Quantum simulation of the Abelian-Higgs lattice gauge theory with ultracold atoms

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Quantum simulation of the Abelian-Higgs lattice gauge theory with ultracold atoms
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2017 New J. Phys. 19 063038
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NewJ.Phys.19(2017)063038 https://doi.org/10.1088/1367-2630/aa6f37
PAPER
QuantumsimulationoftheAbelian-Higgs latticegaugetheorywith
OPENACCESS ultracoldatoms
RECEIVED
17February2017 DanielGonzález-Cuadra1,2,3,ErezZohar2 andJIgnacioCirac2
REVISED 1 ICFO—TheInstituteofPhotonicSciences,Av.C.F.Gauss3,E-08860,Castelldefels(Barcelona),Spain
7April2017
2 Max-Planck-InstitutfürQuantenoptik,Hans-Kopfermann-Straße1,D-85748Garching,Germany
ACCEPTEDFORPUBLICATION 3 Authortowhomanycorrespondenceshouldbeaddressed.
25April2017
E-mail:[email protected]
PUBLISHED
30June2017
Keywords:latticegaugetheory,quantumsimulation,ultracoldatoms
Originalcontentfromthis
workmaybeusedunder
thetermsoftheCreative Abstract
CommonsAttribution3.0
WepresentaquantumsimulationschemefortheAbelian-Higgslatticegaugetheoryusingultracold
licence.
Anyfurtherdistributionof bosonicatomsinopticallattices.ThemodelcontainsbothgaugeandHiggsscalarfields,andexhibits
thisworkmustmaintain interestingphasesrelatedtoconfinementandtheHiggsmechanism.Themodelcanbesimulatedby
attributiontothe
author(s)andthetitleof anatomicHamiltonian,byfirstmappingthelocalgaugesymmetrytoaninternalsymmetryofthe
thework,journalcitation
andDOI. atomicsystem,theconservationofhyperfineangularmomentuminatomiccollisions.Byincluding
auxiliarybosonsinthesimulation,weshowhowtheAbelian-HiggsHamiltonianemergeseffectively.
Weanalyzetheaccuracyofourmethodintermsofdifferentexperimentalparameters,aswellasthe
effectofthefinitenumberofbosonsonthequantumsimulator.Finally,weproposepossible
experimentsforstudyingthegroundstateofthesystemindifferentregimesofthetheory,and
measuringinterestinghighenergyphysicsphenomenainrealtime.
1.Introduction
Thestudyofnaturalsystemsgovernedbythelawsofquantummechanicsisacomplicatedtask.Mostofthe
usualtheoreticalandnumericaltechniquesemployedtotacklethebehaviorofquantumsystemsbecomerapidly
overwhelmedoncetheirsizestartsincreasing.Inparticular,thecomputationalresourcesrequiredtosimulatea
quantummechanicalsystemusingaclassicalcomputerscaleexponentiallywiththenumberofitsconstituents.
ThissituationledRichardFeynmantothinkoftheconceptofaquantumsimulator[1].Accordingtohisidea,
suchadevicewouldbegoverneditselfbythelawsofquantummechanicsand,exploitingthisfact,couldbeused
tostudyotherquantummechanicalsystemsmoreefficiently.Feynman’sideawasformalizedasanalgorithm
thatcouldbeprocessedbyauniversalquantumcomputer[2].Thisisreferredtoasadigitalquantumsimulation,
sinceitisbasedonapproximatingthesimulatedsystem’sdynamicsbyapplyingadiscretesetofoperations
(quantumgates)onahighly-controllablequantumdevice.Quantumcomputersareveryinterestingfroma
theoreticalpointofview,since,inprinciple,theycouldsimulateanyotherphysicalsystemusingonly
polynomialresources,providedthatcertainlocalityconditionsarefulfilled[2].Fromtheexperimentalside,
however—andinspiteofthegreatadvancesoverthelastdecadesregardingthemanipulationofmicroscopic
quantumsystems—apracticalimplementationofafullyoperationalquantumcomputerisstillalong-
termgoal.
Notwithstanding,havingthesecontrollabledevicessowellstudiedandreachablemakesquantum
simulationareality,bymeansofsimplerquantumdevicesthatcanperformcalculationsbeyondthescopeof
classicalmachines.Inparticular,theso-calledanalogquantumsimulators,althoughmuchmorelimitedthanan
universalsimulator,canbeusedtostudyabroadrangeofquantumsystemswithinthereachoftoday’s
technologicalprogress[3,4].Analogsimulatorsactbymappingthedegreesoffreedomofthesimulatedsystem
tothoseofthesimulatingone.Thelattercanbecontrolledinthelaboratoryanditsdynamicscanbetailored(in
particular,thecorrespondingHamiltonian)tobeequivalenttothoseofthesystemwearetryingtostudy.This
©2017IOPPublishingLtdandDeutschePhysikalischeGesellschaft
NewJ.Phys.19(2017)063038 DGonzález-Cuadraetal
allowsustoobtaininformationaboutsystemsthatcannotbeaccessedexperimentally,byinvestigatingothers
forwhichstatepreparationandmeasurementsaremucheasiertasks.
