Non-Abelian gauge fields

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2013 J. Phys. B: At. Mol. Opt. Phys. 46 130201

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J. Phys. B: At. Mol. Opt. Phys. 46 (2013) 130201 (3pp) doi:10.1088/0953-4075/46/13/130201

EDITORIAL

Non-Abelian gauge fields

Fabrice GerbierLaboratoire KastlerBrossel, CNRS, ENS,UPMC, 24 rue Lhomond,75005 Paris, France

Nathan GoldmanCenter for NonlinearPhenomena and ComplexSystems, Universite Librede Bruxelles (U.L.B.),1050 Brussels, Belgium

Maciej LewensteinICFO—Institut deCiencies Fotoniques,Parc Mediterrani de laTecnologia,08860 Barcelona, SpainICREA—InstitucioCatalana de Recerca iEstudis Avancats,08010 Barcelona, Spain

Klaus SengstockInstitut fur Laserphysik,Universitat Hamburg,D-22761 Hamburg,Germany

Building a universal quantum computer is a central goal of emerging quantum technologies,which has the potential to revolutionize science and technology. Unfortunately, this futuredoes not seem to be very close at hand. However, quantum computers built for a specialpurpose, i.e. quantum simulators, are currently developed in many leading laboratories.Many schemes for quantum simulation have been proposed and realized using, e.g., ultracoldatoms in optical lattices, ultracold trapped ions, atoms in arrays of cavities, atoms/ions inarrays of traps, quantum dots, photonic networks, or superconducting circuits. The progressin experimental implementations is more than spectacular.

Particularly interesting are those systems that simulate quantum matter evolving in thepresence of gauge fields.

In the quantum simulation framework, the generated (synthetic) gauge fields may beAbelian, in which case they are the direct analogues of the vector potentials commonlyassociated with magnetic fields. In condensed matter physics, strong magnetic fields lead to aplethora of fascinating phenomena, among which the most paradigmatic is perhaps thequantum Hall effect. The standard Hall effect consists in the appearance of a transversecurrent, when a longitudinal voltage difference is applied to a conducting sample. Forquasi-two-dimensional semiconductors at low temperatures placed in very strong magneticfields, the transverse conductivity, the ratio between the transverse current and the appliedvoltage, exhibits perfect and robust quantization, independent for instance of the material orof its geometry. Such an integer quantum Hall effect, is now understood as a deepconsequence of underlying topological order. Although such a system is an insulator in thebulk, it supports topologically robust edge excitations which carry the Hall current. Therobustness of these chiral excitations against backscattering explains the universality of thequantum Hall effect. Another interesting and related effect, which arises from the interplaybetween strong magnetic field and lattice potentials, is the famous Hofstadter butterfly: theenergy spectrum of a single particle moving on a lattice and subjected to a strong magneticfield displays a beautiful fractal structure as a function of the magnetic flux penetrating eachelementary plaquette of the lattice.

When the effects of interparticle interactions become dominant, two-dimensional gasesof electrons exhibit even more exotic behaviour leading to the fractional quantum Hall effect.In certain conditions such a strongly interacting electron gas may form a highly correlatedstate of matter, the prototypical example being the celebrated Laughlin quantum liquid. Evenmore fascinating is the behaviour of bulk excitations (quasi-hole and quasi-particles): theyare neither fermionic nor bosonic, but rather behave as anyons with fractional statisticsintermediate between the two. Moreover, for some specific filling factors (ratio between theelectronic density and the flux density), these anyons are proven to have an internal structure(several components) and non-Abelian braiding properties. Many of the above statementsconcern theoretical predictions—they have never been observed in condensed matter systems.For instance, the fractional values of the Hall conductance is seen as a direct consequence ofthe fractional statistics, but to date direct observation of anyons has not been possible intwo-dimensional semiconductors. Realizing these predictions in experiments with atoms,ions, photons etc, which potentially allow the experimentalist to perform measurementscomplementary to those made in condensed matter systems, is thus highly desirable!

Non-Abelian gauge fields couple the motional states of the particles to their internaldegrees of freedom (such as hyperfine states for atoms or ions, electronic spins for electrons,etc). In this sense external non-Abelian fields extend the concept of spin–orbit coupling(Rashba and Dresselhaus couplings), familiar from AMO and condensed matter physics.They lead to yet another variety of fascinating phenomena such as the quantum spin Halleffect, three-dimensional topological insulators, topological superconductors and superfluids

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J. Phys. B: At. Mol. Opt. Phys. 46 (2013) 130201 Editorial

of various kinds. One also expects here the appearance of excitations in a form of topologicaledge states that can support robust transport, or entangled Majorana fermionsin the case of topological superconductors or superfluids. Again, while many kinds oftopological insulators have been realized in condensed matter systems, a controlled way ofcreating them in AMO systems and studying quantum phase transitions between variouskinds of them is obviously very appealing and challenging.

