FromRotatingAtomicRingstoQuantumHallStates M. Roncaglia,1,2 M. Rizzi,2 and J. Dalibard3 1DipartimentodiFisicadelPolitecnico, corsoDucadegliAbruzzi24, I-10129, Torino, Italy 2Max-Planck-Institutfu¨rQuantenoptik, Hans-Kopfermann-Str. 1, D-85748, Garching, Germany 3LaboratoireKastlerBrossel,CNRS,UPMC,E´colenormalesupe´rieure,24rueLhomond,75005Paris,France Considerable efforts are currently devoted to the preparation of ultracold neutral atoms in the emblematic stronglycorrelatedquantumHallregime. Theroutesfollowedsofaressentiallyrelyonthermodynamics,i.e. imposingtheproperHamiltonianandcoolingthesystemtowardsitsgroundstate. Inrapidlyrotating2Dhar- monic traps the roleof the transverse magnetic field is played bythe angular velocity. For particlenumbers significantlylargerthanunity,therequiredangularmomentumisverylargeanditcanbeobtainedonlyforspin- ningfrequenciesextremelyneartothedeconfinementlimit;consequently,therequiredcontrolonexperimental parametersturnsouttobefartoostringent. Hereweproposetofollowinsteadadynamicpathstartingfrom thegasconfinedinarotatingring. Thelargemomentofinertiaofthefluidfacilitatestheaccesstostateswith alargeangularmomentum,correspondingtoagiantvortex. Theinitialring-shapedtrappingpotentialisthen 1 adiabaticallytransformedintoaharmonicconfinement,whichbringstheinteractingatomicgasinthedesired 1 quantumHallregime. Weprovideclearnumericalevidencethatforarelativelybroadrangeofinitialangular 0 frequencies, the giant vortex state is adiabatically connected to the bosonic ν =1/2 Laughlin state, and we 2 discussthescalingtomanyparticles. n a PACSnumbers:73.43.-f,05.30.Jp,03.75.Kk J 8 While coherence between atoms finds its realization in Like in solid-state physics, most of the preparation pro- 2 Bose–Einsteincondensates[1–3],quantumHallstates[4]are cedures employed so far in rotating atomic ensembles ap- ] emblematicrepresentativesofthestronglycorrelatedregime. proached the GS by cooling down the system with a fixed s a ThefractionalquantumHalleffect(FQHE)hasbeendiscov- Hamiltonian. By contrast we explore in this paper an alter- g ered in the early 1980s by applying a transverse magnetic nativemethodthatconsistsinstartingfromaneasilyprepara- - fieldtoatwo-dimensional(2D)electrongasconfinedinsemi- ble state (typically uncorrelated), following a dynamic route t n conductor heterojunctions [5]. Since then, FQHE has never by changing an external parameter, and eventually obtaining a stoppedtointriguethescientificcommunityduetonon-trivial the desired state. This strategy has been successful for the u q transport properties and exotic topological quantum phases experimentalinvestigationofthesuperfluidtoMottinsulator . [6]. Such interest has also influenced the research in ultra- transitioninopticallattices[10]. Weproposetoimplementit t a coldatomicgases,whichinthelastdecadehavebeensuccess- toreachQuantumHallstateswiththefollowingsteps: (i)We m fullyexploitedasahighlycontrollableplaygroundforquan- engineer a Mexican-hat trapping potential by superposing a - tum simulations of many-body physics [3]. The large versa- standardharmonictrapwiththerepulsivepotentialcreatedby d tility of these setups allows one to confine atoms in 2D har- a“plug”laserbeam,whichisfocusedatthecenterofthetrap. n o monic traps and to impose an effective magnetic field either (ii) By stirring the gas, we prepare the N bosonic atoms in c by rapid rotation [7, 8] or by laser-induced geometric gauge agiantvortexstate, correspondingtothelowestenergystate [ potentials[9].Inprinciple,suchopportunityshouldallowone of the Mexican-hat potential for a given angular momentum 1 toexperimentallyexplorethebosonicversionofQHE,evenif L. (iii) The stirring is removed and the plug is adiabatically v unfortunatelyithasbeenhithertoelusive. switchedoff. (iv)Inthefinalharmonictrap,weobtaintheGS 3 From a theoretical point of view, a variety of interesting withtheinitiallyimpartedangularmomentumL,thankstoro- 9 groundstates(GS’s)havebeenidentifiedforBosegasesasa tational symmetry. We show that if L=N(N−1) then the 5 function of the effective magnetic field [7, 8]. At zero field, 2-body contact interactions drive the gas into the celebrated 5 . i.e., without rotation, the particles undergo Bose–Einstein bosonicν =1/2Laughlinstate[4]. 1 condensation [1, 2] and the atomic ensemble is superfluid. 