Geometric Phase in Su(n) Interferometry

The purpose of this contribution is twofold: to review how an SU(N) transformation can be experimentally realized using optical elements, and to show how such an experimental realization can be used to investigate the cyclic evolution of a state over the manifold SU(N)/U(N-1). The bulk of the results will be presented explicitly for SU(3) (see [1] for further details) and SU(4), although it will become clear that the method can be applied to any SU(N). Recall that cyclic evolution of a wave function yields the original state plus a phase shift, and this phase shift is a sum of a dynamical phase φd and a geometric (or topological, or quantal, or Berry) phase φg shift [2,3]. The geometric phase shift is important not just for quantum systems but also for all of wave physics. Thus far, controlled geometric–phase