Berry phase in homogeneous K\"ahler manifolds with linear Hamiltonians

TL;DR

This paper derives exact formulas for the total and geometric phases in cyclic quantum systems with linear Hamiltonians on homogeneous K"ahler manifolds, linking quantum phases to classical symplectic geometry.

Contribution

It provides explicit calculations of quantum phases on K"ahler manifolds for linear Hamiltonians, connecting quantum and classical geometric structures.

Findings

Total and geometric phases can be computed exactly.

Geometric phase equals the symplectic area enclosed by classical motion.

Results apply to systems with Lie group symmetries.

Abstract

We study the total (dynamical plus geometrical (Berry)) phase of cyclic quantum motion for coherent states over homogeneous K\"ahler manifolds X=G/H, which can be considered as the phase spaces of classical systems and which are, in particular cases, coadjoint orbits of some Lie groups G. When the Hamiltonian is linear in the generators of a Lie group, both phases can be calculated exactly in terms of {\em classical} objects. In particular, the geometric phase is given by the symplectic area enclosed by the (purely classical) motion in the space of coherent states.