The various systems listed so far correspond to static gauge fields, which are externallyimposed by the experimentalists. Even more fascinating is the possibility of generatingsynthetically dynamical gauge fields, i.e. gauge fields that evolve in time according to aninteracting gauge theory, e.g., a full lattice gauge theory (LGT). These dynamical gaugefields can also couple to matter fields, allowing the quantum simulation of such complexsystems (notoriously hard to simulate using ‘traditional’ computers), which are particularlyrelevant for modern high-energy physics. So far, most of the theoretical proposals concernthe simulation of Abelian gauge theories, however, several groups have recently proposedextensions to the non-Abelian scenarios.

The scope of the present focused issue of Journal of Physics B is to cover all of thesedevelopments, with particular emphasis on the non-Abelian gauge fields. The 14 papers inthis issue include contributions from the leading theory groups working in this field; webelieve that this collection will provide the reference set for quantum simulations of gaugefields. Although the special issue contains exclusively theoretical proposals and studies, itshould be stressed that the progress in experimental studies of artificial Abelian andnon-Abelian gauge fields in recent years has been simply spectacular. Multiple leadinggroups are working on this subject and have already obtained a lot of seminal results.

The papers in the special issue are ordered according to the date of acceptance. The issueopens with a review article by Zhou et al [1] on unconventional states of bosons withsynthetic spin–orbit coupling.

Next, the paper by Maldonado-Mundo et al [2] studies ultracold Fermi gases withartificial Rashba spin–orbit coupling in a 2D gas. Anderson and Charles [3], in contrast,discuss a three-dimensional spin–orbit coupling in a trap. Orth et al [4] investigate correlatedtopological phases and exotic magnetism with ultracold fermions, again in the presence ofartificial gauge fields. The paper of Nascimbene [5] does not address the synthetic gaugefields directly, but describes an experimental proposal for realizing one-dimensionaltopological superfluids with ultracold atomic gases; obviously, this problem is well situatedin the general and growing field of topological superfluids, in particular those realized in thepresence of non-Abelian gauge fields/spin–orbit coupling.

Graß et al [6] consider in their paper fractional quantum Hall states of a Bose gas withspin–orbit coupling induced by a laser. Particular attention is drawn here to the possibility ofrealizing states with non-Abelian anyonic excitations. Zheng et al [7] study properties ofBose gases with Raman-induced spin–orbit coupling.

Kiffner et al [8] in their paper touch on another kind of system, namely ultracoldRydberg atoms. In particular they study the generation of Abelian and non-Abelian gaugefields in dipole–dipole interacting Rydberg atoms.

The behaviour of fermions in synthetic non-Abelian gauge potentials is discussed byShenoy and Vyasanakere [9]. The paper starts with the study of Rashbon condensates (i.e.Bose condensates in the presence of Rashba coupling) and also introduces novel kinds ofexotic Hamiltonians. Goldman et al [10] propose a concrete setup for realizing arbitrarynon-Abelian gauge potentials in optical square lattices; they discuss how such syntheticgauge fields can be exploited to generate Chern insulators.

Zygelman [11], similarly as Kiffner et al [8], discusses in his paper non-Abelian gaugefields in Rydberg systems. Marchukov et al [12] return to the subject of spin–orbit coupling,and investigate spectral gaps of spin–orbit coupled particles in the realistic situations ofdeformed traps.

The last two papers, in contrast, are devoted to different subjects. Edmonds et al [13]consider a ‘dynamical’ density-dependent gauge potential, and study the Josephson effect ina Bose–Einstein condensate subject to such a potential. Last, but not least, Mazzucchi et al[14] study the properties of semimetal-superfluid quantum phase transitions in 3D latticeswith Dirac points.

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J. Phys. B: At. Mol. Opt. Phys. 46 (2013) 130201 Editorial

References

[1] Zhou X, Li Y, Cai Z and Wu C 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134001[2] Maldonado-Mundo D, Ohberg P and Valiente M 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134002[3] Anderson B M and Clark C W 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134003[4] Orth P P, Cocks D, Rachel S, Buchhold M, Le Hur K and Hofstetter W 2013 J. Phys. B: At. Mol.

Opt. Phys. 46 134004[5] Nascimbene S 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134005[6] Graß T, Julia-Dıaz B, Burrello M and Lewenstein M 2013 J. Phys. B: At. Mol. Opt. Phys.

46 134006[7] Zheng W, Yu Z-Q, Cui X and Zhai H 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134007[8] Kiffner M, Li W and Jaksch D 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134008[9] Shenoy V B and Vyasanakere J P 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134009

[10] Goldman N, Gerbier F and Lewenstein M 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134010[11] Zygelman B 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134011[12] Marchukov O V, Volosniev A G, Fedorov D V, Jensen A S and Zinner N T 2013 J. Phys. B: At.

Mol. Opt. Phys. 46 134012[13] Edmonds M J, Valiente M and Ohberg P 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134013[14] Mazzucchi G, Lepori L and Trombettoni A 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134014

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