We present the first scheme for producing and measuring an Abelian geometric phase shift in a three-level system where states are invariant under a non-Abelian group. In contrast to existing experiments and proposals for experiments, based on U(1)-invariant states, our scheme geodesically evolves U(2)-invariant states in a four-dimensional SU(3)/U(2) space and is physically realized via a three-channel optical interferometer.
Sanders, B. C., de Guise, H., Bartlett, S. D., & Zhang, W. (2001). Geometric phase of three-level systems in interferometry. Physical Review Letters, 86(3), 369-372. https://doi.org/10.1103/PhysRevLett.86.369