Superfluidity in atomic Fermi gases

0.27 MB·

by  Yi Yu

Superfluidity in atomic Fermi gases
YiYua,QijinChenb,∗
aCenterforMeasurementandAnalysis,ZhejiangUniversityofTechnology,18ChaowangRd.,Hangzhou310014,China
bZhejiangInstituteofModernPhysicsandDepartmentofPhysics,ZhejiangUniversity,38ZhedaRd.,Hangzhou310027,China
1
1Abstract
0
In a trapped atomic Fermi gas, one can tune continuouslyvia a Feshbach resonance the effective pairing interaction between
2
fermionicatomsfromveryweaktoverystrong. Asa consequence,thelowtemperaturesuperfluidityevolvescontinuouslyfrom
n
the BCS type in the weak interaction limit to that of Bose-Einstein condensation in the strong pairing limit, exhibiting a BCS-
a
JBEC crossover. In this paper, we review recent experimentalprogressin atomic Fermi gases which elucidates the nature of the

4superfluidphaseastheinteractioniscontinuouslytuned. Ofparticularinterestistheintermediateorcrossoverregimewherethe
1s-wavescatteringlengthdiverges.Wewillpresentanintuitivepairingfluctuationtheory,andshowthatthistheoryisinquantitative
agreementwithexistingexperimentsincoldatomicFermigases.
]
s
Keywords:
a
gAtomicFermigases,superfluidity,BCS-BECcrossover,pairingfluctuations
-PACS:03.75.Hh,03.75.Ss,74.20.-z
t
n
a
u
1. Introduction temperature[4]. ThediscoveryofhighTcsuperconductivityin
q
1986gaveastrongboosttotheinterestinBCS-BECcrossover
.
t Ultracold atomic Fermi gases have been a very exciting,
a [6, 7, 8, 9, 10, 1]. It was suggested that the unusual pseudo-
rapidly developing field, which has emerged within the past
m gap phenomena in the cuprate superconductorsmight have to
severalyears,bridgingcondensedmatterandatomic,molecular
dowithBCS-BECcrossover. Experimentaleffortsinthisarea
-
dandopticalphysics[1]. UsingaFeshbachresonanceinamag- fell far behind, because it had been difficult to find a system
nneticfield,onecantunetheeffectivepairinginteractionstrength
where the attractive pairing interaction is tunable. Thanks to
obetween fermionic atoms from very weak to very strong [2].
thelasercoolingandtrappingtechniquein1990’s,oneisable
c
Astheinteractionstrengthvaries,thenatureofthelowtemper-
[ tocreate“artificial”many-bodysystemsoffermionicatomsin
ature superfluidity of these Fermi gases evolves continuously
a laboratory. The existence of a Feshbach resonance in these
1fromtheBCStypeintheweakcouplinglimittoBose-Einstein
Fermigasesmakesitpossibletotunetheinteractionstrength.
v
condensation (BEC) in the strong pairing limit, exhibiting a
6 For ease of control, the Feshbach resonances for the two
BCS-BECcrossover,whichhasbeenofgreattheoreticalinter-
4 widely studied species, 6Li and 40K, are both very wide. The
8estsince1960’s[3,4,5,1]. Ofparticularinterestistheunitary
interaction in both cases are of the short-range, s-wave type.
2regime,wherethe s-wave scatteringlengthdiverges. Thisisa
They are often taken to be a contact potential in theoretical
.strongly correlated regime where no small parameter is avail-
1
treatments.
0ableforperturbativeexpansions. Ithasbeenexpectedthatthis
1regime provides a prototype for studying both high T super- The first experimental realization of BCS-BEC crossover
c
1conductorsandstronglyinteractingFermigaseswhicharealso wasachievedin2004byJinandcoworkers,[11,12]andalmost
v:ofinteresttonuclearandastro-physicists. thesametimebytheGrimmgroup[13]andtheKetterlegroup
i Inthispaper,wefirstreviewexperimentalprogressinatomic [14]. Due to the difficulty in tuningtemperatureT, the Fermi
X
Fermigases,withanemphasisonrecentradiofrequencyspec- gaseswereeitherinthesuperfluidornormalstateatgiveninter-
artroscopymeasurements.Thenwewillpresentapairingfluctua- action strength (or the magnetic detuning). Continuousvaria-
tiontheorycomparewithexperiment.Weshowthatthistheory tionofthesystemasafunctionoftemperaturewasfirstrealized
successfullyexplainexperimentalmeasurements. bytheThomasgroup[15]atunitarity.Incollaborationwiththe
theory groupat Chicago [16], Thomaset al [17] observedfor
thefirsttimecontinuousphasetransitionfromthenormaltosu-
2. Experimentalprogress
perfluidstateinaunitary6Ligas. Onecouldargue,ofcourse,
ThefirsttheoreticalstudyofBCS-BECcrossoverdatesback thatthevortexmeasurementoftheKetterlegroupprovidedthe
to1960’s,althoughitdidnotgetmuchattentionuntilthesem- mostdefinitivesmokinggunforasuperfluidstate. [18]
inal work of Leggett in 1980 on BCS-BEC crossover at zero Besides the interaction strength, another great tunability is
populationimbalancebetweenthetwofermionicspeciestobe
∗Correspondingauthor:[email protected] paired [19]. It adds a whole new dimension to the phase dia-
PreprintsubmittedtoPhysicaC:SuperconductivityanditsApplications January17,2011
gramandmakesthephysicsmuchricher. Italso generatesin-
terest[20]inpossibleobservationofthe Larkin-Ovchinnikov-
Fulde-Ferrell(LOFF)state[21]. Experimentalworkinpopula-
tionimbalancedFermigaseswaspioneeredbytheHuletgroup
[22] and the Ketterle group [23]. Experiment in the extreme Figure1: Schematicdiagramsforthefermionicself-energyΣ(K). Thedotted
population imbalancedlimit by the Ketterle groupmanifested and (red) double lines represent the condensate and finite momentum pairs,
[24] the importance of Hartree-like correlationeffects besides respectively.
BCS-typeofpairing.
Unlikeanelectronsystem,ithasbeendifficulttomeasurethe
mation,givenbythegrandcanonicalHamiltonian
excitation gap in the Fermi gas superfluid. Amongall experi-
mentaltechniques,Radio frequency(RF) spectroscopy[13] is H− µ N = (ǫ −µ )a† a
σ σ k σ k,σ k,σ
arguablythemostdirectprobe. UsingatunableRFfieldtoex- Xσ Xk,σ
cite oneofthetwo pairingatomsfromalowerhyperfinestate + U(k,k′)a† a† a a , (1)
q,k,k′ q/2+k,↑ q/2−k,↓ q/2−k′,↓ q/2+k′,↑
(level2)toahigherhyperfinelevel3whichdonotparticipate
P
in pairing, a higher frequency will be needed if the atoms in whereǫ =~2k2/2misthefreeatomdispersion.Thedifference
k
level2arepaired. Suchafrequencyshift(detuning)providesa betweenEq.(1)anditsBCScounterpartisthatBCSkeepsonly
goodmeasureoftheexcitationgap. Previousmeasurementby the q = 0 term in the interactions. The inclusion of finite q
Grimm and coworkers[13], and later repeatedby the Ketterle termsallowsincoherent,finite momentumpairing. Forclarity
group [24], was done in a momentum integrated fashion. At ofpresentation,wewilltakeacontactpotential,U(k,k′) = 1,
low T, the RF spectra displayeddouble-peakstructure, with a and use a 4-momentum notation, K = (k,iω ), Q = (q,iΩ),
n l
sharppeakatzerodetuningandabroadpeakatpositivedetun- = T , and set ~ = 1. Population imbalance can be
K k n
ing. This double-peakfeature was nicely interpreted [25, 26] PdescribedPbyPµ↑ , µ↓. However,herewewillonlypresentthe
as transitionsfrom unpairedatoms the trap edge (correspond- equationsforthecaseofequalspinmixture. Generalizationto
ing to the sharp peak)andfroma distributionof pairedatoms populationimbalancecanbefoundinRef.[33].
intheinnerpartofthetrap(broadpeak). However,doubtwas Weassumethat(i)thefermionicselfenergyΣhasapairing
castabouttheoriginofthetwopeaksastowhethertheyreflect origin,(ii)pairscanbeeithercondensedorfluctuatingwithafi-
pairingofboundstateeffects[24]orsimplyaresultoftrapin- nitemomentum,and(iii)condensedandnoncondensedpairsdo
homogeneity[27]. Recently, attentionwasalso drawnto final notmixatthelevelofT-matrixapproximation.Figure1shows
stateeffectsboththeoretically[28,29]andexperimentally[24]. diagrammaticallythecontributionstotheself-energy,wherethe
A big step in the RF technique was the recent momentum- double(red)linesindicatefinitemomentumpairsandthedot-
resolved RF spectroscopy experiment in 40K by the Jin tedlineindicatesthecondensate. Thesubscripts“sc”and“pg”
group [30]. With momentum resolution, RF spectroscopy is stand for superfluid condensate and pseudogap contributions,
equivalent to the angle-resolved photoemission spectroscopy respectively.
