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

preprint

posted on 2023-11-30, 17:54 authored by Mark Kremer, Lucas Teuber, Alexander Szameit, Stefan ScheelGeometric 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.