0 1 Differentlyfromarigidbody,asuperfluidofNparticlesreacts Singleparticlephysics 1 to rotation with the formation of quantized vortices, whose : numberN increaseswiththerotationfrequency.Atlargefill- v φ i ingfactorν=N/Nφ (cid:38)10anorderedvortexlatticeisformed. In experiments with rotating atomic gases, particles are X Forν<10,thelatticemeltsbecauseofquantumfluctuations, usually trapped by a harmonic potential and stirred by time- r which signals the breakdown of the mean-field description varying magnetic field or auxiliary laser beams [7]. In the a and the access into the FQHE regime. The filling factor is framerotatingatangularspeedΩΩΩ=Ωz,theHamiltonianofa now better defined as ν =N/mmax with mmax the maximum singleparticleintheharmonictrapoffrequencies(ω,ω,ωz) angularmomentumoccupiedbysingleparticles.FQHEstates canbewrittenas areobtainedforvaluesofν oforderunity,whichcorrespond 1 M M toverylargetotalangularmomentaL∝N2. Htrap=2M(p−A)2+ 2 (ω2−Ω2)(x2+y2)+ 2 ωz2z2, (1) 2 Εm1002468!"!!""!!!"""!!!!""""!!!!!"""""!!!!!!"""""!!!!!!"""""!!!!!"""""!!!!!"""""!!!!!"""""!!!!!"""""!!!!!"""""!!!!!"""""!!!!!"""""!!!!!"""""!!!!!"""""!!!!!"""""!!!!!"""""!!!!!"""""!!!!!"""""!!!!!"""""!!!!!"""""!!!!!"""""!!!!!"""""!!!!!"""""!!!!!"""""!!!!!"""""!!!!!""""!!!!!""""!!!!""""!!!! "5 0 5 10 15 20 25 m (a) (b) (c) Figure1: Mexicanhatpotential:(a)Bosonicatomsareconfinedinacombinedtrapwith(i)anisotropicharmonicconfinementand(ii)the dipolepotentialcreatedbyablue-detuned,gaussianlaserbeamthatplugsthetrapcenterandpushestheparticlesawayfromthispoint. The resultingpotentialexhibitsaMexican-hatshape. (b)Underfastrotation,thesingle-particleenergyspectrumexhibitsaLandauLevelpicture (reddashes),wherelevelsarearrangedinquasi-degeneratemanifoldsasinthecaseofapurelyharmonictrapping(blueasterisks). Thelevel plothasbeendrawnforδ=0.09,α=3.0andw2=8.0,parametersthatwewilluseinthemany-bodyproblemforN=9.Thesingle-particle minimumofenergycanbeadjustedtoanydesiredvalueonangularmomentumbytuningthewaistandthepoweroftheplugbeam(m=8in thepresentcase).(c)InordertoenterFQHEregimeofthemany-bodyinteractingsystem,weproposetoswitchofftheplugbeam,eventually recoveringtheusualparabolicform. withA=MΩΩΩ×r=MΩ(−y,x,0). Inthefollowingwesup- Fig.1. Thiscanbedonebyshiningthecenteroftheharmonic posethatallrelevantenergiesaremuchsmallerthanh¯ω , so trapwithalaserbeampreparedinacircular,GaussianTEM z 00 that the motion along the z direction is frozen and the prob- mode[11].Whenthelaserfrequencyischosenlargerthanthe lem is effectively two-dimensional. In the limit of centrifu- atomicresonancefrequency(‘bluedetuning’),thelaserbeam gal deconfinement Ω → ω, the system is formally equiva- creates a repulsive dipole potential proportional to the light lent to bosons of charge q = 1 in uniform magnetic field intensity. The beam is chosen to be perpendicular to the xy B=∇∇∇×A=2MΩz. Fromnowon,weexpressenergiesand planeandthedipolepotentialisoftheform (cid:112) lengths in units of h¯ω and h¯/Mω, respectively. It is well U (x,y)=αexp(cid:2)−2(x2+y2)/w2(cid:3), (2) known [7] that the problem can be rewritten as two decou- w pledharmonicoscillatorsH =(2a†a+1)+δ(b†b−a†a), trap where w is the laser waist and α is proportional to the laser intermsofladderoperatorsa,b,andδ =1−Ω/ω isthefre- intensity. Thesumoftheharmonicpotential(x2+y2)/2and quencyoffset. Everystateislabeledbytheoccupationnum- ofU (x,y) has a bump in x=y=0 in the laboratory frame bern ,n ofthetwomodes,anditisdenotedas|ψ (cid:105).Note w a b nb,na whenα >w2/4. thatagaugefieldsimilartotheoneenteringinto(1)canalso At moderate intensities of the plug, as the ones employed be induced by geometric phases instead of rotation [9]. The inourpreparationscheme,theclassificationofsingle-particle schemeoutlinedinthepresentpapershouldworkequallywell energy eigenstates in terms of LL remains valid (see Fig. 1 in this case, the only significant difference being that (1) is and Methods). In the LLL the single-body energies are in now the single-particle Hamiltonian in the laboratory frame, goodapproximation: insteadoftherotatingframe. In the limit δ (cid:28)1, the quantum number na identifies dif- (cid:18) 2 (cid:19)−(m+1) ε =mδ+α 1+ . (3) ferentmanifoldscalledLandauLevels(LL).WithineachLL, m w2 thestates(labeledbyn )arequasi-degenerateduetothesmall b separationenergyδ. Thequantitym=n −n istheangular At fixed laser parameters α and w, the angular momentum b a momentumoftheparticle. InthelowestLandaulevel(LLL), m that minimizes εm is a decreasing function of the rotation na =0 and the one-body eigenfunctions assume the simple frequencyoffsetδ. Wedenotebyδm thevalueforwhichthe expression ψm,0(z) = √1 zme−|z|2/2, where z now denotes levelcrossingεm+1=εm occurs. TheLLLstatewithangular πm! momentummisthusthelowestenergystatewhenδ ischosen the position in the complex plane (z=x+iy), with energies intheintervalδ <δ <δ ,whosewidthis m m−1 E =mδ andangularmomentumm. m The first key feature of our proposition is to replace the (cid:18) 2 (cid:19)2(cid:18) 2 (cid:19)−(m+2) I =δ −δ =α 1+ . ordinary harmonic potential with a Mexican-hat one, like in m m−1 m w2 w2 3 Later on, we will be interested in choosing a specific value GS with L=L +1 (the interaction energy is zero in both Lau m=(cid:96)andinmaximizingthewidthI ofthestabilitywindow. cases),whichisaverystringentrequirement. (cid:96) This can be done, at fixed intensity α, by choosing w2 =(cid:96). HereweproposeadifferentpointofviewwheretheFQHE The central rotation frequency in the stability window for (cid:96) regime can be tackled from a dynamical perspective, with a thencorrespondsto two-stepprocedure. Thefirststepisaddressedinthissection and it consists in the preparation of a giant vortex state of N 1 (cid:96)+1(cid:18) 2(cid:19)−((cid:96)+2) particlesintheMexican-hatpotentialofFig. 1a,withthede- δc= (δ +δ )=2α 1+ . (4) (cid:96) 2 (cid:96)−1 (cid:96) (cid:96)2 (cid:96) siredangularmomentumL=LLau. Thispreparationiseasier thanthedirectproductionoftheLaughlinstate,thankstothe Noticethatforlargevaluesof(cid:96),wegetδc∝α(cid:96)−1,thusifwe favourableparametersensitivityoftheMexican-hatpotential. (cid:96) wanttokeepitsizable,wehavetochooseα ∝(cid:96). Morepreciselytheincreasedmomentofinertiaofthegasen- ablesonetoreachL∝N2 inarelativelybroadintervalofΩ. The second step involves the adiabatic transformation of the Many-bodyphysics giantvortexstateintheLaughlinstate,anditwillbeanalyzed inthenextsection. In the context of cold bosonic gases in the LLL subspace, Inthelimitcaseofnointeractions,everysingleparticlein two-particlesinteractionscanbemodelledbythecontactpo- theMexicanhatpotentialshouldrotateatangularmomentum tential (cid:96)=L/N. Thebosonsthencondenseinthegiantvortexstate H =c ∑δ(2)(z −z ), (5) (cid:34) (cid:35) 2 2 i j N i<j Ψ(N)((cid:96))= ∏z(cid:96) e−∑j|zj|2/2, (7) v i i=1 whose strength is given by the adimensional parameter c = √ 2 8πas/az, where as is the 3D s-wave scattering length and similartothosealreadyobservedin[13]andtheoreticallyan- (cid:112) az = h¯/Mωz is the size of the ground state in the strongly alyzedin[14]. Asshownintheprevioussection,thewindow confineddirection[7,8].WithinthekernelofH2theν=1/2 ofstabilityI[(cid:96)Lau]isoptimizedforw2=(cid:96)Lau. ForN =9the Laughlinstate constraint α >w2/4 imposes α >2. We choose in the fol- lowingα =3,whichleadstoδ ∈(0.081,0.101). (cid:34) (cid:35) Lau ΨLau= ∏(zi−zj)2 e−∑j|zj|2/2 (6) leaIdninthgetopreasegnrocuenodfisntateterawctiitohnsL,t=heLintercvaanlobfevdaeluteersmfoinreδd i<j Lau either from a Bogoliubov analysis or from exact diagonal- has the lowest total angular momentum L =N(N−1), or ization. The main role of the interactions is to deplete the Lau equivalentlytheangularmomentumperparticle(cid:96) =N−1. contribution of the mean angular momentum (cid:96) in favour of Lau We first recall the practical difficulties to attain the FQHE neighbouring ones (cid:96)±q, with q(cid:28)(cid:96). The Bogoliubov anal- regime via a thermodynamic route for a pure harmonic con- ysis is well suited for strong plugs (α > 1) where the de- finement in the xy plane. In a typical experiment with 87Rb pletion is small, while for intermediate regimes a full many- atoms(a =5nm),alongitudinalfrequencyω /2π =50kHz body numerical treatment is needed (see Methods). Thanks s z givesa =50nmandaninteractionparameterc =0.5.Then, to the angular momentum conservation, the exact diagonal- z 2 already for a modest number of particles, the Laughlin state ization can be performed in each L sector separately and the isreachedonlyforrotationfrequenciesΩextremelycloseto conjugatevariableδ simplyyieldstheenergyshiftsLδ. The the centrifugal limit ω. For N =9 and c =0.5, we find us- phase diagram as a function of (α, δ) is presented in Fig.2b 2 ingexactnumericaldiagonalizationthattheLaughlinstateis for N =9, c =0.5. It strongly supports the sketch drawn 2 the GS only for δ <5.510−3 (see figure 2a). This very before for non-interacting particles. In particular the com- Lau lowthresholdmakesitdifficulttotransferthedesiredangular puted ground state for a Mexican-hat potential with α = 3 momentum to the gas. Indeed when the stirrer consists in a possessestherequiredangularmomentumL fortheinter- Lau rotatinganisotropicpotentialε(x2−y2)/2,thecorresponding valδ ∈(0.084,0.105),veryclosetotheoneinabsenceof Lau anisotropyεmustbechosensmallerthanδ toavoidadynami- interaction.Thiscorrespondstoa∼10%frequencydifference calinstabilityofthecenter-of-massmotion[12].Onehasthus