Amongtherelevantplatformsthatcanserveasquantumsimulators,bothanaloganddigital,manyatomic
andopticalsystemsstandoutduetotheirremarkableexperimentalcontrollability.Someexamplesinclude
ultracoldatoms[5–9],trappedions[10–14],photonicsystems[15]andRydbergatoms[16].Ultracoldatomsin
opticallattices,inparticular,presentthepossibilityofrecreatingmanydifferentinteractions—suchasnearest-
neighbor,long-rangeforces,on-siteinteractions,etc—allowingforthesimulationofbothcondensedmatter
andhighenergyphysicsmodels.Solid-statesystems,suchasquantumdots[17–19]orsuperconductingcircuits
[20,21],alsoshowprominentresultsthatmaketheminterestingcandidatestoperformquantumsimulations.
Usingtheseideas,manycondensedmatterHamiltonianshavebeenconsideredforquantumsimulations,
someofthemevenrealizedexperimentally.Someexamplesincludespinsystems[14,22–24],suchastheIsingor
Heisenbergmodels;theBose–Hubbardmodel[5,25];theTonks–Girardeaugas[26],orcopper-oxide
superconductors[18].Externalgaugepotentialscanbesimulatedaswell,allowingforthestudyofthefractional
quantumHalleffect[27]andothertopologicalphenomena[28–31].
Quantumsimulationsofhighenergyphysicsmodels,althoughmoredemandingthantheircondensed
mattercounterparts,arealsopossible.SomeexamplesincludethesimulationoftheDirac[32–34]andMajorana
equations[35],theCasimirforce[36],theSchwingermechanism[37]ortheoscillationsofneutrinos[38,39].
Simulationsofquantumfieldtheories[40,41],gravitationaltheories[42]andblackholes[43,44]havebeen
proposedinrecentyears,someofthemrealizedexperimentallyaswell[45].
Withinhighenergyphysics,gaugetheoriesareparticularlyrelevant,sincetheylieinthecoreoftheStandard
Modelofparticlephysics[46–49];dynamicalgaugedegreesoffreedomareintroducedinquantumfieldtheories
toexplaintheinteractionsbetweenthebasicconstituentsofmatter,suchasquarksorelectrons.Many
techniqueshavebeendevelopedtostudysuchtheories.Someofthemrelyonperturbativeexpansionsaround
smallcouplingconstants.However,thesemethodslosetheirvalidityifonetriestoapplythemforthestudyof
non-perturbativephenomena,wheretherelevantcouplingconstantpresentslargevalues.Thisisthecaseforthe
interactionstrengthbetweenseparatedquarks,whichgrowswiththedistancebetweenthem(runningcoupling)
[46,47].Atshortdistances,theinteraction’scouplingconstantpresentssmallvalues(asymptoticfreedom)[50],
andtheperturbativemethodscanbeapplied.Atlongdistances,however,thegrowthintheinteractionstrength
givesrisetotheconfinementmechanism,implyingthatnofreequarkscanbefoundinnature,aclaimthatis
supportedbytheexperiments[47].Todealwiththisandothernon-perturbativephenomena,thelattice
formulationofgaugetheorieswasdeveloped[51,52],obtainingaproperframeworktoperformnumerical
simulationsonthesetheories—usingMonteCarlomethods,inparticular[53].
Thesetechniquesprovidedagreatadvanceintheunderstandingofparticlephysicsduringthelastdecades.
However,theypresentsomelimitationswhentheyareappliedtocertaincases.Anexampleisthesignproblem,
whichappearsinregimeswithafinitechemicalpotentialforfermionicparticles[54],andbecomesproblematic
forthestudyingofdifferentphasesofQCD,suchasthequark-gluonplasmaorthecolor-superconductingphase
[55].Theanalysisofreal-timedynamicsisalsolacking,sinceMonteCarlosimulationsonlyallowtocalculate
Euclideanspace–timecorrelationfunctionsinimaginarytime(afteraWickrotation).
Inthiscontext,twodifferentapproacheshavebeenrecentlyproposedtostudylatticegaugetheoriesinregimes
thatcannotbeaccessedusingprevioustechniques.Oneinvolvestheapplicationofmethodsbasedontensornetworks
[56–69].Anotherconsistsonperformingquantumsimulationsusinglowenergyquantumsystems.Incontrasttothe
caseofcondensedmattermodels,latticegaugetheoriesaremorecomplicatedtosimulateusinglowenergysystems,
suchasultracoldatoms,sincethegaugeandLorentzsymmetriesarenotnaturallypresentinthelatter.Inaddition,
simulatingbothgaugeandmatterfieldsusuallyrequirestheuseofbosonicandfermionicatoms,whichincreasesthe
experimentalcomplexity.Anyattempttosimulatethesestheoriesmusttakethesefactsintoaccount[70,71].
Someexamplesofsimulationproposalsforgaugetheoriesusingultracoldatomsincludeboththecontinuous
[72]andlatticeversionofQED,focusingespeciallyonthelattercase.Inparticular,analogsimulationsforcQED
wereproposedbothintheabsence[73,74]andpresenceofdynamicalmatter[75–78],wheregaugeinvariance
emergesasaneffectivesymmetry.RealizationsofU(1)gaugetheorieswithdynamicalorbackgroundHiggsfields,
usingeffectivegaugeinvariance,werediscussedin[79–81].In[70,82,83]thegaugesymmetryisobtainedexactlyby
mappingittoaninternalsymmetryoftheatomicsystem.Otherquantumsystemsthathavebeenconsideredfor
quantumsimulationsoflatticegaugetheoriesaresuperconductingcircuits[84,85],trappedions[86,87]and
Rydbergatoms[16,88].Discretesymmetrygroupshavebeenconsideredaswell,bothforanalog[70,89,90]and
digitalsimulations[91,92],aswellasnon-abeliantheories[93–100].Ageneralformalismfordigitalsimulationsof
latticegaugetheorieswithultracoldatomscanbefoundin[91].
