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Physical Review A

Detecting non-Abelian geometric phases with three-level Λ systems

Yan-Xiong Du, Zheng‐Yuan Xue, Xin-Ding Zhang, Hui Yan

Published Sep 16, 2011• 4 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.84.034103

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

Marie Ericsson, Erik Sjöqvist

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.

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

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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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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 \ensuremathΛ\ensuremath{\Lambda}\ensuremathΛ 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 propose an experimentally feasible scheme to disclose the noncommutative effects induced by a light-induced non-Abelian gauge structure with trapped ions. Under an appropriate configuration, a true non-Abelian gauge potential naturally arises in connection with the geometric phase associated with two degenerated dark states in a four-state atomic system interacting with three pulsed laser fields. We show that the population in the atomic state at the end of a composed path formed by two closed loops C 1 and C 2 in the parameter space can be significantly different from the composed counter-ordered path. This population difference is directly induced by the noncommutative feature of non-Abelian geometric phases and can be detected unambiguously with current technology.
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.
We analyze the influence of three pulsed laser fields interacting with a four-state atomic system for which there occur two degenerate population-trapping dressed states. We present a simple expression for the geometrical phase acquired by such an atom during a stimulated Raman adiabatic passage (STIRAP) process using a delay between Stokes and pump pulses, during which the two Rabi frequencies adiabatically traverse a closed path in the parameter space. We describe techniques that can produce (geometrical) state-vector phases that are independent of the longitudial velocity with which atoms move across a laser beam, and that are insensitive to radiative decay from the intermediate excited state. The geometrical phase can be changed by changing the relative delay of the pulses. The geometrical phase can be measured by observing the population in atomic states. We show that when the pump and Stokes pulses are properly timed, high population transfer from the initial state to the target state is achieved. In the adiabatic limit the robustness of the population transfer is equivalent to that of STIRAP.
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.
A popular quantum system, in which the Pancharatnam-Berry non-Abelian geometric phase has been generated and exploited, is the atomic tripod system. It is conveniently created by linking a single atomic state with three other states by three electromagnetic fields. Such a linkage pattern naturally emerges between the magnetic sublevels of two atomic levels with angular momenta J=0J=0J=0 and J=1J=1J=1, although tripod implementations between other suitable sublevels are also used. Here we go beyond the limitation of a tripod system and show that it is possible to generate the non-Abelian geometric phase in a quantum system composed of NNN lower and $N\ensuremath{-}2$ upper sublevels. The theoretical instrument is the Morris-Shore transformation which reveals the existence of two uncoupled (dark) states composed of the lower sublevels only. A possible physical implementation is the atomic transition $J\ensuremath{\leftrightarrow}J\ensuremath{-}1$, with JJJ arbitrary, which is driven, as in the case of tripod system, by three electromagnetic fields of different polarizations. This generalization considerably broadens the range of systems that can be used to generate a geometric phase, with the the same experimental complexity as in the tripod system. Specific calculations of the non-Abelian geometric phase are presented for $J=\frac{3}{2}\ensuremath{\leftrightarrow}J=\frac{1}{2}$ and $J=2\ensuremath{\leftrightarrow}J=1$ systems. A method for measuring the geometric phase is proposed.
The Hamiltonians of two- and three-level atomic systems interacting with nearly resonant electro-magnetic fields are described. The geometric phase related to the σ z operator of a two-level system is analysed and it is shown that this phase is well defined also for an open circuit. For a closed circuit this phase becomes equivalent to twice the Berry phase. For three-level atomic system the eigenenergies of the Hamiltonian are obtained by solving a cubic equation. Simplified forms for these eigenenergies and a condition for degeneracy are obtained. For non-degenerate three-level system geometric phases related to Cartan subalgebra are analysed under the adiabatic approximation and their relations with the three-level Berry phases are clarified. We show how to use the general theory of non-Abelian connections to the Hamiltonian of a three-level atomic system which includes a degeneracy.