Quasi-angular momentum of Bose and Fermi gases in

Quasi-angularmomentumofBoseandFermigasesinrotatingopticallatticesBrandonM.Peden,RajivBhat,MeretKramer,andMurrayJ.HollandJILA,NISTandDepartmentofPhysics,UniversityofColorado,BoulderCO80309-0440,USAE-mail:[e-mail]-angularmomentumisintroducedtolabeltheeigenstatesofaHamiltonianwithadiscreterotationalsymmetry.ThisconceptisrecastinanoperatorialformwherethecreationandannihilationoperatorsofaHubbardHamiltoniancarryunitsofquasi-angularmomentum.Usingthisformalism,thegroundstatesofultracoldgasesofnon-interactingfermionsinrotatingopticallatticesarestudiedasafunctionofrotation,andtransitionsbetweenstatesofdierentquasi-angularmomentumareidentied.Inaddition,previousresultsforstrongly-interactingbosonsarere-examinedandcomparedtotheresultsfornon-interactingfermions.Quasi-angularmomentumcanbeusedtodistinguishbetweenthesetwocases.Finally,anexperimentallyaccessiblesignatureofquasi-angularmomentumisidentiedinthemomentumdistributionsofsingle-particleeigenstates.arXiv:0707.0307v2[cond-mat.other]18Oct2007Quasi-angularmomentuminrotatingopticallattices21.IntroductionRecentexperimentalandtheoreticalstudiesofquantumgasesofultracoldatomshavebeensuccessfulinreplicatingthebehaviorofawideclassofcondensedmattersystems.Inparticular,Bose-Einsteincondensation(BEC)[1,2,3]andthedegenerateFermigas[4]havebothbeenobserved,andtherehavebeenextensivestudiesofsuperuidity,vortexformation[5],andtheBCS-BECcrossover[6,7].TheMott-insulatortosuperuidphasetransitionintheBose-Hubbardmodelhasbeenobservedinasystemofultracoldbosonsinanopticallatticeof[8,9].Inaddition,experimentsobservingnovelphasesofultracoldmatterinatomicmixturesofBoseandFermigasesinopticallatticeshavebeenperformed[10,11,12,13].Alongtheselines,focushasturnedtostrongly-correlatedeectsinsystemsfoundoutsidetherealmofcondensedmatter.Onepromisinglineofresearchintherealizationofthesesystemsisthatofcoupledarraysofopticalcavities,whichallowsingle-siteaddressing[14,15,16].AnotherisamappingbetweentheHamiltonianofelectronsconnedintwodimensionsinthepresenceofaconstanttransversemagneticeldandthatofarotatingatomicgas.ThishasledtovarioustheoreticalstudiespredictingfractionalquantumHalleect(FQHE)behaviorinrotatingBEC's[17,18,19,20].However,experimentallyreachingtheparameterregimesnecessarytoobservethisbehaviorisdicult[21].Opticallatticesoerasolutiontothisproblembyenhancingcorrelations.Opticallatticesareformedasstandingwavesofcounter-propagatinglaserbeamsandactasalatticeforultracoldatoms.Thesesystemsarehighlytunable:latticespacinganddepthcanbevariedbytuningthefrequencyandintensityofthelasers,andinteractionsbetweenatomscanbetunedviaaFeshbachresonance[22].SomerecentstudieshavemadedirectconnectionsbetweenFQHEphysicsandstrongly-interactingbosonsinopticallatticesinthepresenceofaneectivemagneticeld[23,24,25,26,27].Connectingthesetheoreticalstudiestoexperimentrequiresidentifyingobservablesthatcanactasexperimentalsignaturesforthepredictedphysics.Forinstance,linearresponsetheoryhasbeenappliedtocurrentowinthepresenceofa\potentialgradientinordertoobserveFQHEphysics[26],noisecorrelationanalysishasbeenappliedtobosonsinarotatingringlattice[28],andBraggspectroscopyhasbeenmentionedasaprobeforvortexstates[29].Acommonobservableinultracoldatomicgasesisthemomentumdistribution,measuredintime-of-ightexperiments.Inthispaper,wepresentanalyticalresultsforstrongly-interactingbosonsinrotatingopticallatticesthatconrmnumericalresultsinapreviouspaper[30]andextendthemtosystemsofnon-interactingfermions.Theseresultsarebasedonthenotionofquasi-angularmomentum,whichisaquantumnumberforsystemswithadiscreterotationalsymmetry.Quasi-angularmomentumisanalogoustoquasi-momentumforperiodictranslationallyinvariantsystems[31]andhasbeenpreviouslyusedinthecontextofrotatingringlattices[32,33,34]andcarbonnanotubes[35]tolabeleigenstates.Aformalismforthisquantumnumberforsecond-quantizedsystemsispresentedhereandappliedtogasesofbosonsandfermionsinrotatingopticallattices.Quasi-angularmomentuminrotatingopticallattices3Theresultspresentedidentifytransitionsbetweenstatesofdierentsymmetryforthegroundstateofarotatingsystem.Apossibleavenueforexperimentallydetectingtheseresultsviathemomentumdistributionofthegroundstateisalsopresented.Thequasi-angularmomentum,m,ofthegroundstateofanultracoldquantumgasofstrongly-interactingbosonsornon-interactingfermionsinarotatingringlatticeofNsitescanbemonitoredasafunctionofrotationspeed.Forthecaseofstrongly-interactingbosons,mcyclesthroughthevaluesm=nlmodN:l=0;1;2;:::,wherenisthenumberofparticles.Forthecaseoffermions,mcyclesthroughthevaluesm=nlmodN:l=0;1;2;:::foroddnumbersofparticlesandm=n(l+1=2)modN:l=0;1;2;:::forevennumbersofparticles.Asystemofnon-interactingfermionsistherebydistinguishablefromasystemofhard-corebosons.Similarbehaviorobainsforsystemsoffermionsintwo-dimensionalsquarelattices.Signaturesofthisquantumnumberinexperimentareobservableviathemomentumdistributionofthegroundstate.Forsingle-particlesystemsintwo-dimensionalsquarelatticegeometries,suchsignaturesincludetheexistenceofapea