Comment on `Detecting non-Abelian geometric phases with three-level $\Lambda$ systems'

TL;DR

This paper critically examines previous claims of non-Abelian geometric phases in three-level $ ext{Lambda}$ systems, demonstrating that such phases are trivial or negligible under realistic conditions, challenging prior interpretations.

Contribution

The authors clarify that the non-Abelian geometric phase in $ ext{Lambda}$ systems is trivial or too small to detect, correcting earlier claims of its non-Abelian nature.

Findings

Non-Abelian geometric phase is trivial in the adiabatic approximation.

Exact treatment shows the phase is very small and indistinguishable from dynamical phases.

Previous proposals for detecting non-Abelian phases in such systems are not feasible.

Abstract

In their recent paper, Yan-Xiong Du et al. [Phys. Rev. A 84, 034103 (2011)] claim to have found a non-Abelian adiabatic geometric phase associated with the energy eigenstates of a large-detuned $\Lambda$ three-level system. They further propose a test to detect the non-commutative feature of this geometric phase. On the contrary, we show that the non-Abelian geometric phase picked up by the energy eigenstates of a $\Lambda$ system is trivial in the adiabatic approximation, while, in the exact treatment of the time evolution, this phase is very small and cannot be separated from the non-Abelian dynamical phase acquired along the path in parameter space.