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Physical Review A

Non-Abelian geometric phase in four-waveguide arrays

Bao-Long Weng, Dong-Mei Lai, Xin-Ding Zhang

Published May 3, 2012• 4 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.85.053801

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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 propose an optical lattice scheme which would permit the experimental observation of Zitterbewegung (ZB) with ultracold, neutral atoms. A four-level tripod variant of the setup for stimulated Raman adiabatic passage (STIRAP) has previously been proposed for generating non-Abelian gauge fields. Dirac-like Hamiltonians, which exhibit ZB, are simple examples of such non-Abelian gauge fields; we show how a variety of them can arise, and how ZB can be observed, in a tripod system. We predict that the ZB should occur at experimentally accessible frequencies and amplitudes.
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 report on an experimental demonstration of light transfer in an engineered triple-well optical waveguide structure which provides a classic analog of coherent tunneling by adiabatic passage (CTAP) recently proposed for coherent transport in space of neutral atoms or electrons among tunneling-coupled optical traps or quantum wells [A. D. Greentree et al., Phys. Rev. B 70, 235317 (2004); K. Eckert et al., Phys. Rev. A 70, 023606 (2004)]. The direct visualization of CTAP wave-packet dynamics enabled by our simple optical system clearly shows that in the counterintuitive passage scheme light waves tunnel between the two outer wells without appreciable excitation of the middle well.
The generation of non-Abelian geometric phases from a system of evanescently coupled waveguides is considered within the framework of nonorthogonal coupled-mode theory. Here, we study an experimentally feasible tripod arrangement of waveguides that contain dark states from which a nontrivial U(2)U(2)U(2) mixing can be obtained by means of an adiabatic parameter variation. We investigate the influence of higher-order contributions beyond nearest-neighbor coupling as well as self-coupling on the stability of a non-Abelian U(3)U(3)U(3) phase generated from an optical tetrapod setup. Our results indicate that, despite the mode nonorthogonality, the symmetry of dark states protects the geometric evolution of light from distortion.
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.