic interferometer to measure Berry flux in momentum space. We demonstrate the interferometer for a graphene-type hexagonal lattice, where it has allowed us to directly detect the singular $\pi$-Berry flux localized at each Dirac point. This interferometer enabled us to determine the distribution of Berry curvature with high momentum resolution. In addition, I will present results on extending these ideas to two-band models, where Berry phases generalize to Wilson loops and give rise to even richer geometric structures. This work can form the basis for a general framework to fully characterize topological band structures and can also facilitate holonomic quantum computing through controlled exploitation of the geometry of Hilbert space.