betweenΩandω,whichisnotablylargerthanthetypicalstir- torestricttoextremelyweakstirrers,withεinthe10−3range. reranisotropyneededtosetagasinrotation.Thisensuresthat Howeveronemustalsochooseε (cid:29)u,whereuisthestrength thepreparationofthegiantvortexstatewithL=L should Lau ofthestaticanisotropydefectu(x2−y2)/2,otherwisethegas beratherrobust. cannot be effectively set in rotation. Unfortunately, in real- It is important to stress that the Mexican-hat potential is istic traps the typical values of u are at least of the order of employedjustforthescopeofinjectingtherightquantityof 10−3. Consequently it is quite problematic to fulfil simulta- angularmomentumL ,andnotforproducingtheLaughlin Lau neouslythesevariousconstraints. Lastly, wementionthatin state itself. The situation considered here is thus completely suchathermodynamicalprocedurethetemperaturehastobe different from former proposals suggesting to find a tradeoff keptbelowδ, i.ethegapbetweentheLaughlinstateandthe betweenΩandα thatoptimizesthefidelitywiththeLaughlin 4 (a) 0.4 0.20 4 0.010 45 Lgs 56 0 0.3 7 9 8 00..000068 63 ∆00..1105 5433221145627486 1/2∆ N 0.2 910 ∆ / c2000...111000048 α = 0 ∆ 63 0.1 72 0.096 0.004 (cid:179)8910 4 5 6 7 8 9 10 72 N 0.05 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.002 α / N 0.000 0.00 (b) x 10-3 Α(cid:61)0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 25 Α (a) (b) 20 N Fiziagtuioren2in: PahtrausnecdatieadgrLaLmLfboarsNis=(m9≤an2dNc=2=180).i5s.pEexrfaocrtmdeiadgsoenpaal-- 〉/ 15 4 2 ratelyforeachsectorofL,thankstorotationalinvariance. Energies H 10 5 〈 6 are then shifted by the total angular momentum term Lδ to draw 7 thephaseboundaries. (a)Inabsenceoftheplugbeam, α =0, the 5 8 9 windowofstabilityforLaughlinangularmomentum(LLau=72)is 0 10 narrowandextremelyclosetodeconfinementlimit. (b)Conversely, 0.0 0.1 0.2 0.3 0.4 0.5 theregionwithLLauopensupanddriftsawayfromδ=0astheplug α / N power α is ramped up (at constant w2=8); the same happens for (c) otherlargeangularmomentaaroundit. Forlargevaluesofα (typi- 60 50 [0.0,0.5] callylargerthanunity),theGSfoundwhenvaryingδ areessentially 102 40 [0.1,0.5] T30 non-correlatedstates,whereallatomsaccumulateinthesamegiant 20 vortexstate.ConsequentlythetotalangularmomentumLundergoes 101 10 0 jumpsofsizeN,correspondingtotheadditionofonefluxquantum 4 5 6 7 8 9 10 toeachparticle. Fα 4 N 1 5 6 7 10-1 8 state[15]. 9 10 10-2 0.0 0.1 0.2 0.3 0.4 0.5 α / N Adiabaticevolution Figure 3: Finite size scaling analysis. Data collapse of (a) the Once the gas has gained the desired angular momentum gap and (b) interaction energy (cid:104)H2(cid:105) as a function of α for an ini- L=N(N−1) via equilibrating in the giant vortex state (7), tialvalueofµ=c2N1/2/(2π3/2)∼0.15,correspondingtoc2=0.5 the stirrer at frequency Ω can be suppressed. The situation forN=9. Theinsetinplot(a)showsthefinitesizescalingofthe becomesrotationallysymmetricandthetotalangularmomen- Laughlin gap in the harmonic case (α =0). Our estimates gives ∆ =0.097(1) in the thermodynamical limit. (c) Scaling of the tum is thus conserved. Then, the slow removal of the laser Lau plugwillresultinaredistributionofparticlesaroundthemean function Fα ≡∆−2|(cid:104)Ψ0|(∂Uw/∂α)|Ψ1(cid:105)| whose integral provides anestimationfortheadiabatictime(inset). Atsizableenoughα’s angularmomentum(cid:96)Lau byrepulsiveinteractions. Suchare- ((cid:38)0.1N)thecurvescollapsefromabove,givingatotaltimeT which distributionreachesaparadigmaticformintheunpluggedhar- isapproximatelyconstantwithN(”×”points).Alasthepresenceof monictrap,wheretheLaughlinstate(6)hasnointeractionen- apronouncedbumpforsmallα’sleadstoT ∝N (”+”points). Al- ergyanymore. Fromatechnicalpointofview,wenotethatin ternativestrategiesthatleadtoloweradiabatictimesarediscussedin absenceofstirringwecannowlookfortheGSofthegasin Fig.5. the laboratory frame, within the subspace of the L that had z beenimpartedtothecloudduringthestirringphase. knownfromexactdiagonalization,thechangingrateoflaser ThesystemwillfollowtheinstantaneousGSΨ iftheun- 0 intensityα and/oritsrescaledcrosssection σ ≡w2/(N−1) plugging path can be followed slowly enough to satisfy the adiabatic condition (cid:12)(cid:12)(cid:104)Ψ0|(∂H/∂t)(cid:12)(cid:12)Ψj(cid:11)(cid:12)(cid:12) (cid:28) ∆2j, where Ψj withtimet isdeterminedbythecondition represents an excited eigenstate of energy Ej of the instan- F dα+F dσ (cid:28)dt , (8) α σ taneous Hamiltonian H, and where ∆ =E −E [16]. We j j 0 have checked numerically that the most stringent constraint where F ≡∆−2|(cid:104)Ψ |(∂U /∂x)|Ψ (cid:105)| is the matrix element x 0 w 1 originates