When a neutral atom moves in a properly designed laser field, its center-of-mass motion may mimic the dynamics of a charged particle in a magnetic field, with the emergence of a Lorentz-like force. In this Colloquium we present the physical principles at the basis of this artificial (synthetic) magnetism and relate the corresponding Aharonov-Bohm phase to the Berry's phase that emerges when the atom follows adiabatically one of the dressed states of the atom-laser interaction. We also discuss some manifestations of artificial magnetism for a cold quantum gas, in particular in terms of vortex nucleation. We then generalise our analysis to the simulation of non-Abelian gauge potentials and present some striking consequences, such as the emergence of an effective spin-orbit coupling. We address both the case of bulk gases and discrete systems, where atoms are trapped in an optical lattice.

Abstract. 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.arXiv:1308.6533v3 [cond-mat.quant-gas]

We describe a simple technique for generating a cold-atom lattice pierced by a uniform magnetic field. Our method is to extend a one-dimensional optical lattice into the "dimension" provided by the internal atomic degrees of freedom, yielding a synthetic 2D lattice. Suitable laser-coupling between these internal states leads to a uniform magnetic flux within the 2D lattice. We show that this setup reproduces the main features of magnetic lattice systems, such as the fractal Hofstadter butterfly spectrum and the chiral edge states of the associated Chern insulating phases.PACS numbers: 37.10. Jk, 03.75.Hh, 05.30.Fk Intense effort is currently devoted to the creation of gauge fields for electrically neutral atoms [1][2][3][4]. Following a number of theoretical proposals in presence [5][6][7][8][9][10][11][12][13] or in absence of optical lattices [14][15][16][17][18][19][20], synthetic magnetic fields have been engineered both in vacuum [21][22][23][24][25] and in periodic lattices [26][27][28][29]. The addition of a lattice offers the advantage to engineer extraordinarily large magnetic fluxes, typically of the order of one magnetic flux quantum per plaquette [5-7, 10, 11], which are out of reach using real magnetic fields in solid-state systems (e.g. artificial magnetic fields recently reported in graphene [30][31][32]). Such cold-atom lattice configurations will enable one to access striking properties, such as Hofstadter-like fractal spectra [33] and Chern insulating phases, in a controllable manner. Existing schemes for creating uniform magnetic fluxes require several laser fields and/or additional ingredients, such as tilted potentials [6,10], superlattices [11], or lattice-shaking methods [9,13,[34][35][36][37]. Experimentally, strong staggered magnetic flux configurations have been reported [26,27], and very recently also uniform ones [28,29]. Besides, an alternative route is offered by optical flux lattices [38][39][40][41].In all of these lattice schemes, the sites are identified by their location in space. This need not be the case: the available spatial degrees of freedom can be augmented by employing the internal atomic "spin" degrees of freedom as an extra, or synthetic, lattice-dimension [42]. Here we demonstrate that this extra dimension can support a uniform magnetic flux, and we propose a specific scheme using a 1D optical lattice along with Raman transitions within the atomic ground state manifold (Fig. 1). The flux is produced by a combination of ordinary tunneling in real space and laser-assisted tunneling in the extra dimension creating the necessary Peierls phases. Our proposal therefore extends the toolbox of existing techniques to create gauge potentials for cold atoms.The proposed scheme distinguished by the naturally sharp boundaries in the extra dimension, a feature which greatly simplifies the detection of chiral edge states resulting from the synthetic magnetic flux [43][44][45][46][47]. We demonstrate that the chiral motion of these topological edge states can be directly visualiz...

We show that the adiabatic motion of ultra-cold, multi-level atoms in spatially varying laser fields can give rise to effective non-Abelian gauge fields if degenerate adiabatic eigenstates of the atomlaser interaction exist. 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 describe a new class of atom-laser coupling schemes which lead to spin-orbit-coupled Hamiltonians for ultracold neutral atoms. By properly setting the optical phases, a pair of degenerate pseudospin (a linear combination of internal atomic) states emerge as the lowest-energy eigenstates in the spectrum and are thus immune to collisionally induced decay. These schemes use N cyclically coupled ground or metastable internal states. We focus on two situations: a three-level case and a four-level case, where the latter adds a controllable Dresselhaus contribution. We describe an implementation of the four-level scheme for 87 Rb and analyze its sensitivity to typical laboratory noise sources. Last, we argue that the Rashba Hamiltonian applies only in the large intensity limit since any laser coupling scheme will produce terms nonlinear in momentum that decline with intensity.