(ARPES) for an electron system, In fact, it is cleaner than To tackle this problem, we use a Green’s function method.
ARPESinthatARPESisonlyatwo-dimensionalprobe,which We derive the equations of motion for one- and two-particle
isoftenplaguedbytheexistenceofsurfacestates,surfacecon- Green’s functionsG and G , which will involve higher order,
2
taminations, work function, and the complication of energy three particle Green’s functions G : iG˙ = [H,G] ∼ G,G ,
3 2
dispersion in the third dimension. In comparison, of course, iG˙ = [H,G ] ∼ G,G ,G . We then truncate the equations
2 2 2 3
the signal-to-noise ratio in a Fermi gas experiment is much of motion at the level ofG , factorizeG into a sum of prod-
3 3
lower, as limited by the (low) total number of atoms in the ucts of G and G , and treat G and G on equal footing. For
2 2
gas. although the trap inhomogeneity adds complication to G , we focus on the particle-particle channel, neglecting the
2
the interpretation of the spectrum. Like ARPES, momentum- particle-hole channel which normally only provides a chemi-
resolvedRFspectroscopymeasuresthefermionspectralfunc- calpotentialshift. Weemphasizethatitistheparticle-particle
tion, A(k,ω), which is of central importancein characterizing channelthatgivesrisetosuperfluidity. Aftersomelengthybut
thesystem. straightforwardderivation,weobtaintheselfenergy:
Σ(K) = Σ (K)+Σ (K), (2)
sc pg
3. TheoreticalFormalism Σ (K) = −∆2G (−K), (3)
sc sc 0
Σ (K) = t (Q)G (Q−K), (4)
pg pg 0
In this section, we now presenta simple pairingfluctuation X
Q
theory, which was first developed [10] to explain the pseudo-
gapphenomenainhighTcsuperconductors.Fermigasesinthe where
presenceofaFeshbachresonancecanbeeffectivelydescribed U
t (Q)= (5)
by a two-channelmodel [2]. It has now been known that the pg X1+Uχ(Q)
closed-channelfraction[31,32]isverysmallforboth6Liand Q
40K,throughouttheBCS-BEC crossover. Therefore,forthese is the (pseudogap)T-matrix,and χ(Q) = G (Q−K)G(K)
K 0
systems,aone-channelmodelisoftenusedasagoodapproxi- isthepairsusceptibility. HereG0 isthebaPreGreen’sfunction.
2
40 0.5 Equations(9)and(7),alongwiththenumberequation
V) (a) ∆ (b) B E C n=2 G(K), (10)
me TF XK
s ( /c
p ∆ T formaclosedsetofequationsforthehomogeneouscase,which
Ga ∆pg sc CS can be used to solve for µ, Tc, and the gaps at T ≤ Tc. Tc is
B
determinedby setting ∆ = 0. Typicalbehaviorsof the gaps
sc
0 0
0 0.5 1 1.5 -2 0 2 4 areshowninFig.2(a).
T/T 1/k a To address Fermi gases in a trap, we use the local density
F F
approximation,byreplacingµ → µ−V (r). Thenthenum-
trap
Figure2: Typicalbehaviorof(a)thetemperaturedependenceofthegapsina ber equation becomes N = d3rn(r). In Fig. 2(b) we show
pseudogappedsuperfluid,andof(b)Tcasafunctionof1/kFainatrap,where the BCS-BEC crossoverbehaRviorof Tc in a trap. Here 1/kFa
kFisthenoninteractingFermimomentum,andaisthes-wavescatteringlength. parametrizestheinteractionstrength.
The RF response can be derived using the linear response
theory.TheRFinteractionisdescribedby
A detailed derivation of this result can be found in Ref. [34].
