Symmetry-protected non-Abelian geometric phases in optical waveguides with nonorthogonal modes

The generation of non-Abelian geometric phases from a system of evanescently coupled waveguides is extended towards the framework of nonorthogonal coupled-mode theory. Here, we study an experimentally feasible tripod arrangement of waveguides that contain dark states from which a nontrivial U(2)-mixing can be obtained by means of an adiabatic parameter variation. We investigate the influence of higher-order contributions beyond nearest-neighbour coupling as well as self-coupling on the stability of a U(3)-phase generated from an optical tetrapod setup. Our results indicate that, despite the mode nonorthogonality, the symmetry of dark states protects the geometric evolution of light from distortion.