Comprehensive Analysis of Geometric Phase for SU(3) Representations

TL;DR

This work develops a geometric-phase framework for three-level open quantum systems by embedding mixed states in the eight-dimensional SU(3) Bloch sphere and linking Pancharatnam, Berry, and Aharonov–Anandan concepts. It extends the Mukunda–Simon kinematic approach to non-unitary evolution, expressing the mixed-state geometric phase in terms of Bargmann invariants and a Mead–Berry connection, while ensuring gauge and reparameterization invariance. The study combines depolarization models—including a Lindblad master equation for dephasing and a non-resonant random-atom bath—to obtain explicit phase expressions and elucidate decoherence effects on the Bloch eight-vector and Stokes-like polarizations. These results highlight persistent geometric memory in mixed states, reveal how population inversions influence the phase, and set the stage for future generalizations to multi-mode and SU(3) polarization dynamics in more complex photonic systems.

Abstract

Geometric Phase in Quantum Mechanics is generally formulated entirely in terms of geometric structure of the Complex Hilbert Space. We will exploit this fact in case of mixed states for three level open systems undergoing depolarization using the eight dimensional Poincare sphere in the SU(2) Polarisation picture and non unit vector rays in H3 within the limit of pure state approach may be found to be in agreement with the Pancharatnam Phase, Berry Phase and Aharonov-Anandan Phase.