Analysis of the Pancharatnam-Berry phase of vector vortex states using the Hamiltonian based on the Maxwell-Schr\"odinger equation

We derived the Berry connection of vector vortex states (VVSs) from the "true" Hamiltonian obtained through the Maxwell--Schr\"odinger equation for an inhomogeneous anisotropic (IA) medium, and we experimentally demonstrated measurement of the corresponding Pancharatnam--Berry (PB) geometrical phase of VVSs. The PB phase (PBP) of VVSs can be divided into two phases: homogeneous and inhomogeneous PBPs. Homogeneous and inhomogeneous PBPs are related to the conventional PBP and the spatially-dependent geometric phase given by an IA medium such as a polarization converter, respectively. We theoretically detected that inhomogeneous PBP accumulation originates from the gauge dependence of the index of the hybrid-order Poincar\'e sphere, which provides an alternate method for understanding optical spin--orbital angular momentum conversion. The homogeneous PBP, which is explicitly observed for the first time, has implications for quantum state manipulation and information processing.