Non-Abelian quantum holonomy of hydrogen-like atoms

TL;DR

This paper investigates the non-Abelian and Abelian properties of Uhlmann holonomy in hydrogen-like atoms with spin-orbit coupling under a changing magnetic field, revealing the role of entanglement and phase relations.

Contribution

It demonstrates the non-Abelian nature of holonomy for subsystems and the Abelian nature for the whole system, linking entanglement to gauge structure and phase properties.

Findings

Holonomy for orbital and spin subsystems is non-Abelian.

Holonomy of the entire system is Abelian.

The phase of the Wilson loop differs from the mixed-state geometric phase.

Abstract

We study the Uhlmann holonomy [Rep. Math. Phys. 24, 229 (1986)] of quantum states for hydrogen-like atoms where the intrinsic spin and orbital angular momentum are coupled by the spin-orbit interaction and subject to a slowly varying magnetic field. We show that the holonomy for the orbital angular momentum and spin subsystems is non-Abelian, while the holonomy of the whole system is Abelian. Quantum entanglement in the states of the whole system is crucially related to the non-Abelian gauge structure of the subsystems. We analyze the phase of the Wilson loop variable associated with the Uhlmann holonomy, and find a relation between the phase of the whole system with corresponding marginal phases. Based on the result for the model system we provide evidence that the phase of the Wilson loop variable and the mixed-state geometric phase [E. Sj\"oqvist {\it et al.} Phys. Rev. Lett. 85, 2845 (2000)] are in general inequivalent.