Recently,thefirstexperimentalrealizationofaquantumsimulationofalatticegaugetheorywasperformed
[101],wherethereal-timedynamicsoftheSchwingermodelweresimulatedonafew-qubittrapped-ion
quantumcomputer.Thisexperimentopenedthedoorforstudyinghighenergyphysicsbeyondthecapabilities
ofclassicalsimulations.
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NewJ.Phys.19(2017)063038 DGonzález-Cuadraetal
ThepresentworkisdevotedtothequantumsimulationoftheAbelian-Higgslatticegaugetheory,which
containsbothgaugeandHiggsscalarfields,usingultracoldbosonicatomsconfinedinopticallattices.Thiswill
bedonebymappingatomicsymmetriestolocalgaugeinvariance.Thepaperisorganizedasfollows.Insection2,
weintroducetheAbelian-Higgsmodel,whichisusedastoymodelinhighenergyphysicstostudytheHiggs
mechanism.Thismodeladmitsadiscretizeddescriptionasalatticegaugetheory,whosephasediagramshows,
apartfromthementionedmechanism,aconfinementphaseforthescalarHiggsfield.Weconsiderthe
Hamiltonianformulationofthemodel,moresuitedforquantumsimulationpurposes.Insection3,we
introducethenecessaryingredientstosimulatetheAbelian-HiggsHamiltonianusingultracoldatomsinoptical
lattices.First,thebasicpropertiesofthesimulatingsystemareexplained,startingwiththesecond-quantized
Hamiltonianthatdescribeanatomicsystemintheultracoldregime,aswellasthemostimportantexperimental
techniquesthatareneededtocontrolandmanipulateit.Then,weshowhowthedegreesoffreedomand
interactionsofthesimulatedmodel—and,inparticular,thelocalgaugesymmetry—canbemappedontothe
atomicsystem.Withthehelpofauxiliary‘hard-core’bosons,weshowhowthecompleteAbelian-Higgs
Hamiltonianemergeseffectivelyuptosomesmallcorrections.Finally,insection4,weintroducepossible
experimentstomeasureinterestinghighenergyphysicsphenomena,suchasthedynamicalbreakingofelectric
fluxlinesbetweenscalarcharges,effectthatisrelatedtoconfinement[102].
2.Thesimulatedmodel
Thespontaneousbreakingofasymmetryreferstoasituationcharacterizedbyagroundstatewhichisnot
invariantunderthesamesetoftransformationsasthetheory’sLagrangianorHamiltonian.TheHiggs
mechanism[103–106]describesthisphenomenoninthepresenceofa(local)gaugesymmetry.Itcanbeapplied,
forinstance,totheSU(2) ´ U(1)gaugegroup,associatedtotheelectroweakinteractionintheStandardModel
ofparticlephysics,explaininghowthecorrespondinggaugebosons,WandZ,acquireanon-zeromass[107].
Thebasicfeaturesofthemechanismcanbestudied,however,inamuchsimplercase,theAbelian-Higgsmodel
[46,108],involvingcomplexscalarfieldscoupledtoabeliangaugefields(appendixA.1).
TheAbelian-Higgsmodelcanbeformulatedasalatticegaugetheory[109–115],thisis,asafieldtheoryona
discretizedspace–time,invariantunderU(1)localtransformations.Thephasediagramofthetheoryshowsvery
interestingphenomena,relatedtotheHiggsmechanismandtheconfinementphenomenon(appendixA.2).
Forquantumsimulationpurposes,itismoreconvenienttoexpressthetheoryinaHamiltonian
formulation.Thismakesthemappingbetweenthesimulatingandthesimulatedsystem—inthiscase,ultracold
atomsinopticallattices—morestraightforward,sincethelatterisdescribedusuallyintermsofasecond-
quantizedmany-bodyHamiltonian.Here,weshallintroducetheAbelian-Higgsmodeldirectlyinthe
Hamiltonianformulation.Weobtainedtheformerbyapplyingthetransfermatrixmethodtothestandard
latticegaugeactionusedinthehighenergyphysicsliterature.Thedetailsofthesecalculationscanbefoundin
appendixB.
TheAbelian-Higgstheoryinvolvestwotypesoffields:gaugeandHiggsfields.Wewillformulateitonad
dimensionalspatialsquarelattice,withcontinuoustime,andwillfixthelatticespacinga=1.
