Measurement of a Topological Edge Invariant in a Microwave Network

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 nonzero 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, are discussed.

Popular Summary

Topological insulators are exotic phases of matter that exhibit many unusual features, such as the existence of surface states that are immune to scattering. We show that an electromagnetic analog of a topological insulator can be realized with a classical “microwave network” consisting of coaxial cables attached to directional couplers.

We employ a network divided into identical subunits, all operating at 5 GHz. The coherent electromagnetic waves propagating through the network play the role of quantum wave functions. We are able to directly observe a “topological edge invariant,” a quantity that demonstrates the distinction between a topological insulator and a conventional insulator, which is normally not possible in quantum systems. This invariant is observed through the winding of the phase of waves reflected from the network, which is found to occur despite the presence of loss. We are able to switch between topologically trivial and nontrivial states by switching the order of the output ports.

Since our network components are subject to loss, our setup is not an ideal topological pump. We expect that future studies will focus on electromagnetic networks with lower levels of loss.

Figure 1

(a) Schematic of a periodic 2D directed network. Wave amplitudes undergo phase delay $\varphi$ along each directed link (black arrows) and couple through a $2\times 2$ unitary matrix at each node (blue boxes). (b) In a “topological pump” setup, a 2D system is rolled into a cylinder, twisted boundary conditions with tunable twist angle $k$ are applied, and a transport measurement is performed in the axial direction (cyan arrows). (c) Topological pump setup for a network model, with a cylinder comprising one unit cell in the azimuthal direction and two cells in the axial direction.

Figure 2

Experimental setup. Each of the identical units, labeled “Cell 1” and “Cell 2,” corresponds to one cell in the topological pump geometry of Fig. 1. A pair of phase shifters in each cell (pink boxes), with shifts of $+k$ and $-k$, respectively, implement the twisted boundary condition. The couplers (blue rods) are depicted in the strong-coupling configuration. The weak-coupling configuration is achieved by swapping each coupler’s outputs. The overall input and output amplitudes are ${\alpha }_{\pm }$ and ${\beta }_{\pm }$. Their scattering parameters are measured with a network analyzer.

Figure 3

Scattering matrix eigenvalues measured across one cell (left) and across two cells in series (right). Arrows indicate the direction of motion with increasing $k$. Circles show experimental data and solid curves show theoretical calculations using the scattering parameters measured for each network component individually (including losses). Nonzero winding numbers can be observed in the strong-coupling two-cell case. The unit circle is indicated by dashed curves.

Figure 4

Arguments of the complex $S$ matrix eigenvalues for the two-cell network, as the phase shift $k$ is tuned through $2\pi$. Empty circles show the strong coupling measurement data, filled circles show the weak coupling data, and solid curves show theoretical calculations. For strong coupling, the two eigenvalues have winding numbers $\pm 1$, corresponding to the bulk band structure being topologically nontrivial.

Figure 5

Complex projected band structure of a hypothetical lossy network (blue lines), consisting of an infinite strip 5 unit cells wide. Each link has real tunable phase delay $\varphi$, each coupler has loss $e^{-\gamma }$ in each output port where $\gamma =0.25$, and each edge of the strip obeys lossy boundary conditions $r_{\pm }=\exp (-0.2\pi )$, chosen to be roughly the same as in the experiment. The coupling strength is $\theta =0.4\pi$, approximately equal to the “strong coupling” (topologically nontrivial) configuration in the experiment. The band structure for the lossless network ($\gamma =0$ and $|r_{\pm }|=1$) is plotted for comparison (red lines). The lossy network exhibits edge states whose dispersion relations are almost identical to the lossless network’s topological edge states, and have lower loss than all other states (blue dashes).