Microwave Realization of the Chiral Orthogonal, Unitary, and Symplectic Ensembles

Random matrix theory has proven very successful in the understanding of the spectra of chaotic systems. Depending on symmetry with respect to time reversal and the presence or absence of a spin 1/2 there are three ensembles, the Gaussian orthogonal (GOE), Gaussian unitary (GUE), and Gaussian symplectic (GSE) one. With a further particle-antiparticle symmetry the chiral variants of these ensembles, the chiral orthogonal, unitary, and symplectic ensembles (the BDI, AIII, and CII in Cartan's notation) appear. We exhibit a microwave setup based on a linear chain of evanescently coupled dielectric cylindrical resonators allowing us to study all three chiral ensembles experimentally. In all cases the predicted repulsion behavior between positive and negative eigenvalues for energies close to zero could be verified.