Thegaugefieldsresideonthelinksofthelattice.Theyarerepresentedbytheunitaryoperators
Uˆn(q,k)º e-iqqˆn,k,residingonthelinks(joiningtheverticesnandn + kˆ,withkˆaunitvectorinoneofthed
orthogonaldirections).qisanintegerdenotingtheU(1)representation;wewillomititfromnowonandfocus
onthecaseq=1.Oneachlinkweintroduceanotheroperator,Eˆ ,withthecommutationrelations
n,k
[Eˆ , U† ] = d d U† . (1)
n,k n¢,k¢ k,k¢ n,n¢ n,k
Uˆ andUˆ† areunitaryoperatorsactingasladderoperatorswithrespecttoEˆ ,whichhasadiscretespectrum,
n,k n,k n,k
Eˆ ∣mñ = m ∣mñ , m Î . (2)
n,k n,k n,k n,k n,k
Wewillcallthelattertheelectricfieldoperator.Thecommutationrelations(1)definehowtheoperatorsUˆ and
n,k
Uˆ† actontheeigenstatesofEˆ ,
n,k n,k
Uˆ ∣mñ = ∣m - 1ñ , Uˆ† ∣mñ = ∣m + 1ñ . (3)
n,k n,k n,k n,k n,k n,k
TheHiggsfields,fˆn º e-ijˆn,whichresideontheverticesofthelattice,areunitaryoperatorsaswell.Their
Hilbertspacesareverysimilartotheseofthegaugefields;wedefinetheoperatorQˆ —referredasthecharge
n
operator—obeyingthecommutationrelations
[Qˆ , fˆ †] = d fˆ†, (4)
n n¢ n,n¢ n
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NewJ.Phys.19(2017)063038 DGonzález-Cuadraetal
Figure1.Theplaquettetermcontainstheproductoffourgauge-fieldtermscorrespondingtothefourlinksofaplaquette.
havingQˆ ,again,adiscretespectrum.fˆ†andfˆ areunitaryoperatorsthatactasraisingandloweringoperators,
n n n
respectively,
fˆ ∣Qñ = ∣Q - 1ñ , fˆ†∣Qñ = ∣Q + 1ñ , (5)
n n n n n n
where∣Qñ isaneigenstateofthechargeoperatorQˆ .
n n
AlltheseingredientsappearintheAbelian-HiggsHamiltonian,
Hˆ = 2R12 åQˆn2- R22 å[fˆn†Uˆn†,kfˆn+kˆ + h.c.] + g22 åEˆn2,k - 21g2 å(Uˆn,iUˆn+iˆ,kUˆn†+kˆ,iUˆn†,k + h.c.),
n n,k n,k n,ik
(6)
wherethesumsinnandkrunovertheverticesandlinksofaspatiallatticeinddimensions,andthecoupling
constantsRandgareintroducedtorecoverthecontinuousAbelian-Higgsmodelinthelimita  0.The
Hamiltonian(6)iscomposedofanon-interactingpartwithlocalterms(chargeandelectricfield)forthevertices
andlinksofthelattice,andaninteractingpartthatincludesaplaquette-typeterm(figure1)thatcreateselectric
fieldexcitationsalongthefourlinksofaplaquette,andnearest-neighborvertexcouplings,mediatedbyan
excitationonthelinkthatjoinsthem.Thesecondrowinequation(6)corresponds,inparticular,tothepure-
gaugeKogut–SusskindHamiltonian[116].
IntheHamiltonianformulationoflatticegaugetheories,thegaugeisfixedinthetemporaldirection,
q = 0(appendixB).However,itisnotfixedinthespatialdirections.Therefore,theHamiltonianisstillgauge
n,0
invariantunderarestrictedsubsetoflocaltransformationsappliedtotheverticesofthespatiallattice,
Vn = eian. (7)
Inthisformalism,thegaugetransformationsaregenerated,likeanyothercontinuoussymmetry,by
operatorsGˆ thatcommutewiththeHamiltonianofthetheory,
n
[Hˆ, Gˆ ] = 0, (8)
n
defined,inthiscase,locallyforeachvertex.InthecaseoftheAbelian-HiggsHamiltonian(6),thegeneratorsGˆ
n
areexpressedas
Gˆn = å(Eˆn,k - Eˆn-kˆ,k) - Qˆn. (9)
k
ItiseasytocheckthatsuchoperatorcommuteswiththeHamiltonianforeveryvertexn.
TheeigenvaluesoftheoperatorsGˆ arecalledstaticchargesq ,taking,aswell,integervalues.TheHilbert
n n
spaceofthesystemisdividedintosectors,eachoneassociatedtoadifferentstaticchargeconfiguration,
 = ⨁({q }), (10)
n
{q }
n
suchthat
Gˆ ∣y({q })ñ = q ∣y({q })ñ, "∣y({q })ñ Î ({q }). (11)
n n n n n n
SincetheHamiltoniancommuteswiththegeneratorsofthegaugetransformations,thedynamicscannot
mixdifferentsectors.Iftheinitialstatebelongstoacertainsector,thesystemwillremaininthesamesector
throughitstimeevolution.TheaboveequationisknownastheGauss’slaw,anditiscrucialforthestructureof
thephysicalstatesofthesystem.
Ingeneral,thetotalHilbertspaceofthesystemwouldbetheproductovertheverticesandlinksofthelattice
ofeachinfinitedimensionallocalHilbertspace.However,Gausslaw(9)imposesaconstraintontheallowed
stateswheretheHamiltonian(6)acts.Letusfirstconsiderthepure-gaugeKogut–SusskindHamiltonian,forthe
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NewJ.Phys.19(2017)063038 DGonzález-Cuadraetal
Figure2.Thepure-gaugeexcitationsontopoftheemptystateconsistofclosedelectricfluxlines,createdbytheplaquetteinteraction
termintheHamiltonian(6).Sincetheplaquettetermisgauge-invariant,thesestatesaregaugeinvariantaswell.Thisisclearobserving
thattheGausslaw(9)isfulfilledoneachvertex,forcingtheclosedloopstructure.Inthefigure,positivevaluesoftheelectricfieldare
representedwitharrowspointingrightorupwards.
