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Cold atom dynamics in non-Abelian gauge fields

A. Goÿ

Published Jan 1, 2007• 0 citations
OpenAlex

Abstract

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.

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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.
Gauge fields are central in our modern understanding of physics at all scales. At the highest energy scales known, the microscopic universe is governed by particles interacting with each other through the exchange of gauge bosons. At the largest length scales, our Universe is ruled by gravity, whose gauge structure suggests the existence of a particle-the graviton-that mediates the gravitational force. At the mesoscopic scale, solid-state systems are subjected to gauge fields of different nature: materials can be immersed in external electromagnetic fields, but they can also feature emerging gauge fields in their low-energy description. In this review, we focus on another kind of gauge field: those engineered in systems of ultracold neutral atoms. In these setups, atoms are suitably coupled to laser fields that generate effective gauge potentials in their description. Neutral atoms 'feeling' laser-induced gauge potentials can potentially mimic the behavior of an electron gas subjected to a magnetic field, but also, the interaction of elementary particles with non-Abelian gauge fields. Here, we review different realized and proposed techniques for creating gauge potentials-both Abelian and non-Abelian-in atomic systems and discuss their implication in the context of quantum simulation. While most of these setups concern the realization of background and classical gauge potentials, we conclude with more exotic proposals where these synthetic fields might be made dynamical, in view of simulating interacting gauge theories with cold atoms.
We demonstrate how to create artificial external non-Abelian gauge potentials acting on cold atoms in optical lattices. The method employs atoms with k internal states, and laser assisted state sensitive tunneling, described by unitary k x k matrices. The single-particle dynamics in the case of intense U2 vector potentials lead to a generalized Hofstadter butterfly spectrum which shows a complex mothlike structure. We discuss the possibility to realize non-Abelian interferometry (Aharonov-Bohm effect) and to study many-body dynamics of ultracold matter in external lattice gauge fields.
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.
Gauge fields play a prominent role in modern physics from electromagnetic theory to standard model of particle physics. These are cutting edge mathematical tools to understand fundamental forces and interaction between sub-atomic particles. Though gauge fields are considered natural tools of study in high-energy physics, they can be extended to low-energy condensed matter physics to study the behaviour of quantum particles in special environments like electrons in a magnetic field. Apart from naturally occurring gauge fields, controllable artificial gauge fields can be used to tailor physical effects on quantum systems like neutral atoms, qubits, photons, anions etc. This thesis reports the study of laser-induced artificial non-Abelian gauge fields interacting with an ultracold atomic wave packet in free space. In sharp contrast to the Abelian gauge fields, spatially uniform non-Abelian gauge fields can induce non-inertial motion of a particle. This motion is locked to the pseudo-spin state of the system and induces spin-orbit coupling. These non-Abelian gauge fields are experimentally realized by adiabatic evolution of the state of the atoms in the degenerate subspace of the coupling Hamiltonians. We produce SU(2) and SU(3) non-Abelian gauge fields using doubly degenerate dark state subspace of tripod and triply degenerate dark state subspace of double-tripod atom-light coupling schemes, respectively. Using the SU(2) gauge field in one dimension, we demonstrate the function of an atomtronic Datta-Das transistor (DDT) where spin polarized atoms enter the gate region and exit with their spin orientation controlled by the gate parameter analogous to the gate voltage in conventional field effect transistor. We demonstrate the geometric nature of the spin rotation in our DDT by showing insensitivity of the result to the input velocity and velocity dispersion of the atomic cloud. We have also implemented a Ramsey interferometric sequence to extract the phase of the output state of the DDT showing that the spin rotation by the DDT is coherent. In two dimensions, the dynamics of the wave packet in SU(2) gauge field show Zitterbewegung oscillations revealing spin Hall effect characteristics of the system. Whereas the dynamics of the wave packet in SU(3) gauge field show oscillations with multiple frequencies emulating color-orbit coupling. These frequencies correspond to the energy separation between the eigenstates of the Hamiltonian. Using the SU(3) Hamiltonian, we show two paths of reaching the same final state from the initial state demonstrating the existence of additional ladder operators in the SU(3) system in addition to the well-known raising and lowering operators of angular momentum represented by SU(2) symmetry. With this study, we realize a system that can be used to study the rich physics of SU(3) Hamiltonians on table top experiments.
Gauge theories are ubiquitous in physics. Many intriguing phenomena in condensed matter physics owe to the action of the electromagnetic field, which is an Abelian gauge theory. The numerical treatment of many-body systems is inherently complex due to the exponentially growing size of the Hilbert space. While in one dimension an area law guarantees that numerical methods on classical computers can deal with strongly correlated systems, in higher dimensions the quantum simulation comes as the panacea for the many-body problem. The present thesis comprises the elaboration of experimentally feasible methods for the quantum simulation of dynamical Abelian gauge fields with ultra-cold gases of neutral atoms and the theoretical analysis of the related model Hamiltonians. As neutral atoms do not interact with external vector potentials like charged particles would do, the gauge fields have to be artificially engineered. The elements of a gauge theory that need to be replicated on a quantum simulator vary depending on the subject of investigation. The key ingredient at the root of many condensed matter phenomena, from the quantum Hall effect to superconductivity and chiral topological insulators, is the Berry phase. Whilst artificial static gauge fields have been widely explored, much remains to do regarding the realization of artificial dynamical gauge fields. In Chapter 3 we present a method based on the amplitude modulation of a one-dimensional optical lattice, which allows for an unprecedented degree of control over a wide range of parameters. The method also comprises the generation of a density-dependent complex phase, fundamental to the creation of anyonic pseudo-particles. The anyons are amenable of observation through interferometric measurement, realizable with the same experimental set-up. With regard to gauge theories, the Berry phase is just the visible tip of the iceberg. Below the waterline, there is more to consider in order to comprehensively reproduce a gauge theory, like the electric and magnetic fields in quantum electrodynamics. Moreover, a full account for the inherent symmetry is crucial to investigate phenomena proper of non-Abelian gauge theories in the context of high-energy physics, such as confinement. For this collection of topics, one can turn to lattice gauge theories. In Chapter 5, we consider a class of lattice gauge theories particularly suitable for quantum simulation, the Quantum Link Model. The study of the Abelian U(1) Quantum Link Model on a ladder geometry reveals a highly non-trivial phase diagram, featuring a symmetry-protected topological phase. In both Chapters, innovative solutions for the experimental realization of the model Hamiltonians are designed and proposed. To gain numerical access to the ground-state properties and the dynamics of the systems investigated we make use of state-of-the-art numerical methods based on Tensor Networks. The elements of the numerical analysis carried out throughout this thesis are presented in Chapter 6. In the last part we offer an outlook on research perspectives related to the topics discussed in the thesis.