Noncyclic continuous Pancharatnam-Berry phase in dual-beam interference

The geometric phase for classical electromagnetic light beams, in its original formulation as introduced by Pancharatnam, concerns fields experiencing cyclic, discrete in-phase polarization-state changes. A similar phase was later recognized by Berry to govern the behavior of adiabatic quantum systems, with consequent extensions to nonadiabatic and noncyclic evolutions of the quantum state. However, no optical counterpart for the noncyclic, adiabatic (continuous) evolution has been demonstrated. Here we employ a modified Young's two-pinhole setup with controlled pinhole polarizations and intensities to generate on interference an arbitrary continuous spatial evolution of the polarization state, an optical analogue to the adiabatic case. The customized arrangement allows separating at any point the accumulated dynamical and geometric phases from the total phase, enabling a detailed study of the noncyclic Pancharatnam-Berry phase in a continuous transformation. Our theoretical and experimental results are in excellent agreement and consistent with the geodesic rule for noncyclic evolutions. The geometric phase of optical fields experiencing discrete cyclic polarization-state transformations was introduced by Pancharatnam and later extended to adiabatic cyclic evolutions of quantum systems by Berry. Here, adiabatic noncyclic evolution of the Pancharatnam-Berry phase in an optical context is demonstrated, both theoretically and experimentally.