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arXiv:1211.6862 · Physical Review A

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

Marie Ericsson, Erik Sjöqvist

Published Mar 29, 2013• 1 citation
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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 three-level $\ensuremath{\Lambda}$ 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 $\ensuremath{\Lambda}$ 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

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In this reply, we address the comment by Ericsson and Sjoqvist on our paper [Phys. Rev. A {\bf 84}, 034103 (2011)]. We point out that the zero gauge field is not the evidence of trivial geometric phase for a non-Abelian SU(2) gauge field. Furthermore, the recalculation shows that the non-Abelian geometric phase we proposed in the three-level $Λ$ system is indeed experimentally detectable.

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Reply to "Comment on 'Detecting Non-Abelian Geometric Phases with Three-Level Atoms' "

Yan-Xiong Du, Zheng-Yuan Xue, Xin-Ding Zhang +1 more

•Jan 1, 2013•0 citations
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 \ensuremathΛ\ensuremath{\Lambda}\ensuremathΛ 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 \ensuremathΛ\ensuremath{\Lambda}\ensuremathΛ 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.
In this reply, we address the comment by Ericsson and Sjoqvist on our paper [Phys. Rev. A {\bf 84}, 034103 (2011)]. We point out that the zero gauge field is not the evidence of trivial geometric phase for a non-Abelian SU(2) gauge field. Furthermore, the recalculation shows that the non-Abelian geometric phase we proposed in the three-level ΛΛΛ system is indeed experimentally detectable.
We show that a non-Abelian gauge potential in two nearly degenerated dressed states may be induced by two laser beams interacting with a three-level $\ensuremath{\Lambda}$ atomic system. We demonstrate that the populations of the atomic states at the end of a composed path formed by two closed loops are dependent on the order of those two loops, showing an unambiguous signature of the non-Abelian geometric phase. Through numerical calculations, we show that the non-Abelian feature of the geometric phases can be tested under realistic conditions.
We unveil that even an infinitesimal deviation of the initial state from the eigenstates of the initial Hamiltonian can yield a significant change in the geometric phase accompanying adiabatic evolution without degeneracies of the energy levels. This leads to the counterintuitive result that the values of two geometric phases computed from the general definition of the Berry phase [Samuel and Bhandari, Phys. Rev. Lett. 60, 2339 (1988)] can dramatically differ from each other in the adiabatic limit even if their evolution paths are sufficiently close in a pointwise manner. The result of this work indicates that the widely accepted theory of geometric phases is not very consistent in the context of adiabatic evolution involving superposed eigenstates.
In a recent letter [Phy. Rev. Lett. 95, 080502 (2005)], it is claimed that based on a new kind of quantum mechanical phase of wave function which is neither dynamical nor geometrical a new kind of phase gate for quantum computation is discovered. The origin of this phase is the adiabatic evolution of a dark state. I would like to point out that these claims are questionable. The so called new phase gate was based on a phase which was nothing new but has been known for decades. It is something which has no physical meaning and is not observable. It was also pointed out by [quant-ph/0509150v1] that [Phy. Rev. Lett. 95, 080502 (2005)] was incorrect and the phase is the known geometric phase. Here we present a totally different argument.