Optimal Design Strategy of Non-Abelian Geometric Phases Based on Quantum Metric

When quantum systems evolve through Hilbert space, they acquire geometric phases [1]. For many applications, this can be expressed by a line integral over an (Abelian) vector potential, which leads to a scalar phase. However, systems with degenerate energy levels experience matrix-valued geometric phases [2], known as Non-Abelian holonomies. While Non-Abelian structures naturally arise in Yang Mills theories, they have also proven to be cruicial for topological quantum computation [3]. In our work we use an integrated photonic structure to realize a non-Abelian phase utilizing an Abelian gauge field, namely electro-magnetic wavepackets.