Measurement of a topological edge invariant in a microwave network (1408.1808v2)

Abstract: We report on the measurement of topological invariants in an electromagnetic topological insulator analog formed by a microwave network, consisting of the winding numbers of scattering matrix eigenvalues. The experiment can be regarded as a variant of a topological pump, with non-zero winding implying the existence of topological edge states. In microwave networks, unlike most other systems exhibiting topological insulator physics, the winding can be directly observed. The effects of loss on the experimental results, and on the topological edge states, is discussed.

Measurement of a Topological Edge Invariant in a Microwave Network

The paper "Measurement of a Topological Edge Invariant in a Microwave Network" by Wenchao Hu and collaborators reports an experimental investigation into the measurement of topological invariants within a microwave network emulating a topological insulator. Utilizing scattering matrix eigenvalues, the paper focuses on the integer winding numbers characteristic to the response of the microwave network's boundaries, effectively establishing an innovative method to explore topological phenomena in engineered systems.

Overview and Results

The authors construct a network using microwave components, such as cables, phase shifters, and directional couplers. Their configuration permits a controlled investigation of the system's topological properties by varying these components, particularly focusing on a parameter denoted as $\theta$, which describes the coupling strength. The paper delineates the system into two phases: one being topologically nontrivial (anomalous Floquet insulator) and another being trivial. The network is effectively subject to a topological pump protocol, conceptually similar to Laughlin's thought experiment for the quantum Hall effect—interpreting edge responses as indicators of topological protection.

The salient outcome is that strong numerical evidence supports the existence of topological edge states when $\theta > \pi/4$, visibly shown by non-zero winding numbers in the reflection eigenvalues. This assertion is demonstrated by experimentally tuning the boundary twist angle $k$ through $2\pi$, while maintaining a phase within the bulk band gap. Remarkably, the work indicates that even in the presence of material loss, the network's topological behavior, characterized by these winding numbers, remains observable and distinct.

Implications and Future Directions

The significance of this research lies in its methodology, providing a tangible route to measure the topology of band structures in photonics, which traditionally relied heavily on indirect observations such as edge propagation phenomena. The potential applications are extensive, including the robustness of photonic circuits and the intricate control of wave paths, valuable for technologies requiring resilient information transmission.

Theoretically, the exploration of network models as analogs for Floquet topological insulators presents a fertile ground for future investigations. As losses have been a tolerant feature in these experimental setups, further reduction and control of losses would likely enhance precision and applicability. Moreover, the cross-compatibility with other topological phases, such as Chern insulators, could be explored by adjusting network configurations and coupling strengths.

The paper serves as a pioneering proof-of-concept for direct edge invariant measurement techniques. Anticipated future research could expand into tunable networks with variable $\theta$ enabling direct observation transitions across insulating phases. Building upon this work, subsequent experiments can explore multidimensional setups or incorporate nonlinear components, probing interactions that are presently beyond the scope of condensed matter realizations of topological insulators.

Conclusion

Hu et al.'s research represents a conclusive step forward in the practical measurement of topological invariants within photonic analogs of topological insulators. By achieving actual measurable changes in the reflection eigenvalues as a function of geometric phase alterations—a haLLMark of topological states—this paper substantiates the feasibility of experimental approaches to topological properties, opening avenues for novel photonic device applications. The rigorous approach to understanding the conditions under which topological protection arises and persists under dissipation represents a substantial contribution to both fundamental and applied physics domains.