from the first excited state Ψ and we thus focus ofthepotentialvariationinx=α,σ. Theminimaltotaltime 1 our discussion on this state. Once the gap ∆ = E −E is T required for adiabaticity will then be the integral of those 1 0 5 (cid:82) functions along the chosen path, T = (F dα+F dσ). In aim,constrainedoptimizationtechniquescanbeimplemented α σ the following, we first consider the case where the waist is using the data of the vector (F ,F ), represented in Fig.5. σ α kept fixed, which is experimentally straightforward since it Experimentally, another effective way of reducing the adia- involvesonlyavariationofthelaserintensity;thenweaddress batic ramp time is to increase the interaction coupling con- thegeneralcaseofchangingofbothα andσ. stantc ,hencethegap,viaeitherFeshbachresonances[17]or 2 WehaveperformednumericalsimulationsforuptoN=10 atighterlongitudinalconfinementωz. Forarampofα only, particles,inaLLLtruncatedsingle-particlebasism≤2N,in ournumericalcalculationswithN=9giveT ≈65,43,20for order to test the validity of the adiabatic approximation (see c =0.33,0.5,1.0,respectively,correspondingtotheempiri- 2 Methods). ForthechosentestcaseofN=9,c =0.5,ramp- calscalinglawT ≈20c−1. 2 2 ing down the intensity from the initial value α =4.5 at con- Finally we briefly address the consequences of some of stantσ =1,weobtainT ≈43(inunitsofω−1). Suchavalue the unavoidable experimental imperfections on the proposed of T is a reasonable time in state-of-art experiments, estab- scheme. The two principal perturbations that we can fore- lishingthefeasibilityofourschemeforN=9,asopposedto see are the imperfect centering of the plug beam and the theprocedureinvolvingapurelyharmonicrotatingtrap. residual trap anisotropy. We model these defects by writ- (cid:48) The exponentially increasing dimension of the Hilbert ing the dipole potential created by the plug beam as U = w space and the strong correlations involved ward off going αexp(cid:2)−2[(x−v)2+y2]/w2(cid:3),andbyaddingthetermu(x2− much further than N =10 particles with the exact diagonal- y2)/2 to the single-particle Hamiltonian to account for the ization method. To infer the behavior of larger samples, we staticanistropicdefect.Herevanduaredimensionlesscoeffi- performed finite size scaling of the relevant energies using cientscharacterisingtheseimperfections. Thesetwocoupling the Bogoliubov approximation (see Methods). Our scheme termsbreaktherotationsymmetry: intheirpresence,thean- requires the preparation of the gas in the ring with α ∝ N gularmomentumisnotaconservedquantityanymoreandthe and σ = 1, for which the chemical potential goes as µ (cid:39) gaswillundergoacascadefromL=L downtostateswith Lau c N1/2/(2π3/2). TheLLLapproximationrequiresµ<2,and noangularmomentum,bypopulatingthefirstexcitedLL.To 2 workingatfixed µ impliesc ∝N−1/2. Wethendeducethat getaconservativeestimate,weimposetheverystringentcon- 2 theenergygapbehaveslike∆∝N−1/2andtheinteractionen- dition that the total angular momentum remains unchanged ergyas(cid:104)H (cid:105)∝N(seeMethods).WehaveplottedinFig.3(a)- overtheadiabaticramptime,andweestimatethecorrespond- 2 (b)thevariationsofthegap∆andtheinteractionenergy(cid:104)H (cid:105) ing constraint on u and v using time-dependent perturbation 2 usingc =1.5N−1/2. Theexpecteddatacollapseiswellveri- theory (see Methods). The constraint on u is certainly the 2 fiedforvaluesofα/N largerthan0.1. mostchallengingone. Wefindthatthemaximaltolerabletrap Afinite-sizescalingcanbeperformedalsoforthequantity anisotropy u (cid:47)2∆ /N ≈0.2c /N. Taking u∼10−3 as max Lau 2 F enteringtheadiabaticcondition(8)andweplottheresult arealistictrapanisotropy,wefindthatourschemeshouldbe α atfixedwaistσ inFig.3c.Thisfunctiontakesitslargestvalues operationalforatomnumbersuptoN =100forc =0.5. max 2 intheintervalα∈(0,α )withα =0.1/N. Wecantherefore c c decompose the adiabatic path into two successive parts. In thefirstpartthepluglaserintensityα isdecreasedfromα = Detection i 0.5/N down to α and this can be down adiabatically in a c relatively short time T1 =(cid:82)ααciFαdα ∼5, independent of the Oneofthesimplesttechniquestoprobecoldatomicsetups numberofparticles(insetofFig.3c). Inthesecondpart(0≤ consists in taking time-of-flight (TOF) pictures [3]. The ab- α ≤αc) the breakdown of the scaling ∆∝N−1/2 imposes a sorption image of the density profile expanded after releas- slowdowninthereductionoftheplugintensity.ThetimeT2= ingtheharmonicconfinementcontainsindeedusefulinforma- (cid:82)αcF dα needed for this second part actually show a linear tionsabouttheinitialsituationinthetrap. Inthespecificcase 