Optimal design strategy for non-Abelian geometric phases using Abelian gauge fields based on quantum metric

Geometric phases, which are ubiquitous in quantum mechanics, are commonly more than only scalar quantities. Indeed, often they are matrix-valued objects that are connected with non-Abelian geometries. Here we show how generalized, non-Abelian geometric phases can be realized using electromagnetic waves travelling through coupled photonic waveguide structures. The waveguides implement an effective Hamiltonian possessing a degenerate dark subspace, in which an adiabatic evolution can occur. The associated quantum metric induces the notion of a geodesic that defines the optimal adiabatic evolution. We exemplify the non-Abelian evolution of an Abelian gauge field by a Wilson loop.