From Rotating Atomic Rings to Quantum Hall States

2.2 MB·English

by  Marco Roncaglia

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
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"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
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