NotethattheT-matrixiseffectivelyarenormalizedpairingin-
teraction. Itsharesexactlythesamepolestructureasthetwo- Hrf =eiΩtZ d3xψ†3ψ2+h.c., (11)
particle Green’s function,G . Througha Taylor expansionof
2
itsdenominator,onecanextractthepairdispersion: andtheresponseKernelby
t−pg1(Q)≈Z(iΩl−Ωq+µpair). (6) D(iΩl)=XG(2)(K)G(3)(K+Q). (12)
K
The superfluid instability is given by 1+Uχ(0) = 0 ∝ µ ,
pair We assume hyperfine level 3 was initially empty. In the ab-
whichistheBECconditionforpairs. Notethatχ(Q)involvesa
sence of final state interactions, as in 40K, we obtain [35] the
mixofbareandfullGreen’sfunctions. Weemphasizethatthis
RFcurrent
is a natural consequence of the equation of motion technique
sleinacdesibtainckvotlovethsethBeCoSp-eforartmoroGfˆ−0g1a.pIteiqsutahtiisonG0inGtfhoermabosfenχctehaotf I(k,ν) = −π1ImDR(ν+µ−µ3)
finitemomentumpairs. 1
QW=e0.fDoceufisnoinngthe superfluid phase where tpg(Q) diverges at = 2πXk A(k,ω)f(ω)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)ω=ǫk−µ−ν. (13)
(cid:12)
InordertoaddressA(k,ω)= −2ImG(cid:12)(k,ω+i0+)properly,we
∆2pg ≡− tpg(Q), (7) need to include the lifetime effects of finite momentum pairs
XQ,0 andaddanincoherenttermiΣ in(andonlyin)Σ , reflecting
0 pg
the residuetermδΣ whichwe dropin solvingthe setof equa-
wehave
tions,i.e.,
Σ (K) = − t (Q) G (−K)+δΣ ∆2
pg  pg  0 Σ (k,ω)= pg −iΣ . (14)
= −∆XQ2 G (−K)+δΣ. (8) pg ω+ǫk−µ+iγ 0
pg 0 WhileaboveT thespectralfunctionwithapseudogapconsti-
c
tuteadoublepeakstructurewithsuppressedspectralweightat
NeglectingtheresiduetermδΣ,Σ takesthesameformasΣ .
pg sc
theFermilevel,belowT ,thereisazeroatω = −(ǫ −µ). As
Thus we have immediately the BCS form of total self energy, c k
∆ increases with decreasing T below T , the spectral peaks
Σ(K) = −∆2G (−K), with ∆2 = ∆2 +∆2 . Thisthenleadsto sc c
0 sc pg sharpenrapidly. Thisisaphasecoherenceeffect. Theparame-
theBCSformofgapequation,
tersγandΣ canbeestimatedfromexperimentalRFspectra.
0
1−2f(E )
1+U k =0, (9)
Xk 2Ek 4. Comparisonbetweentheoryandexperiment
where E = (ǫ −µ)2+∆2 is the quasiparticle dispersion. In Fig. 3, we compare between theory (curves) and experi-
k k
Different frompthe BCS mean-field theory, we emphasize that ment(symbols)(a)the densityprofile[36] and(b)system en-
here ∆2 containscontributionsfrom both condensedandnon- ergy [17] for 6Li in the unitary limit. Both experimental and
condensed pairs so that it in general does not vanish at T . theoreticaldensityprofilesareverysmooth,ingoodagreement
c
Note thatthefiniteqpairsaredifferentfromthe orderparam- witheachother.Alternativetheoriespredictseitherakinkatthe
etercollectivemodes;thelatterrepresentacoherentmotionof edgeofthesuperfluidcoreornonmonotonicradialandtemper-
the condensate. Here ∆2 and ∆2 are loosely proportionalto aturedependence. Theenergycomparisonalsorevealsaquan-
sc pg
thedensityofcondensedandnoncondensedpairs,respectively. titativeagreement. Thefactthattheunitaryandnoninteracting
3
1.5
Theory, noninteracting 20 2
T/T =0.19
F 4 Theory, unitary
noninteracting 10 1
1
n(x)0.5 (a) E/EF2 unitary (b) ω (kHz) −100 ωµ+)/EF−01
(
−20
−2
T=0.29
0 -1 x (1000µm) 1 00 c 0.5 T/TF 1 1.5 −300 5 10 15 −30 0.5 1 1.5
k (µm−1) k/k
F
Figure3:Comparisonof(a)densityprofileand(b)energyE/EF foraunitary
6Ligasbetweentheory(curves)andexperiment(symbols). Alsoshownin(b) Figure4:ComparisonofspectralintensitymapI(k,ν)k2/(2π2)betweenexper-
iscomparisonforthenoninteractingenergy. HereEF =kBTF isthenoninter- iment(left)fromRef.[30]andtheory(right). Thewhitedashedcurveisan
actingFermienergy. experimentalextractedquasiparticledispersion,andthewhitesolidlineisob-
tainedtheoreticallyfollowingthesameexperimentaldataanalysisprocedure.
curvesmergeatT ≈ 0.6T ≫ T manifeststhe presenceof a
F c
pseudogap.Itshouldbenotedthatthereisnofittingparameter [5] P.Nozie`resandS.Schmitt-Rink,J.LowTemp.Phys.59,195(1985).
inourtheoreticalcalculations. [6] Y.J.Uemura,PhysicaC282-287,194(1997).
