Riemann-Silberstein geometric phase in 4D polarization space

Geometric phase is a far-reaching concept in quantum and classical physics. The first discovered geometric phase, the Pancharatnam-Berry (PB) phase, has profoundly shaped nanophotonics through metasurfaces. However, the PB phase arises from SU(2) polarization evolution and is constrained to a 2D polarization space, failing to capture the full polarization degrees of freedom. We generalize geometric phase to the 4D Riemann-Silberstein (RS) space that simultaneously describes electric, magnetic, and hybrid electric-magnetic polarizations. We show that SU(4) polarization evolution can generate a new geometric phase, the RS phase, alongside the PB phase. Unlike the PB phase that typically manifests in circularly polarized light, the RS phase can emerge in arbitrarily polarized light. Together, they enable a high-dimensional geometric framework for light propagation across general interfaces. We reveal that the phase shifts governed by Fresnel equations are direct manifestations of the RS-space geometric phases, integrating a century-old wave theory into this paradigm. We experimentally validate the framework using metasurfaces and achieve high-dimensional wavefront manipulation. Our work offers fundamental insights into the geometric nature of light-matter interactions, with implications for topological and non-Abelian physics in classical wave systems.