Figure3.Twoseparatedstaticchargesarejoined,inthestrongcouplinglimit,byanelectricfluxline.Ifthecouplingconstant
decreases,pure-gaugeloopswillappear.Ifsomelinkintheloopcoincideswiththeelectricfluxlinebetweenthecharges(inthe
oppositedirection),itwillresultinadeformationofthefluxline.
caseR=0,withinthezero-staticchargesector(q = 0"n).Inthestrongcouplinglimit(g  1),theground
n
stateofthesystemistheemptystate.Thisis,everylinkofthelatticeisaneigenstateoftheelectricfieldoperator
witheigenvalueequaltozero.Asthevalueofthecouplingconstantgdecreases,theemptystateisnolongerthe
groundstateofthesystem.Theplaquette-interactiontermcancreateexcitationscomposedofclosedelectric
fluxlines.Thegroundstate,andtherestofexcitedstates,willbesuperpositionofsuchloops,withanincreasing
numberofthemwhentheenergyofthesystemgrows(figure2).
Considernowagauge-invariantsectorwithnon-zerostaticchargesinsomevertices(figure3).Inthiscase,
theemptystateisnotthegroundstateofthesystemeveninthestrongcouplinglimit.Thereasonforthisisthat
Gausslaw(9)requiresanelectricfluxlinetoexistbetweeneverypairofstaticcharges,beingthegroundstatein
thestrongcouplinglimitcharacterizedbytheshortestofthosepaths.Ifthevalueofthecouplingconstantg
decreases,closedloopswillbeformedallaroundthelattice,deformingthefluxlinebetweenthestaticchargesif
theyshareacommonlinkwiththelatter(figure3).
Ifmatterispresentinthetheory,thefluxlineswillnotonlyfluctuatebutcanevenbreak.ForR ¹ 0,the
gauge-matterinteractiontermcancreatechargeexcitationsintwonearest-neighborvertices.Wecanfind,then,
statescharacterizedbyseparatedpairsofdynamicalchargesjoinedviafluctuatingfluxlines,whichwewillcall
mesons.Thedifferencewithrespecttothepure-gaugecaseisthat,now,thefluxlinescanbreak,resultingintwo
new—andshorter—meson-likestates(figure4).ThisphenomenonresemblesthebehaviorofQCD,where,due
totheconfinementmechanism,thefluxlinesbetweenaquark–antiquarkpaircanbreakthroughthecreationof
anewpairfromthevacuum(stringbreaking)[47].
Here,wehavedescribedsomebasicfeaturesofthislatticemodel.Thecompletephasediagramofthetheory
isdescribedinappendixA.2fordifferentdimensionsanddifferentrepresentationsofthesymmetrygroup.
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NewJ.Phys.19(2017)063038 DGonzález-Cuadraetal
Figure4.Ifameson-likeexcitationiscreatedalongtheelectricfluxlineofanexistingmesonicstate,thelattercanbreak,givingbirthto
twoseparatedandsmallermesons.
3.Quantumsimulation
WeshallnowdescribeaquantumsimulationschemeforthelatticeAbelian-Higgstheoryusingultracold
bosonicatomstrappedinopticallattices.Thegoalistoimposecertainconditionsonthishighlycontrollable
atomicsystem,suchthattheHamiltonianthatdescribesitmaps,approximately,theHamiltonianunderstudy
(6).Boththeparametersappearinginthelatter,aswellasthedegreeofapproximation,canbecontrolled
experimentally.Thisallowstostudy,forinstance,thegroundstateofthismany-bodyquantumtheoryinthe
interactingregime,whichisahardtasktoperformusingstandardanalyticalornumericalcalculations.
Thesimulationproposalinvolvesthreemainsteps.First,startingfromthemostgeneralmulti-species
Hamiltonianthatdescribesultracoldbosons,theexperimentalrequirementsthatarenecessaryforthe
simulationwillbepresented.Afterthat,thetransformationsrequiredformappingthedegreesoffreedomofthe
atomicsystemtothoseoftheAbelian-Higgstheorywillbeintroduced.Finally,wewillseehowthedesired
Hamiltonianiseffectivelyobtained,apartfromcorrectionterms,afterincreasingoneoftheenergyscalesofthe
systemcomparedtotherest.Mostofthesecorrectionscanbemadeassmallasrequiredbychangingthe
experimentalparametersoftheatomicsystem,keepingthefreedomtomovethroughthephasespaceofthe
simulatedtheory,andbyincreasingthenumberofatomsusedinthesimulation.