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[8] Randeria,PhysicaB198,373(1994).
as a function of k and single-particle energy ω + µ between
[9] B.Janko´,J.Maly,andK.Levin,Phys.Rev.B56,R11407(1997).
experiment[30]andtheory[35]foraunitary40Kgasatatem- [10] Q.J.Chen,I.Kosztin,B.Janko´,andK.Levin,Phys.Rev.Lett.81,4708
peratureslightlyaboveT . Thewhitedashedcurveistheexper- (1998).
c
imentally extractedquasiparticledispersion, whereasthe solid [11] C.A.Regal,M.Greiner,andD.S.Jin,Phys.Rev.Lett.92,040403(2004).
[12] Q.J.Chen,C.A.Regal,M.Greiner,D.S.Jin,andK.Levin,Phys.Rev.
curveisobtainedtheoreticallyfollowingtheexperimentalpro-
A73,041601(2006).
cedure.Itisevidentthattheoreticalandexperimentalresultsare [13] C. Chin, M.Bartenstein, A.Altmeyer, S.Riedl, S.Jochim, J.Hecker-
rather close to each other. Indeed, as T decreasesfrom above Denschlag,andR.Grimm,Science305,1128(2004).
[14] M.W.Zwierlein, C.A.Stan, C. H.Schunck, S.M.F.Raupach, A.J.
to below T , the spectral intensity map evolves [35] from an
c Kerman,andW.Ketterle,Phys.Rev.Lett.92,120403(2004).
upwarddispersingbranchathighT toabifurcationaroundT ,
c [15] J.Kinast,A.Turlapov,andJ.E.Thomas,arXiv:cond-mat/0409283.
and finallyto a downwarddispersingbranchat T ≪ Tc. This [16] Q.J.Chen,J.Stajic,andK.Levin,Phys.Rev.Lett.95,260405(2005).
result establishes the actual single particle dispersions which [17] J. Kinast, A. Turlapov, J. E. Thomas, Q. J. Chen, J. Stajic, and K.
Levin, Science 307, 1296 (2005), published online 27 January 2005;
contribute to the RF current, revealing that the broad peak in
doi:10.1126/science.1109220.
previous momentum-integrated RF spectra [13] indeed has a
[18] M.W.Zwierlein,J.R.Abo-Shaeer,A.Schirotzek,andW.Ketterle,Na-
pairingorigin. Furthermore,italsoshowsthat,despitethetrap ture435,170404(2005).
inhomogeneity,momentumresolvedRFspectroscopycanstill [19] Q.J.Chen,Y.He,C.-C.Chien,andK.Levin,Phys.Rev.A74,063603
(2006).
provideaquantitativemeasureofthespectralfunctionandsin-
[20] Y.He,C.-C.Chien,Q.J.Chen,andK.Levin,Phys.Rev.A75,021602(R)
gleparticledispersion.ItalsolendssupportforthepresentG0G (2007).
schemesincealternativeNSR-basedtheoriesdonot[37]seem [21] P.FuldeandR.A.Ferrell,Phys.Rev.135,A550(1964);A.I.Larkinand
to generate the two-branch-like feature observed in Ref. [30]. Y.N.Ovchinnikov,Zh.Eksp.Teor.Fiz.47,1136(1964)[Sov.Phys.JETP
20,762(1965)].
The downwarddispersion around (and above)T providesdi-
c [22] G.B.Partridge,W.Li,R.I.Kamar,Y.A.Liao,andR.G.Hulet,Science
rectevidencefortheexistenceofapseudogapaboveTc atuni- 311,503(2006).
tarity. Ourtheoryservesasabasisformomentum-resolvedRF [23] M.W.Zwierlein,C.H.Schunck,A.Schirotzek,andW.Ketterle,Nature
spectroscopyanalysis. (London)442,54(2006).
[24] C.H.Schunck,Y.Shin,A.Schirotzek,M.W.Zwierlein,andW.Ketterle,
In summary, we have presenteda pairingfluctuationtheory
Science316,867(2007).
wherefinitemomentumpairingplaysaprogressivelymoreim- [25] J.Kinnunen,M.Rodriguez,andP.To¨rma¨,Science305,1131(2004).
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[27] E.J.Mueller,arXiv:0711.0182(unpublished).
dogap in the single particle excitation spectrum. This theory
[28] S.BasuandE.Mueller,Phys.Rev.Lett.101,060405(2008).
hasbeensuccessfullyappliedtomultipleexperimentsinatomic
[29] Y.He,C.C.Chien,Q.J.Chen,andK.Levin,Phys.Rev.Lett.102,020402
Fermigases. (2009).
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(2008).
ChinaGrantNo.10974173.
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Phys.Rev.Lett.95,020404(2005).
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4