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arXiv:1910.06446 · Physical review. B./Physical review. B

Seeing topological edge and bulk currents in time-of-flight images

Álvaro Rubio-García, Chris N. Self, Juan José García‐Ripoll, Jiannis K. Pachos

Published Jul 23, 2020• 2 citations
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Abstract

Here we provide a general methodology to directly measure the topological currents emerging in the optical lattice implementation of the Haldane model. Alongside the edge currents supported by gapless edge states, transverse currents can emerge in the bulk of the system whenever the local potential is varied in space, even if it does not cause a phase transition. In optical lattice implementations the overall harmonic potential that traps the atoms provides the boundaries of the topological phase that supports the edge currents, as well as providing the potential gradient across the topological phase that gives rise to the bulk current. Both the edge and bulk currents are resilient to several experimental parameters such as trapping potential, temperature, and disorder. We propose to investigate the properties of these currents directly from time-of-flight images with both short-time and long-time expansions.

DOI
10.1103/physrevb.102.041123

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Here we provide a general methodology to directly measure the topological currents emerging in the optical lattice implementation of the Haldane model. Alongside the edge currents supported by gapless edge states, transverse currents can emerge in the bulk of the system whenever the local potential is varied in space, even if it does not cause a phase transition. In optical lattice implementations the overall harmonic potential that traps the atoms provides the boundaries of the topological phase that supports the edge currents, as well as providing the potential gradient across the topological phase that gives rise to the bulk current. Both the edge and bulk currents are resilient to several experimental parameters such as trapping potential, temperature, and disorder. We propose to investigate the properties of these currents directly from time-of-flight images with both short-time and long-time expansions.
Conventional topological insulators exhibit exotic gapless edge or surface states, as a result of non-trivial bulk topological properties. In periodically-driven systems the bulk-boundary correspondence is fundamentally modified and knowledge about conventional bulk topological invariants is insufficient. While ultracold atoms provide excellent settings for clean realizations of Floquet protocols, the observation of real-space edge modes has so far remained elusive. Here we demonstrate an experimental protocol for realizing chiral edge modes in optical lattices, by creating a topological interface using a potential step that is generated with a programmable optical potential. We show how to efficiently prepare particles in these edge modes in three distinct Floquet topological regimes that are realized in a periodically-driven honeycomb lattice. Controlling the height and sharpness of the potential step, we study how edge modes emerge at the interface and how the group velocity of the particles is modified as the sharpness of the potential step is varied.
Since the experimental realization of synthetic gauge fields for neutral atoms, the simulation of topologically nontrivial phases of matter with ultracold atoms has become a major focus of cold-atom experiments. However, several obvious differences exist between cold-atom and solid-state systems, for instance the small size of the atomic cloud and the smooth confining potential. In this article we show that sharp boundaries are not required to realize quantum Hall or quantum spin Hall physics in optical lattices and, on the contrary, that edge states in a smooth confinement exhibit additional interesting properties, such as spatially resolved splitting and merging of bulk bands and the emergence of robust auxiliary states in bulk gaps to preserve the topological quantum numbers. In addition, we numerically validate that these states are robust against disorder. Finally, we analyze possible detection methods, with a focus on Bragg spectroscopy, to demonstrate that the edge states can be detected and that Bragg spectroscopy can reveal how topological edge states are connected to the different bulk bands.
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.
Topological behavior has been observed in quantum systems including ultracold atoms. However, background harmonic traps for cold atoms hinder the direct detection of topological edge states arising at the boundary because the distortion fuses the edge states into the bulk. We propose experimentally feasible protocols to probe localized edge states and dimerization of ultracold fermions. By confining cold atoms in a ring lattice and changing the boundary condition from periodic to open using an off-resonant laser sheet to cut open the ring, topological edge states can be generated. A lattice in a topological configuration can trap a single particle released at the edge as the system evolves in time. Alternatively, depleting an initially filled lattice away from the boundary reveals the occupied edge states. Signatures of dimerization in the presence of contact interactions can be found in selected correlations as the system boundary suddenly changes from periodic to open and exhibit memory effects of the initial state distinguishing different configurations.
In a recent paper, Lang et al. proposed that edge states and topological phases can be observed in one-dimensional optical superlattices. They showed that the topological phases can be revealed by observing the density profile of a trapped fermion system, which displays plateaus with their positions. However, disorders are not considered in their model. To study the effect of disorders on the topological phases, we introduce random potentials to the model for optical superlattcies. Our calculations show that edge states are robust against the disorders. We find the edge states are very sensitive to the number of the sites in the optical superlattice and we propose a simple rule to describe the relationship between the edge states and the number of sites. The density plateaus are also robust against weak disorders provided that the average density is calculated over a long interval. The widths of the plateaus are proportional to the widths of the bulk energy gaps when there are disorders. The disorders can diminish the bulk energy gaps. So the widths of the plateaus decrease with the increase of disorders and the density plateaus disappear when disorders are too strong. The results in our paper can be used to guide the experimental detection of topological phases in one-dimensional systems.