0 α increasewithN,hamperingthefeasibilityformorethanafew ofbosonsintheLLLregime,thedensityprofileisself-similar tensofbosons. intimeandtheTOFpicturesimplymagnifiestheoriginalpar- The sequence of spatial density profiles during the time ticledistributioninthetrap[18]. Giventhedirectconnection evolution is depicted in Fig.4a. From such a sequence it is between single-particle angular momenta and orbital radius clear that the gas starts feeding the trap center at the end of (seeMethods),aTOFimageallowsonetocomputetheangu- thepath. Analternativestrategytorampdowntheplugcon- larmomentum. Theν =1/2LaughlinstatewithN particles sists on reducing its waist w while maintaining α constant, exhibitsafairlyflatprofileofdensity0.5insidearimofradius √ this being performed in practice (up to the diffraction limit) ∼ N. ObservingsuchTOFimageswouldbealreadyafirst usingamotorizedfocusingopticalelement. Thecorrespond- hintthatonehaseffectivelyreachedtheQHEregime. ingevolutionofthedensityprofileinthetrapisrepresentedin Multi-particlecorrelationsofferevenmoreinsightintothe Fig.4bandinourspecificcaseitcanbecoveredadiabatically many-body state. These correlations are directly accessible inhalftimewithrespecttotheabovesituation(seeFig.5). if one uses a detection scheme that can resolve individual Anaturalextensionofouranalysisistoconsiderasimulta- atomswithsub-micronresolution[19,20]. Alternativelythe neousrampingofα andσ,inordertominimizethetotalevo- two-bodycorrelationfunctioncanbetestedatshortdistances lution time while fulfilling the adiabaticity criterion. To this using the resonant photo-association of spatially close pairs 6 Figure 4: Density profile during adiabatic evolution (N=9). The leftmost panel corresponds to a giant vortex like structure, whereas the rightmost one depicts the flat disk shaped profile of the Laughlin state. In the upper row σ = 1 is kept constant while α = 1.,0.5,0.4,0.3,0.2,0.1,0. The last part of the ramp down procedure 0<α (cid:46)0.1 is the slowest, due to the large value of Fα in this re- gion(seeFig. 3c). Inthelowerrowwesqueezethelaserwaistσ =1.,0.5,0.25,0.125,0.025,0.00625,0.atfixedintensityα=1.:particles spread towards the inner part of the trap in a different way, corresponding in a lower value of F and faster allowed rates of change. For σ systemswithinLLL,densityprofilesaftertrapreleaseandtime-of-flightimagingwillsimplydisplayrescalingsofthesepictures. ofinitialδ. Production,observationandcontrolofanyonsisoneofthe mostintriguingquestsmotivatingtheconsiderableeffortsto- wardsFQHEregime. Anyonsarequasiparticleswiththepe- culiarpropertyofsatisfyingnoncommonbraidingruleswhen moving around each other. In the Laughlin case the anyonic excitationsarequasiholes∏i(zi−zqh)ΨLau,whichcanbepro- ducedandcontrolledbyimpinchinganarrowstronglyrepul- sivelaserbeamatpositionz asputforwardin[22].Afurther qh feature of our proposal is that addressing a giant vortex with (cid:96)≥N permits in principle to get a final state with a whole manifoldofanyonicquasiholesandtostudyitsexoticproper- ties. Figure5:Mapofadiabaticityrequirements.Absolutevalueofthe Acknowledgements vector(F ,F )isplottedinthecolouredmapforN=9, evidenc- σ α ingthelargevalueofFα atlargeσ =w2/(N−1)andsmallα, as We acknowledge fruitful discussions with T. Busch, J.I. wellasthemorefavorableconditionifoneusesareductionintime Cirac, N. Gemelke, M. Haque and G. Juzeliu¯nas. M. Rizzi ofthebeamwaist. ThetwopathsdescribedinthetextgiveT ∼40 (solidblueline)andT ∼20(dashedblueline).Superimposedwhite has received funding from the European Community’s Sev- arrowsrepresentthedirectionsofthevector(F ,F ). Thisplotcan enth Framework Programme (FP7/2007-2013) under grant σ α serveforconceivingmoreintricatepathswiththehelpofoptimiza- agreement no. 247687 (IP-AQUTE). M. Roncaglia was par- tiontechniques. tially supported by the EU-STREP Projects HIP (Grant No. 221889) and COQUIT (Grant No. 233747). J.D. acknowl- edgessupportbyIFRAF,ANR(BOFLproject)andtheEuro- [21].Theamountofproducedmoleculesisindeeddirectlyre- peanUnion(MIDASproject). latedtothecorrelationfunctiong(2)(0),whichisalsoindirect correspondencewiththeinteractionenergy(cid:104)H (cid:105)/c (Fig.3b). 