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arXiv:1203.3308 · Physical Review A

Vortex lattices for ultracold bosonic atoms in a non-Abelian gauge potential

Stavros Komineas, Nigel R. Cooper

Published May 16, 2012• 11 citations
arXivHTMLPDF

Abstract

The use of coherent optical dressing of atomic levels allows the coupling of ultracold atoms to effective nondynamical gauge fields. These can be used to generate effective magnetic fields and have the potential to generate non-Abelian gauge fields. We consider a model of a gas of bosonic atoms coupled to a gauge field with $\text{U}(2)$ symmetry and with constant effective magnetic field. We include the effects of weak contact interactions by applying Gross-Pitaevskii mean-field theory. We study the effects of a $\text{U}(2)$ non-Abelian gauge field on the vortex lattice phase induced by a uniform effective magnetic field, generated by an Abelian gauge field or, equivalently, by rotation of the gas. We show that, with increasing non-Abelian gauge field, the nature of the ground state changes dramatically, with structural changes of the vortex lattice. We show that the effect of the non-Abelian gauge field is equivalent to the introduction of effective interactions with nonzero range. We also comment on the consequences of the non-Abelian gauge field for strongly correlated fractional quantum Hall states.

DOI
10.1103/physreva.85.053623

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The use of coherent optical dressing of atomic levels allows the coupling of ultracold atoms to effective nondynamical gauge fields. These can be used to generate effective magnetic fields and have the potential to generate non-Abelian gauge fields. We consider a model of a gas of bosonic atoms coupled to a gauge field with U(2)\text{U}(2)U(2) symmetry and with constant effective magnetic field. We include the effects of weak contact interactions by applying Gross-Pitaevskii mean-field theory. We study the effects of a U(2)\text{U}(2)U(2) non-Abelian gauge field on the vortex lattice phase induced by a uniform effective magnetic field, generated by an Abelian gauge field or, equivalently, by rotation of the gas. We show that, with increasing non-Abelian gauge field, the nature of the ground state changes dramatically, with structural changes of the vortex lattice. We show that the effect of the non-Abelian gauge field is equivalent to the introduction of effective interactions with nonzero range. We also comment on the consequences of the non-Abelian gauge field for strongly correlated fractional quantum Hall states.
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 present thesis studies a variety of cold atomic systems in artificial gauge fields. In the first part we focus on fractional quantum Hall effects, asking whether interesting topological states can be realized with cold atoms. We start by making a close connection to solid-state systems and first consider fermionic atoms with dipolar interactions. Assuming the system to be in the Laughlin state, we evaluate the energy gap in the thermodynamic limit as a measure for the robustness of the state. We show that it can be increased by additionally applying a non-Abelian gauge field squeezing the Landau levels. We then switch to bosonic systems with repulsive contact interactions. Artificial magnetic fields for cold bosons have extensively been discussed before in the context of rotating Bose gases. We follow a different approach where the gauge field is due to an atom-laser coupling. Thus, transitions between different dressed states have to be included. They are shown to break the cylindrical symmetry of the system. Modifying the Laughlin state and the Moore-Read state accordingly, we determine the parameter regimes where they are good representations for the ground state of the system obtained via exact diagonalization. One of the most interesting feature of fractional quantum Hall states is the anyonic behavior of their excitations. We therefore also study quasiholes in the Laughlin state and the modified Laughlin state. They are shown to posses anyonic properties, which become manifest even in small systems. Moreover, the dynamics of a single quasihole causes visible traces in the density of the system which allow to clearly distinguish the Laughlin regime from less correlated phases. In the latter, a sequence of collapses and revivals of the quasihole can be observed, which is absent in the Laughlin regime. Extending our study to bosonic systems with a pesudospin-1/2 degree of freedom, we discuss the formation of strongly correlated spin singlets. Strikingly, at filling v=4/3, the system is described by a state with non-Abelian excitations, which is constructed as the zero-energy ground state of repulsive three-body contact interactions. Systems with internal degrees of freedom also allow for implementing artificial spin-orbit coupling. It is shown to give rise to a variety of incompressible states. In the second part of the thesis, we concentrate on condensed system. Bose-Einstein condensates with spin-orbit coupling are shown to have a degeneracy on the mean-field level, which is lifted by quantum and thermal fluctuations. The system becomes experimentally feasible in three dimensions, where the condensate depletion remains finite, and thus allow for an experimental observation of this order-by-disorder mechanism. Finally, we study the influence of Abelian and non-Abelian gauge fields on the quantum phase transitions of bosons in a square optical lattice. Re-entrant superfluid phases and superfluids at finite momenta are interesting properties featured by such systems.
Synthetic gauge fields for ultracold neutral atoms-engineered using the interaction between laser fields and the atoms' internal 'spin' degrees of freedom-provide promising techniques for generating the large (synthetic) magnetic fields required to reach the fractional quantum Hall (FQH) limit in quantum gases, bosonic or fermionic alike.Because neutral atoms can move in a nearly disorder-free environment and they have extremely simple contact interactions, the resulting FQH states would be revealed in their most essential form.Moreover, bosonic FQH states represent a new frontier and have never been seen in any setting.Going beyond electromagnetismʼs conventional scalar gauge field, it is possible to create more general non-Abelian gauge potentials.When these are spatially uniform, they are equivalent to spin-orbit coupling familiar in material systems, and can lead to cold atom analogs of topological insulators and topological superconductors.In this tutorial, we introduce basic concepts underlying these gauge fields, making connections to the Aharonov-Bohm phase and geometric phase.We focus on the system of neutral atoms 'dressed' by multiple laser beams, where the eigenstates of the resulting Hamiltonian are known as dressed states.Synthetic gauge potentials arise from the unitary transformation required to express these dressed states in terms of the laser-free eigenstates.We discuss stability of laser-dressed atoms corresponding to the adiabatic condition and the probability of non-adiabatic transitions.Adopting both the semiclassical and quantum mechanical approaches, we demonstrate they agree in the suitable limit.We also analyze using both the conventional adiabatic picture and exact picture, where the kinetic energy is neglected in the former and retained in the latter picture.
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.
Abstract We consider a two-dimensional weakly interacting ultracold Bose gas whose constituents are two-level atoms. We study the effects of a synthetic density-dependent gauge field that arises from laser–matter coupling in the adiabatic limit with a laser configuration such that the single-particle zeroth-order vector potential corresponds to a constant synthetic magnetic field. We find a new exotic type of current nonlinearity in the Gross–Pitaevskii equation which affects the dynamics of the order parameter of the condensate. We investigate the rotational properties of this system in the Thomas–Fermi limit, focusing in particular on the physical conditions that make the existence of a quantized vortex in the system energetically favourable with respect to the non-rotating solution. We point out that two different physical interpretations can be given to this new nonlinearity: firstly it can be seen as a local modification of the mean field coupling constant, whose value depends on the angular momentum of the condensate. Secondly, it can be interpreted as a density modulated angular velocity given to the cloud. Looking at the problem from both of these viewpoints, we show that the effect of the new nonlinearity is to induce a rotation to the condensate, where the transition from non-rotating to rotating states depends on the density of the cloud.
This chapter discusses the physics of ultracold gases in ‘artificial’ magnetic fields. The analogy between electronic systems in the presence of external magnetic fields and ultracold gases in rotating traps leads to the familiar concepts of Landau Levels, the integer Hall effect, the fractional quantum Hall effect, and laughing states for neutral particles. The second part analyses the possibility of simulating Abelian and non Abelian Lattice Gauge theories with ultracold gases. Finally, the state-of-the-art in generating artificial magnetic fields for ultracold gases is reviewed.