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arXiv:0801.2935 · New Journal of Physics

Landau levels of cold atoms in non-Abelian gauge fields

A Jacob, Patrik Öhberg, Gediminas Juzeliūnas, Luis Santos

Published Apr 30, 2008• 34 citations
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Abstract

The Landau levels of cold atomic gases in non-Abelian gauge fields are analyzed. In particular we identify effects on the energy spectrum and density distribution which are purely due to the non-Abelian character of the fields. We investigate in detail non-Abelian generalizations of both the Landau and the symmetric gauge. Finally, we discuss how these non-Abelian Landau and symmetric gauges may be generated by means of realistically feasible lasers in a tripod scheme.

DOI
10.1088/1367-2630/10/4/045022

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The Landau levels of cold atomic gases in non-Abelian gauge fields are analyzed. In particular we identify effects on the energy spectrum and density distribution which are purely due to the non-Abelian character of the fields. We investigate in detail non-Abelian generalizations of both the Landau and the symmetric gauge. Finally, we discuss how these non-Abelian Landau and symmetric gauges may be generated by means of realistically feasible lasers in a tripod scheme.
We study ultracold atoms subjected to U(2) non-Abelian potentials: we consider gauge potentials having, in the Abelian limit, degenerate Landau levels and we then investigate the effect of general homogeneous non-Abelian terms. The conditions under which the structure of degenerate Landau levels is preserved are classified and discussed. The typical gauge potentials preserving the Landau levels are characterized by a fictitious magnetic field and by an effective spin-orbit interaction (e.g., obtained through the rotation of two-dimensional atomic gases coupled with a tripod scheme). The single-particle energy spectrum can be analytically determined for a class of gauge potentials, whose physical implementation is discussed. The corresponding Landau levels are deformed by the non-Abelian contribution of the potential and their spin degeneracy is split. The related deformed quantum Hall states for fermions and bosons (in the presence of strong intraspecies interaction) are determined far from and at the degeneracy points of the Landau levels, where non-Abelian states appear. We present a discussion of the effect of the angular momentum, as well as results for U(3) gauge potentials.
The dynamics of ultracold neutral atoms subject to a non-Abelian gauge field is investigated. In particular we analyze in detail a simple experimental scheme to achieve a con- stant, but non-Abelian gauge field, and discuss in the frame of this gauge field the non-Abelian Aharanov-Bohm effect. In the last part of this paper, we discuss intrinsic non-Abelian effects in the dynamics of cold atomic wavepackets. the implementation of these proposals, ultracold atoms would offer a unique testbed for the analysis of non-trivial non- Abelian effects on the quantum dynamics of multicomponent wavepackets. In this paper, we investigate the wave packet dynam- ics of a cloud of ultracold atoms in the presence of a non- Abelian gauge potential. In Sect. 2 we discuss how this un- doubtedly rather exotic scenario can be envisaged in a sam- ple of cold atoms where the internal electronic energy levels are addressed by laser fields with a nontrivial spatial phase and intensity distribution. This setup opens up a number of new scenarios for ultracold gases, allowing for the study of non-Abelian atom optics, which naturally ties together opti- cal and magnetic effects. Remarkably, as shown in Sect. 2, even very simple laser arrangements may induce non-trivial cold-atom dynamics. As a first example of this non-trivial dynamics, we discuss in Sect. 3 a possible optical tweezer experiment including a non-Abelian flux, for which the pop- ulation transfer between internal states crucially depends on the path taken (non-Abelian Aharonov-Bohm effect). This effect resembles indeed what one would expect from scat- tering protons onto a non-Abelian flux line where the proton can be transferred into a neutron (8). The tweezer experiment discussed in Sect. 3 just involves the internal-state dynamics, without exploring the rich dynamics resulting from the in- terplay between external and internal degrees of freedom in non-Abelian gauge fields. Section 4 is devoted to the analy- sis of this interplay. In particular, we show that the dynamics of cold-atom wavepackets can be significantly affected by in- trinsically non-Abelian effects, which are crucially dependent on the initial momentum distribution of the wavepacket. We consider in particular the relevant examples of wavepacket propagation and wavepacket reflection at an atomic mirror. Finally, we conclude in Sect. 5.
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 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.
We show that non-Abelian potentials acting on ultracold gases with two hyperfine levels can give rise to ground states with non-Abelian excitations. We consider a realistic gauge potential for which the Landau levels can be exactly determined: The non-Abelian part of the vector potential makes the Landau levels nondegenerate. In the presence of strong repulsive interactions, deformed Laughlin ground states occur in general. However, at the degeneracy points of the Landau levels, non-Abelian quantum Hall states appear: These ground states, including deformed Moore-Read states (characterized by Ising anyons as quasiholes), are studied for both fermionic and bosonic gases.