2 2 SincetheLaughlinstatebelongstothekernelofH ,itspres- Methods 2 ence will be signaled by a strong suppression of two-body losses. Moreover,inastrictanalogywithsolidstatephysics, LLLapproximation. Freezingthelongitudinaldegreesoffree- wecanimagineanexperimenttomeasureFQHEplateausin dombyalargeωz,thesingle-particleHamiltonian(1)canberewrit- physicalquantities.Namely,byvaryingtherotationaloffsetδ tenindimensionlessunitsas inthegiantvortexpreparationstageitispossibletochangeL Htrap=(2a†a+1)+δ(b†b−a†a), bystepsofN,i.e. movethepenetratingmagneticfluxinunits ofsinglequanta. Removingnowtheplug,thesystemwillfall with a†=−∂z+z¯/2 and b†=−∂z¯+z/2 in terms of z=(x+iy). inasequenceofincompressibleFQHEstates:thefinalg(2)(0) Eigenstates with angular momenta l =m−p and energies εm,p = isexpectedtodisplayplateausatdiscretevaluesasafunction 2p+lδ are built by iteratively applying the ladder operators a† 7 √ and b† on the vacuum ψ0,0≡(cid:104)z|0,0(cid:105)=e−|z|2/2/ π, i.e. |m,p(cid:105)= forLLLstates.Thecoefficientsin(11)accountforpairshavingtotal (a√†)p(b†)m|0,0(cid:105).Theexplicitwavefunctionsare angular momentum m0 with null component in the center of mass p!m! frame.Thisleadstoasparsematrixformfortheinteractions,withan averagefillingperrowgrowingas∼0.2·N2.92.ForN=10particles zm−p p (cid:18)p(cid:19)(cid:18)m(cid:19) ψm,p=ψ0,0·√p!m!q∑=0(−1)q q q q!|z|2(p−q), aannddLle=ssLthLaaun=on9e0,hwouerneCePdU∼-ti1mGeboRnAaMsitnoglseto-croertehe3GHHamzildteosnkiatonps processortodiagonalizeasingleinstanceoftheproblem. which can be rewritten in terms of confluent hypergeometric func- tions U(−p,m−p+1,|z|2). Energy levels for small δ’s are or- Condensate in the ring. Given the noninteracting energies (3) ganized in quasi-degenerate manifolds called Landau Levels (LL), withaminimuminm=(cid:96),theGSofH1=∑mεmdm†dm isgivenby labelledbytheinteger pandseparatedbyanenergygap2. When thegiantvortex(7). Theangularmomentum(cid:96)getsdepletedbythe dealingwithmany-bodyproblems,theusualapproximationistocut insertionoftwobodyinteractions(11)infavourofthenearestones downthesingle-particleHilbertspacetothelowestLL(LLL)p=0 (cid:96)+q,(cid:96)−q,withq(cid:28)(cid:96). ForacondensateofN0 particlesinm=(cid:96), wherewavefunctions(apar√tfromGaussianweight)areanalyticalin themostdominanttermsinH2are z, being ψm,0 =ψ0,0·zm/ m!. This is well justified and valid if c (cid:34) (cid:35) tthheercmhoerme,ictahlepLoLtenmtiiaxlinrgemdauienstowtheleliunntedrearctthioenLtLermgapinvtahleuest2ro;nfgulry- H2= 4√π23(cid:96) N02+N0q∑(cid:54)=02βq†βq+2β−†qβ−q+βlβ−q+β−†qβq† , correlatedFQHEregimeisnegligible[23]. HereweshowthattheLLLapproximationremainsvalidevenin wherenewoperators√βq=d(cid:96)+q havebeendefinedandtheStirling thepresenceofapluglaser(2).ThematrixelementswithintheLLL approximationn!≈ 2πe−nnn+12 employed. Thecouplingc2 gets are renormalizedby(cid:96)−1/2 asacon√sequenceofthewavefunctionlocal- (cid:18) 2 (cid:19)−(m+1) ization on a ring of length 2π (cid:96). Eliminating N0 by the number (cid:104)m,0|Uw(r)|m,0(cid:105)=α 1+w2 , (9) operatorNˆ =N0+∑qβq†βq,theoverallHamiltonianH1+H2reads wtuhrbicahtiopnrotvhiedoeryt.heTheenerrogtyatisohnifatlslydusyemtomUetwricatUtwheonfilrystcoorudpelresofstpateers- H = Nε(cid:96)+g2N2+∑q (cid:0)ε¯q+gN(cid:1)βq†βq withthesameangularmomentumanddifferentLLlabels: g (cid:16) (cid:17) (cid:18) 2 (cid:19)−p +2N∑q βqβ−q+β−†qβq† , (12) (cid:104)m+(cid:104)mp,,p0||UUww||mm++pp,,pp(cid:105)(cid:105) (cid:39)∼ ((cid:104)m√,m0/|Uww2)|mp(cid:104),m0(cid:105),·0|U1w+|mw,20(cid:105). ,(10) wthiethexgp=on2e√nctπ2i3a(cid:96)lcaonndsε¯tqan≡t.εU(cid:96)+nqde−rεth(cid:96)e≈p2roαpqo2s/e(des(cid:96)c)a2lfinogrsαm∝all(cid:96)q=anNd−e1is, increasingNenhancestheimportanceofinteractionswithrespectto DuetotheexponentialdecaywithminEq.(9),thepluglaseraffects single-bodyenergies. mainlylowangularmomenta,localizedinsideacircularareaofra- √ Thequadraticbosonicmodel(12)canbeexactlysolved[25]by dsiimusilarmto<thwe.LLFoLrolanreg,ethwusthaelmenoestrgpyressheirfvtinogf hthigehderistLaLncies2qubiete- theBogoliubovtransformationβq=uqηq−vqη−†qwithu2q−v2q=1, bywhichitreads tween adjacent LLs. The inter-LL terms in the illuminated region √ mwi<thwth2eaLrLeriendduecxedp.bFyotrhmeapllrye,faotnoerismalwlo−w2e,dvatoryuinsegLexLpLonaepnptrioaxlliy- H =Nε(cid:96)+g2N2+12 ∑(cid:0)Λq−ε¯q−gN(cid:1)+∑Λqηq†ηq. q(cid:54)=0 q(cid:54)=0 mationonlyforthosemsuchthatthematrixelements(10)are(cid:28)2. In practice among the exact eigenstates φm,0 of Htrap+Uw, those Due to the positiveness of Λq, the GS |Φ0(cid:105) (an approximation to thatdiffersignificantlyfromtheunperturbedstatesψm,0haveavery the exact GS |Ψ0(cid:105)) is given by the vacuum of quasiparticle ex- smalloccupationinthemany-bodysolution: theglobalLLmixing citations, ηq|Φ0(cid:105) = 0, ∀q. The quasiparticle spectrum is Λq = ∑tiomn[s(1p−res|e(cid:104)ψntme,d0|(φsmee,0F(cid:105)|i2g).(cid:104)1ncm).