Realizing a Gaussian Symplectic Ensemble with Quantum Spin Hall Photonic Topological Insulator Graphs*

Unprecedented wave phenomena have been realized with the creation of photonic topological insulators (PTI).  One feature is the emulation of a spin-1/2 degree of freedom (DOF) in the photonic context.  The bi-anisotropic meta-waveguide (BMW) is a versatile platform for realizing analogs of quantum Hall, quantum spin Hall (QSH), and quantum valley Hall (QVH) effects for electromagnetic waves.  Random Matrix Theory has enjoyed great success in describing the statistics of large nuclei with various symmetries, and has been successfully applied to describing statistical properties of various microwave chaotic systems.  Due to the lack of a spin-1/2 DOF for light, the realization of a Gaussian Symplectic Ensemble (GSE) system was not observed for a long time in classical wave systems.

Here we aim to create a microwave simulator of GSE statistics by utilizing the intrinsic physical properties of the BMW PTIs. We conduct studies of the eigenmodes of closed PTI graphs using numerical simulations.  We construct quantum graph structures with a combination of both QSH and QVH BMW-domains, and the two-fold degeneracy of system eigenmodes are observed from eigenmode simulation results. The level spacing statistics of an ensemble of pyramidal graphs agree well with the GSE statistics.  We also present the results of an experimental realization of the QSH/QVH PTI quantum graph in superconducting Aluminum.