Ultracold atoms in optical lattices form a clean quantum simulator platform which can be utilized to examine topological phenomena and test exotic topological materials. Here we propose an experimental scheme to measure the Chern numbers of two-dimensional multiband topological insulators with bosonic atoms. We show how to extract the topological invariants out of a sequence of time-of-flight images by applying a phase-retrieval algorithm to matter waves. We illustrate advantages of using bosonic atoms as well as efficiency and robustness of the method with two prominent examples: the Harper-Hofstadter model with an arbitrary commensurate magnetic flux and the Haldane model on a brick-wall lattice.
Ultracold atoms in optical lattices form a clean quantum simulator platform which can be utilized to examine topological phenomena and test exotic topological materials. Here we propose an experimental scheme to measure the Chern numbers of two-dimensional multiband topological insulators with bosonic atoms. We show how to extract the topological invariants out of a sequence of time-of-flight images by applying a phase-retrieval algorithm to matter waves. We illustrate advantages of using bosonic atoms as well as efficiency and robustness of the method with two prominent examples: the Harper-Hofstadter model with an arbitrary commensurate magnetic flux and the Haldane model on a brick-wall lattice.
We propose an experimental scheme to directly detect the Chern number of the topological Haldane model realized with optical lattices based on the time-of-flight imaging. The Haldane model can be realized with two-component fermionic atoms loaded in the optical lattices consisting of two sublattices and each of the sublattices hosts atoms with a specific component of the atoms. We find that the Bloch wave function of the lower band can be measured with the time-of-flight imaging and thus the Chern number can be directly obtained from the Bloch function. We numerically simulate the Chern number of the Haldane model and its modified version in both honeycomb lattice and brick-wall lattice, and show that the proposed approach is general and feasible in practical experiments.
Cold-atom experiments in optical lattices offer a versatile platform to realize various topological quantum phases. A key challenge in those experiments is to unambiguously probe the topological order. We propose a method to directly measure the characteristic topological invariants (order) based on the time-of-flight imaging of cold atoms. The method is generally applicable to detection of topological band insulators in one, two, or three dimensions characterized by integer topological invariants. Using detection of the Chern number for the two-dimensional anomalous quantum Hall states and the Chern-Simons term for the three-dimensional chiral topological insulators as examples, we show that the proposed detection method is practical, and robust to typical experimental imperfections such as limited imaging resolution, inhomogeneous trapping potential, and disorder in the system.
We present a novel technique for detecting topological invariants ‐ Chern numbers ‐ from time-of-flight images of ultra-cold atoms. We show that the Chern numbers of integer quantum Hall states of lattice fermions leave their fingerprints as oscillations in the atoms’ momentum distribution. The number of oscillations within the lowest Brillouin zone is simply equal to the Chern number. The ubiquitous signature is strikingly enhanced for anisotropic hopping, where the momentum distribution assumes a sinusoidal form with period determined by the Chern number. Thus, an everyday observable in cold atom experiments can serve as a useful tool to characterize and visualize quantum states with non-trivial topology. Topological insulators (TI) and superfluids are many-body quantum systems with energy gaps in the bulk but topologically protected, gapless excitations on the boundary [1, 2]. Each class of topological phases is characterized by a topological invariant which can be either an integer or a binary quantity. For example, the integer quantum Hall insulators [3] are distinguished from other two-dimensional insulators with the same symmetry by the Chern number, an integer that capture the global properties of all occupied bands [4]. The Chern number coincides with the number of chiral edge modes, and the Hall conductance in units of the conductance quantum. A central problem in exploring new forms of topological matter is how to extract the topological invariants from experiments, and unambiguously determine the topological phase. Experiments with ultracold atoms has brought us opportunities to creating topological phases of matter in parameter regimes unreachable in solids. In addition to band-structure engineering using optical lattices and tuning the atom-atom interaction by Feshbach resonance, artificial magnetic fields for neutral atoms have been produced by either rotation [5] or atom-laser coupling [6, 7], making way to realizing quantum Hall states of cold atoms [8]. Moreover, non-Abelian synthetic gauge fields [7, 9] and spin-orbit coupling [10] have been demonstrated. These developments motivated theories on properties of ultracold topological matter, e.g., the fractional quantum Hall effect of interacting bosons [5, 11, 12], the quantum spin Hall effect of fermions [9], and the quantum anomalous Hall states ofp-orbital fermions [13]. While cold atom systems offer new detection techniques and great tunability, it is challenging to measure the topological invariants directly. Measurements of local density of states (of the edge states) or mass transport are marred with complexity in contrast to solid state systems. Thus, creative methods have to be invented. For example, Ref. [8] proposed to extract the quantized Hall conductance from the derivatives
Topological states of matter emerge as a new type of quantum phases, which can be distinguished by their associated topological invariants, e.g. , Chern numbers. Currently, there has been increasing interest toward the physical detection of the new predicted topological phases. Here, we propose an interferometric approach to directly measure the Chern number of the topological Haldane model in bosonic optical lattices via detecting the associated Zak phases. We show that this interferometric approach can distinguish Zak phases of from 0 in the first Brillouin zone, and thus it provides a new tool to directly detect the Chern number of topological systems. In addition, we demonstrate that this method is feasible under realistic experimental conditions and may generalize to detect topological systems with higher Chern numbers.
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.
We propose an experimental technique for classifying the topology of band structures realized in optical lattices, based on a generalization of topological charge pumping in quantum Hall systems to cold atoms in optical lattices. Time-of-flight measurement along one spatial direction combined with in situ detection along the transverse direction provides a direct measure of the system's Chern number, as we illustrate by calculations for the Hofstadter lattice. Based on an analogy with Wannier function techniques of topological band theory, the method is very general and also allows the measurement of other topological invariants, such as the Z(2) topological invariant of time-reversal symmetric insulators.