(cid:105)]isboundedby1%forallthesimula- (cid:113)(cid:0)ε¯q+gN(cid:1)2−g2N2 and the occupation number of the state q is v2= 1(cid:16)ε¯q+gN−1(cid:17). Thedepletionisthefractionofparticlesout- Numerics. LLLapproximationreducestheparticles’degreesof q 2 Λq freedomtoone,theangularmomentumm,andthemanybodysys- sidethecondensate, temisthendescribedbytheFockbasis|n n ...(cid:105).TheHilbertspace   advearitloabrleeafliozreanusimngerleicspawrtiitchloeuitsaffufercthtienrgc0tuht1etoco0rre≤ctmde≤sc2riNptiionnoor-f N−NN0 = N1 q∑(cid:54)=0v2q= 21Nq∑(cid:54)=0(cid:113)(cid:0)q2q+2+Q2Q(cid:1)22−Q4−1 (13) theLaughlinstateanditslowestexcitations; occupationnumberin higherm’sneverexceedsnegligibleamounts. Evenwiththisstrong with Q2 =gN(eλ)2/2α. The expression for v2 converges as q−4 q reduction,thedimensionofthelargestHilbertsubspaceconsidered forhighmomenta,andnoinfrareddivergenciesappearsinceangular forN particlesgrowsas∼1.75·100.74N−2 (∼4·105 forN=10). momenta are quantize√d in integers. For large Q, Eq.(13) becomes Within the LLL approximation, central contact interactions of the 1−N /N≈QlogQ/( 2N)whichvanishesforN→∞onlyifQ∝ 0 form(5)couldbewrittenintermsofasingleHaldanepseudopoten- N1−ζ,withζ >0.InthesamelimittheGSenergyturnsouttobe tial[24]: (cid:18) gN gQ g(cid:19) H2= 4cπ2 m0∑m1m221m0(cid:115)(cid:18)mm01(cid:19)(cid:18)mm02(cid:19)dm†0−m1dm†1dm2dm0−m2 . (11) andthechemicalpEot≈enNtial ε(cid:96)+ 2 −3√2+4 Here we introduced the second quantization operator dm†,p, which µ= ∂E ≈gN(cid:34)1−e(cid:18)gN(cid:19)1/2(cid:35), (14) createsaparticleinψm,p,andusedthesimplifiednotationdm†,0≡dm† ∂N 6 α 8 whoseleadingtermistheexpectedresultfromtheGross–Pitaevskii WithrespecttoH(0),thestate|Ψ1LL(cid:105)hasthesameenergyas|Ψ (cid:105) trap −2 0 approachintheLLL.Workingatconstantµ<2,asrequiredbyLLL andthecouplingbetweenthesetwostatesisthusthedominantes- √ approximation, implies a scaling c2 =CN−1/2, withC=2 π3µ caperoutefrom|Ψ0(cid:105). Toestimatethecorrespondingrate,wecon- (see Fig.3). Moreover, α ∝N implies that Q∝N1/2, ensuring the centrateonthelastpartoftheadiabaticevolutionandwetake|Ψ0(cid:105) vanishingofthedepletionfraction. Theradialconfinementisrather equaltotheLaughlinstate,whereL=N(N−1). Thecouplingma- strong, since the standard deviation ∆q grows onl√y as N1/4, as de- trixelementisthen ducedfromthecalculation(∆q)2=∑qq2v2q/N≈ 2Q3/(3N). The energy gap to the first excitation |Φ1(cid:105)=η1†η−†1|Φ0(cid:105) with Γ1−L2L=|(cid:104)Ψ1−L2L|Hu|Ψ0(cid:105)|= 2u(cid:114)∑m(cid:104)Ψ0|nm,0|Ψ0(cid:105)= u2√L≈ u2N. thesameLreads m √ (cid:113) 2 2µ ∆=Λ1+Λ−1=2gN (Q−2+1)2−1≈ Q Since|Ψ1−L2L(cid:105)and|Ψ0(cid:105)havethesametrappingenergy,thedetuning |E1LL−E | originates solely from the difference in interaction en- and vanishes as N−1/2 just as c , i.e. the energy scale of the final −2 0 2 ergy.MorepreciselyalowerboundforthisdetuningistheLaughlin Laughlin state (Fig.3). Within such Bogoliubov analysis, it is also gap∆ ≈0.1c .Hence,forN=10andc =0.5,thenon-resonant possibletodeterminethescalingofmanyotherinterestingquantities: conditLiaounΓ1LL(cid:28)2 |E1LL−E |issatisfiedi2fthedefectamplitudeu e.g.theinteractionenergy(cid:104)H2(cid:105)=c2∂E/∂c2scalesasN. ismuchless−t2han10−−22. For0u=10−2,weexpectthepopulationin Robustness against trap defects. The main experimental de- |Ψ (cid:105)todecayinatimeontheorderof1/Γ1LL∼20. 0 −2 fects that may hinder our protocol are a residual static quadrupole anisotropy u and an off-centering v of the plug beam, which are Theoff-centeringdefectHvexpandedatfirstorderinvconnects described by the single-particle potentials Hu =u(x2−y2)/2 and |Ψ0(cid:105) only with states |Ψα(cid:105) whose energy detuning is equal to 1. Hv=αexp(cid:2)−2[(x−v)2+y2]/w2(cid:3)−Uw, respectively. Bothterms Thislargedetuningisfavourabletominimisethedepartureratefrom break the rotation symmetry and couple manifolds corresponding |Ψ (cid:105).Moreover,theinfluenceofthisdefectfadesawaytogetherwith 0 to different total angular momenta. We consider first the cou- theplugduringtimeevolution. Hence,repeatingasimilaranalysis pling Hu since it turns out to have the largest impact for prac- asforHu,weeventuallyfindthattheconditionΓα (cid:28)|Eα−E0|is tical conditions. Its second-quantized expression can be written safelyfulfilledwhenv<1(inunitsofthetraplength),whichisnot Hu=Hu(0)+Hu(1)+Hu(2)with averystringentconditioninpractice. Hu(0) = 4u ∑(cid:112)(m+1)(m+2)(dm†+2,pdm,p+H.c.), m,p Hu(1) = 2u ∑(cid:112)m(p+1)(dm†−1,p+1dm,p+H.c.), m,p [1] Anderson, M. H., Ensher, J. R., Matthews, M. R., Wieman, Hu(2) = 4u ∑(cid:112)(p+1)(p+2)(dm†,p+2dm,p+H.c.), C.E.&Cornell,E.A. ObservationofBose–EinsteinConden- m,p sationinaDiluteAtomicVapor. 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