The Adiabatic Phase and Pancharatnam's Phase for Polarized Light

TLDR

Pancharatnam showed that cyclic changes in light’s polarization produce a geometric phase on the Poincaré sphere, a phenomenon rooted in the non‑transitivity of polarization states and analogous to the Aharonov‑Bohm effect, and distinct from the phase shift of circularly polarized light in coiled fibers. The authors employ spinor algebra and 2×2 Hermitian matrices, together with Maxwell’s equations for a twisted dielectric, to derive the precise relationship between Pancharatnam’s geometric phase and the adiabatic phase of slowly cycled quantum systems. They establish a precise correspondence between Pancharatnam’s geometric phase and the adiabatic phase observed in slowly cycled quantum systems.

Abstract

Abstract In 1955 Pancharatnam showed that a cyclic change in the state of polarization of light is accompanied by a phase shift determined by the geometry of the cycle as represented on the Poincaré sphere. The phase owes its existence to the non-transitivity of Pancharatnam's connection between different states of polarization. Using the algebra of spinors and 2 × 2 Hermitian matrices, the precise relation is established between Pancharatnam's phase and the recently discovered phase change for slowly cycled quantum systems. The polarization phase is an optical analogue of the Aharonov-Bohm effect. For slow changes of polarization, the connection leading to the phase is derived from Maxwell's equations for a twisted dielectric. Pancharatnam's phase is contrasted with the phase change of circularly polarized light whose direction is cycled (e.g. when guided in a coiled optical fibre).