3.1.Ultracoldatomsinopticallattices
3.1.1.AtomicHamiltonian
Ultracoldatomicgasesconsistofneutralatomstrappedandcooleddowntoalmostabsolutezerotemperature,
wherequantumeffectsplayasignificantrole.Theyrepresentoneofthemostrelevantplatformsforthestudyof
collectivequantumphenomenainregimesthatareeitherhardtoaccessexperimentally,orwherenumerical
simulationsdonotprovidesufficientlygoodresults.Inparticular,theso-calledopticallattices—structures
madeofcounter-propagatinglasers—canbepreparedinawaythatemulatesaperiodiclatticestructurewith
atomsboundtothevertices[5,7,9].Withintheselattices,interactionsamongtheatomscanbetunedinorderto
recreateHamiltonianmodelsdescribingmany-bodysystems,bothincondensedmatterandinhighenergy
physics.Inthiswork,wewillconsiderthemostgeneralmulti-speciesbosonicHamiltonian[9],
Hˆ = ååta,bbˆ† bˆ + å å Ua,b,d,gbˆ† bˆ† bˆ bˆ , (12)
i,j i,a j,b i,j,k,l i,a j,b k,d l,g
a,b i,j i,j,k,la,b,d,g
wherethelatinindicesdenotethedifferentsitesofanopticallattice,andthegreekindicesrepresentdifferent
bosonicspecies(eitherdifferentkindsofatomsoratomswithdifferentinternallevels).Theoperatorsbˆ and
i,a
bˆ† arebosoniccreationandannihilationoperators,respectively,fulfillingthecommutationrelations
i,a
[bˆ , bˆ† ] = d d . (13)
i,a j,b ij a,b
Theparametersta,bandUa,b,d,gareconstrainedbyconservationlawsandtheirrespectivesymmetries,and
i,j i,j,k,l
canbefine-tunedusingexperimentaltechniquessuchas(optical)Feshbachresonances[7,117–119].Thefirst
termin(12)correspondstoahoppingprocessofindividualatomsalongthelatticesites,whilethesecondisa
two-bodycollisionprocess.Forthelatter,theconservationofthetotalangularmomentumFimplies,in
particular,thatthefollowingconditionmustbefulfilledineachcollision,
m (a) + m (b) = m (d) + m (g), (14)
F F F F
wherem (a)representsthehyperfineangularmomentumstatefortheatomicspeciesα.
F
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NewJ.Phys.19(2017)063038 DGonzález-Cuadraetal
Figure5.(a)Twosuperimposedopticallattices,orientedwitha45anglebetweentheirlinks,arerequiredtosimulateboththematter
andgaugedegreesoffreedom.Theverticesofonelattice(blueinthefigure)representtheverticesofthesimulatedone,whereasthe
verticesoftherotatedlattice(green)correspondwiththelinksofthelatter.The‘matter’latticeisfilledwithfourtypesofbosons(c),
twospeciespersite,alternatingonevenandoddvertices.Twoofthesebosonicspeciesrepresentdynamicalmatter(speciescandd),
andtheothertwoareauxiliarybosonsrequiredforthesimulation(specieseandf).The‘gauge’lattice,ontheotherhand,isfilledwith
twotypesofbosonsoneachsite(speciesaandb),withnodistinctionforevenandoddlinks.Thecharacteristicsoftheselatticesmust
beconfiguredsuchthatthereisnointeractionsbetweenbosonsfromdifferentsitesofthe‘gauge’lattice—using,forexample,deep
potentials—butallowingforinteractionsbetweenthemandthebosonsfromthenearest-neighborverticesofthe‘matter’lattice.This
isthecaseifthecorrespondingWannierfunctionshaveanon-vanishingintersection(b).
3.1.2.Experimentalrequirements
Oursimulationproposalfocusesonthe2+1dimensionalcase,whichisthefirst‘non-trivial’dimension,sinceit
containsplaquetteinteractions.Inparticular,weneedtwosuperimposedopticallatticestoperformalattice
gaugetheorysimulation,whoseminimacorrespondtotheverticesandlinksofthesimulatedlattice,respectively
(figure5(a)).Fillingandcombiningtwosuchlatticesispossibleinthetwo-dimensionalcase,however,it
becomesmorecomplicatedinthree-dimensions.Allthetheoreticalcalculationsarevalid,nevertheless,for
d + 1dimensions,withd  2.
TosimulatethecompleteAbelian-HiggsHamiltonian(6)weneedsixdifferentbosonicspecies(figure5(c)).
Twoofthem,aandb,areplacedonthelinksofthesimulatedlatticeandrepresentthegaugedegreesoffreedom.
Anotherpairofbosonssimulatesthescalarfieldsonthevertices(dynamicalmatter),onespeciesontheodd
vertices,c,andtheotheroneontheevenones,d.Thelasttwospecies,eandf,aredistributed,again,onoddand
evenvertices,respectively,actingasauxiliaryparticlesrequiredtoobtaintheplaquetteinteractionsinan
effectiveway,aswewillsee.Eachpairofbosonicspeciescorrespondstothesametypeofatom,buttheirinternal
leveldifferinthevalueofthehyperfineangularmomentum,m .Wewilldenotebothtypesofdynamicalbosons
F
withη,andbothauxiliaryoneswithχ,inthosesituationswhereadistinctionbetweenthehyperfineangular
momentumlevelisnotimportant.
Dyn.η Aux.χ
Odd c e
Even d f
3.2.PrimitiveHamiltonian
Fordimensionsd + 1,withd > 1,theHamiltoniansthatdescribelatticegaugetheories,suchastheAbelian-
Higgsone(6),includeplaquette-typeinteractionsmadeoutoftheproductoffourunitaryoperators(figure1).
Suchfour-bodyinteractionsarenotfoundintheultracoldatomicHamiltonian.Inordertosimulatethem,we
havetoobtainthesetypeoftermseffectively[70,71],aswewillexplaininlatersections.
