Investigation of Two Dimensional Floquet Topological Insulator Using Microwave Network
Date of Issue2017-12-27
School of Electrical and Electronic Engineering
Topological insulators are one of the most profound discoveries in condensed matter physics during the past few decades. They are “topologically distinct” from conventional insulators, because they are insulating in the bulk while supporting metallic states on surfaces. The most extraordinary physical property of certain classes of 2D topological insulators is the existence of uni-directional transmission along the edge, which is robust to imperfections and exhibits no back-reflection. In 2005, Haldane and Raghu introduced the topological insulator concept into pho- tonics by theoretically proposing a photonic analogue of the quantum Hall effect in photonic crystals. Wang et al. experimentally confirmed this idea by observing a uni-directional transmission line in a gyromagnetic photonic crystal operating in the microwave frequency range. “Topological photonics” has also been realized with resonator lattices and waveguide lattices subsequently. However, in all these experiments, edge propagation measurements serve exclusively as the proof of topological nontrivial system due to the lack of a direct analog of the Hall conductance or similar linear response-based quantity. During my PhD career, my research has focused on the investigation of two-dimensional Floquet topological insulators using microwave networks. The first part of my work involves experimentally measuring a topological edge invariant which consists of the integer winding numbers of scattering matrix eigenvalues in a microwave network. The second part of my study concerns the relation between topological edge invariants and exceptional points by introducing controllable loss and gain into the microwave network. The network model we used to measure topological edge invariants is a two-dimensional network which is mapped into a microwave network using Laughlin’s topological pump idea. The experiment setup is a two-port network system with variable phase shifters. By measuring the scattering matrix of the two-port network, we can observe the winding behavior of the eigenvalues. We implemented this experiment using microwave components at 5 GHz and successfully observedthe non-zero winding behavior for non-trivial topological system and zero winding behavior for trivial topological system, as predicted by theoretical arguments but never previously found in an experiment. Due to the existence of loss, the system we used to measure topological edge in-variants is non-Hermitian; furthermore, in the standard theory of the topological pump, we can never observe rigorously non-zero winding in a finite system, since there is always a gap in the projected band structure. Moreover, non-Hermitian effects introduce the novel physical feature of exceptional points into the band-structure which is easy to study by introducing controllable loss and gain into our experiment setup. Hence, we re-implemented our experiment setup at 900 MHz and added digital variable attenuators into the system. By controlling gain and loss, we demonstrate, theoretically and experimentally, a direct relationship between a Hermitian topological invariant and exceptional point winding numbers. In summary, my PhD research has concentrated on the investigation of two dimensional Floquet topological insulators using microwave networks. Two major works have been finished, one is the experimental measurement of topological edge invariants in the form of scattering matrix eigenvalue winding numbers in a microwave network. The other is theoretically and experimentally proving the relation between Hermitian topological invariant and exceptional point winding numbers.