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arXiv:0709.4329 · New Journal of Physics

Detecting unambiguously non-Abelian geometric phases with trapped ions

Xin-Ding Zhang, Z D Wang, Liang-Bin Hu, Zhi-Ming Zhang, Shi-Liang Zhu

Published Apr 15, 2008• 16 citations
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Abstract

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.

DOI
10.1088/1367-2630/10/4/043031

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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.
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.
A theoretical scheme is investigated to detect the noncommutative feature of non-Abelian geometric phases with superconducting Josephson circuits.The results show that,two degenerate dark states are generated in the considered system,and the geometric phases are naturally induced through adjusting external fluxes.By designing two composite evolutions and considering the population difference associated with the same quantum state,the noncommutative feature of the geometric phase can be shown directly. Further analyses on the feasibility and advantages indicate that the present scheme possesses the convenient operations and good quantum coherence.Thus the proposal provides an effective approach to experimentally study the noncommutative property of non-Abelian geometric phase.
In this paper we present a scheme to simulate the four-level stimulated Raman adiabatic passage in a four-waveguide array. The engineered tripod-well optical-waveguide structure provides a classic analog of coherent transport in cold atoms coupled with optical traps. We show that a non-Abelian gauge structure emerges in this adiabatic passage. Furthermore, we demonstrate that this waveguide structure may help us to see ``directly'' the noncommutative feature of the non-Abelian geometric phases.
We show that $N\ensuremath{-}1$ degenerate dark states can be generated by coupling NNN-fold degenerate ground states and a common excited state with NNN laser fields. Interferences between light waves with different frequencies can produce laser fields with time-dependent amplitudes, which can induce not only U(N)\mathrm{U}(N)U(N) non-Abelian vector fields but also the scalar ones for the adiabatic motion of atoms in such laser fields. As an example, a time-periodic gauge potential is produced by applying specific laser fields to a tripod system. Some features of the Landau levels and the ground-state phase diagram of a rotating Bose-Einstein condensate for a concrete gauge field are also discussed.
The adiabatic motion of ultra-cold, multi-level atoms in spatially varying laser fields creating a dark state can give rise to effective gauge potentials. We show that these potentials lead to effective magnetic fields if the light fields possess a relative orbital angular momentum. This allows to study quantum-Hall like effects in ultra-cold atomic gases in various geometries. If the atom-light interaction creates several degenerate adiabatic eigenstates, the associated gauge potentials are non-Abelian. A pair of such degenerate dark states emerges e.g. if laser fields couple three internal states of an atom to a fourth common one under pairwise two--photon-resonance conditions. For this so-called tripod scheme we derive general conditions for truly non-Abelian gauge potentials and discuss special examples. In particular we show that using orthogonal laser beams with orbital angular momentum an effective magnetic field can be generated that has a monopole component.
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.