WeshallcallprimitiveHamiltoniantheonefromwhichthedesiredAbelian-HiggsHamiltoniancanbe
obtainedeffectively,actingonlyonalow-energysector.Ithastheform
Hˆ = låNˆc(Nˆc- 1) +å[cˆ†Uˆ† cˆ + h.c.] + måEˆ2 + m¢åQˆ2+¢å[fˆ†Uˆ† fˆ + h.c.].
n n n n,k n+kˆ n,k n n n,k n+kˆ
n n,k n,k n n,k
(15)
7
NewJ.Phys.19(2017)063038 DGonzález-Cuadraetal
Thefirstrowcontainstheoperatorscorrespondingtoauxiliarybosonslivingontheverticesofthelattice.cˆ†
n
andcˆ arethebosoniccreationandannihilationoperators,respectively,andNˆc = cˆ†cˆ isthecorresponding
n n n n
numberoperator.Thesecondrowcontainsallthetermsof(6)exceptfortheplaquetteinteraction.Theprimitive
HamiltonianisinvariantaswellunderU(1)localtransformationsappliedontheverticesofthelattice.
WewillnowfocusonbringingtheatomicHamiltoniantotheprimitive(intermediate)form(15).Although
onlyanapproximateHamiltoniancanbeobtained,thelevelofapproximationcanbecontrolledbyusinga
differentnumberofatomsinthesimulation.
3.3.Operatortransformations
Thefirststeptowardsaquantumsimulationistoconstructaone-to-onemappingbetweenthedegreesof
freedomofboththesimulatingandthesimulatedsystems.Here,thisisachievedbydefiningtheoperatorsthat
appearintheAbelian-HiggsHamiltonian(6)intermsofbosoniccreationandannihilationoperators
correspondingtothedifferentatomicspecies.Forafinitenumberofatomstrappedontheverticesandlinksof
theopticallattice,thecorrespondenceisnotexact,inthesensethatthedefinedoperatorswillactonlyona
truncatedHilbertspace.However,theapproximationwillimprovewhenthenumberofatomsincreases,asthe
connectionbetweenbothsystemsbecomesexactintheinfinitenumberofatomslimit.
Ontheverticesofthelattice(bothevenandodds),weintroducetheoperatorsQˆ ,fˆ andfˆ†usingthe
n n n
second-quantizedoperatorshˆ andhˆ†forthedynamicalbosons,
n n
hˆ = fˆ (N + Qˆ )1 2, hˆ†= (N + Qˆ )1 2fˆ† (16)
n n 0,v n n 0,v n n
and
Qˆ = hˆ†hˆ - N , (17)
n n n 0,v
whereN isthedensityofηbosons.Withtheserelations,boththecanonicalcommutationrelationsforthe
0,v
bosonicoperatorsandthecommutationrelations(4)arefulfilled.Accordingtothisdefinition,theoperatorQˆ
n
isboundedfrombelowby-N .Forthisreason,itisnotexactlyequivalenttothedynamicalchargeoperator
0,v
thatappearsin(6),however,wecanthinkaboutitasthelatteractingonatruncatedHilbertspace—theone
spannedbythebasis∣mñwithm Î Ç[-N , ¥)—whereitisequivalenttothedynamicalchargeoperator.
0,v
Forthisreason,weusethesamenotationforboth,understanding,inthefollowing,thatwearereferringtothe
operatoractingonthetruncatedspace.
AnotherproblemconcerningthefinitenessofN isthat,byimposingthecommutationrelations(4),hˆ
0,l n
andhˆ†cannotbeunitary,whichalsomeansthattheyarenottheexponentialofaself-adjointphasejˆ ,
n n
fˆn = e-ijˆn.Thisisrelatedtothequantumphaseoperatorproblem[120–124],andliesinthefactthatQˆnis
boundedfrombelow.Aunitaryoperatoriswell-definedonlyinthelimitN  ¥,suchthatthecharge
0,v
operatorisnolongerbounded.
Weareworkingwithalarge,butfinite,numberofatomsN oneachvertex.Thereasonitisstillfine,froma
0,v
physicalpointofview,tousethesedefinitions—althoughtheyarenotformallycorrect—isthatinallthe
interactiontermsoftheHamiltonianthebosonicoperatorsappearinpairs,correspondingtonearest-neighbor
vertices,hˆ†hˆ .Therefore,onlyphasedifferencesbetweendifferentverticesarerelevant.Thephasedifference
n n+kˆ
isawell-definedquantity,sinceitiscanonicallyconjugatetoanumberoperator,Nˆ = Nˆc - Nˆd,whichisnot
boundedfrombelow.
Theoperatorsassociatedtothegaugefields,Eˆ andUˆ canbeobtainedfromthebosonicoperatorsonthe
n,k n,k
linksifwefirstrepresenttheminasimilarwayastheatomicoperatorsonthevertices,
aˆ = Uˆa ⎜⎛N0,l + Eˆa ⎟⎞1 2, bˆ = Uˆb ⎜⎛N0,l + Eˆb ⎟⎞1 2, (18)
n,k n,k⎝ n,k⎠ n,k n,k⎝ n,k⎠
2 2
whereUˆa andUˆb areloweringoperators,and
n,k n,k
Eˆa = aˆ† aˆ - N0,l, Eˆb = bˆ† bˆ - N0,l. (19)
n,k n,k n,k 2 n,k n,k n,k 2
Again,ifthecommutationrelations(1)arefulfilled,thecanonicalrelationsforthebosonicoperatorswillbe
satisfied.
