Comment on "Detecting non-Abelian geometric phases with three-level Λ systems"

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 three-level Lambda system. They further propose a test to detect the noncommutative 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. DOI: 10.1103/PhysRevA.87.036101