Geometric phases for SU(3) representations and three level quantum systems

Abstract

A comprehensive analysis of the pattern of geometric phases arising in unitary representations of the group SU(3) is presented. The structure of the group manifold, convenient local coordinate systems and their overlaps, and complete expressions for the Maurer-Cartan forms are described. Combined with a listing of all inequivalent continuous subgroups of SU(3) and the general properties of dynamical phases associated with Lie group unitary representations, one finds that nontrivial dynamical phases arise only in three essentially different situations. The case of three level quantum systems, which is one of them, is examined in further detail and a generalization of the SU(3) solid angle formula is developed.