Thetotalnumberofbosonsoneachlink,N isaconservedquantity,
0,l
N = aˆ† aˆ + bˆ† bˆ , (20)
0,l n,k n,k n,k n,k
since,asweshallsee,onlyproductsofbosonicoperatorsoftheformaˆ† bˆ andbˆ† aˆ willappearinthe
n,k n,k n,k n,k
Hamiltonian.ThisimpliesthatEˆa = -Eˆb º Eˆ .Weusethelatterastheelectricfieldoperatoronthelink
n,k n,k n,k
(n, k),whichcanalsobeexpressedas
8
NewJ.Phys.19(2017)063038 DGonzález-Cuadraetal
E = 1(aˆ† aˆ - bˆ† bˆ ). (21)
n,k 2 n,k n,k n,k n,k
Weconstructtheunitaryoperatorsactingonthelinksas
Uˆ† = Uˆa†Uˆb , (22)
n,k n,k n,k
satisfyingthecorrectcommutationrelations(1)withtheelectricfield(21).Notethat,inthiscase,theoperator
E isnotboundedfrombelow.Uˆ andUˆ† are,therefore,well-definedunitaryoperators,althoughtheyonly
n,k n,k n,k
actonatruncatedHilbertspacethatgrowswithN .
0,l
Finally,theproductoftwobosonicoperatorsonthelinkscanbewrittenas
aˆ† bˆ =Uˆ† ⎜⎛N0,l(N0,l + 2) - Eˆ (Eˆ + 1)⎟⎞1 2,
n,k n,k n,k⎝ 4 n,k n,k ⎠
bˆ† aˆ =⎜⎛N0,l(N0,l + 2) - Eˆ (Eˆ + 1)⎟⎞1 2Uˆ , (23)
n,k n,k ⎝ n,k n,k ⎠ n,k
4
whereweusedthepropertyUˆ f(Eˆ) = f(Eˆ + 1)Uˆ (seeappendixC).Inthefollowing,wewillassumethis
property—similarlyfortheoperatorsonthevertices,fˆ f(Qˆ) = f(Qˆ + 1)fˆ—todealwiththesenon-
commutingoperators.
Inordertosimplifythenotation,wedefinethenon-unitaryoperators
⎛ ⎞1 2
Fˆ† º⎜1 + 1 Qˆ ⎟ fˆ†,
n ⎝ N n⎠ n
0,v
⎛ ⎞1 2
ˆ† º⎜1 - 4 (Eˆ2 - Eˆ )⎟ Uˆ† , (24)
n,k ⎝ N (N + 2) n,k n,k ⎠ n,k
0,l 0,l
whichbecomethedesiredunitaryones,fˆ†andUˆ†,inthelimitsN  ¥andN  ¥,respectively.The
n n 0,v 0,l
connectionwiththeatomicoperatorsissummarizedinthefollowingrelations,
hˆ† aˆ† bˆ
Fˆ† = n , ˆ† = n,k n,k ,
n n,k
N N (N + 2) 4
0,v 0,l 0,l
hˆ bˆ† aˆ
Fˆ = n , ˆ = n,k n,k . (25)
n n,k
N N (N + 2) 4
0,v 0,l 0,l
3.4.Gauge-invariantinteractions
Localgaugeinvarianceisnotafundamentalsymmetryinasystemofultracoldatoms.Itisessential,however,for
thedescriptionoftheinteractionspresentinhighenergyphysics,aswehaveseen.Ifoneaimstocorrectly
describethelatter,gaugesymmetrymustbeimposedonthesimulatingsystem.Thiscanbedoneeitherina
digital[91]orinananalogsimulationframework.Forthelatter,therearetwodistinctwaystoachievethis[71],
onewheregaugeinvarianceisobtainedeffectively,asanemerging,lowenergysymmetry[73,74,76],and
anotherwhereitismapped,exactly,toafundamentalsymmetryofthesystem—suchastheconservationof
hyperfineangularmomentuminatomiccollisions.Inthiscase,asopposedtoaneffectivegaugesymmetry,
gaugeinvarianceisexact,makingthesimulationmorerobustagainstexperimentalimperfections.Usingthe
secondapproach,quantumsimulationsofbothabelianandnon-abelianlatticegaugetheorieshavebeen
proposed[70,93].Here,wewillemployittosimulatetheU(1)gaugeinvarianceoftheAbelian-Higgsmodel.
Inordertoobtainthecorrect‘gauge-invariant’interactions,thehyperfineangularmomentaofthesix
bosonicspeciesmustfulfillcertainconditions(figure6),
m (a) - m (b) = m (c) - m (d) = m (e) - m (f) º Dm. (26)
F F F F F F
Sinceonlycollisionprocessesthatconservethetotalangularmomentumareallowed,thisconditionselectsthe
typeoftermsthatappearintheatomicHamiltonian.Consider,forinstance,theinteractionsbetweena
dynamicalbosononavertexandagaugebosonononeofitsnearest-neighborlinks.Theconservationof
hyperfineangularmomentumallowsbothspecies-changingcollisions(figure7(a)),
dˆn†aˆn†,kbˆn,kcˆn+kˆ + cˆn†+kˆbˆn†+kˆ,kaˆn+kˆ,kdˆn+2kˆ, (27)
9