Thermodynamics of Interacting Fermions in Atomic Traps

0.17 MB·English

by  Qijin Chen

Thermodynamics ofInteracting Fermions inAtomicTraps
Qijin Chen, Jelena Stajic,∗ and K. Levin
JamesFranckInstituteandDepartment ofPhysics, Universityof Chicago, Chicago, Illinois60637
(Dated:February2,2008)
6 Wecalculatetheentropyinatrapped,resonantlyinteractingFermigasasafunctionoftemperatureforawide
0 rangeofmagneticfieldsbetweentheBCSandBose-Einsteincondensationendpoints. Thisprovidesabasisfor
0 theimportanttechniqueofadiabaticsweepthermometry,andservestocharacterizequantitativelytheevolution
2 andnatureoftheexcitationsofthegas.Theresultsarethenusedtocalibratethetemperatureinseveralground
breakingexperimentson6Liand40K.
n
a
J PACSnumbers:03.75.Hh,03.75.Ss,74.20.-z Phys.Rev.Lett.95,260405(2005)

7
1
The claims [1, 2, 3, 4, 5] that superfluidity has been ob- periments[9]andtheory[15,16],and(iii)T-dependentden-

] servedinfermionicatomicgaseshavegeneratedgreatexcite- sity profiles[17]. Finally, (iv)plots of the energyE vs T at
n
ment. Varying a magnetic field, one effects a smooth evo- unitarity[18]yieldverygoodagreementwithexperimentand
o
lutionfromBCSsuperfluiditytoBose-Einsteincondensation serveto calibrate the presentthermometry. Two well-known
c
- (BEC) [6, 7]. In this Letter we use a BCS-BEC crossover weaknesses of the mean field approach(an underestimateof
r
p theory to study the entropy S over the entire experimentally β andanoverestimateoftheinter-bosonscatteringlengthaB
u accessible crossover regime. Our goal is to help establish a in the deep BEC regime) should be noted. The first affects
s methodologyforobtainingthetemperatureT ofastronglyin- E(T)butnotS(T). However,forthesecondweintroducea
.
t teracting Fermi gas via adiabatic sweeps. This addresses an caveat: if the initialendpointof sweep thermometryis suffi-
a
m essential need of the experimentalcold atom community by cientlydeepintheBECregime(say,kFa 0.3)theaccuracy
≤
providingatemperaturecalibrationfortheirgroundbreaking of the finaltemperature,we calculate forthe unitaryregime,
-
d experiments[2, 8, 9]. In the process, we characterize quan- couldbeimprovedbycomputingtheinitialSinthedeepBEC
n titativelytheevolutionoftheexcitationsandshowhowtheir regimeusingapure-bosonmodelwitha setbyhandtothe
B
o character evolves smoothly from fermionic to bosonic, and Petrovresult[19].
c
[ conversely. Because previousthermodynamictheoriesdidnotaddress
Inadiabaticsweeps,thestartingT ateitheraBECorBCS unitarity,ithasnotbeenpossibleuntilnowtoarriveatatem-
3
endpointisestimatedfromthe“known”shapeoftheprofilein perature scale in the experimentally interesting resonant su-
v
thetrappedcloud. Then,thetemperature(nearunitarity,say) perfluidregime. Carretal. [20, 21]calculatedS attheBCS
0
9 is obtained by equatingthe entropybefore the sweep to that andweaklyinteracting,deep-BECendpoints. Thelattertrue
0 in the strongly interacting regime after the sweep. Conven- Boselimitwhichtheyconsidereddoesnotappeartobeappro-
1 tionally,thetemperaturescalewhichappearsinthesuperfluid priate to current collective mode experiments, [3, 4], which
1
phasediagram[2,8]involvesanisentropicsweepbetweenthe show [13, 14] that for physically accessible (i.e., near-BEC)
4
unitaryandthenon-interactingFermigasregimes.Thedirec- fields,fermionsareplayinganimportantrole.Thus,theBCS-
0
/ tion of the sweep is irrelevant in these reversible processes. Leggettgroundstateappearstobemoreappropriatethanone
t
a Theimportantexperimentalphasediagramsplottheconden- derivingfromBose-liquid-basedtheory. Williamsetal. [22]
m satefraction,N /N nearunitarityvsthisFermigas-projected calculatedSforaBCS-BECcrossovertheoryusingamixture
s
- temperature,Teff. ofnoninteractingfermionsandbosons[22]. Thiswork,omits
d
In this paper, our thermodynamical calculations are used the importantandself consistentlydeterminedfermionicex-
n
torelatetheactualphysicaltemperaturesT toT , where,in citationgap∆whichisanessentialcomponentfordescribing
o eff
c general, T is significantlygreater thanTeff. A calculationof thethermodynamicsoffermionicsuperfluids.
: N (T)issimultaneouslyundertaken[10,11]whichprovides Our thermodynamical calculations focus on this self-
v s
i animportantself-consistencyconditiononthethermodynam- consistentlydetermined∆;theyarebased,forcompleteness,
X
ics, since the same excitations appear in both, Moreover, a ona two-channelHamiltonian[11, 23, 24]. Here ∆ appears
ar calculationofNshastobedonewithproperattentionpaidto in the fermionic dispersion Ek = p(ǫk µ)2+∆2. (We
collectivemodesandgaugeinvariance[12]. Hereweaddress define ǫ = ~2k2/2m as the kinetic ene−rgy of free atoms,
k
thevariouscondensatefractionsfoundexperimentally[1,2], andµthe fermionicchemicalpotential.) Importantly,this∆
(with emphasis on 6Li), as a function of Teff, in the experi- provides a measure of bosonic degrees of freedom. In the
mentalrangeofmagneticfields. fermionicregime(µ>0),∆isjusttheenergyrequiredtodis-
OurworkisbasedontheBCS-Leggettgroundstate [6, 7] sociatethepairsandtherebyexcitefermions. AtfiniteT,the
anditsfiniteT extension[11].Fourdifferentclassesofexper- closed-channelmolecularbosonsandtheopen-channelfinite
iments have been successfully addressed in this framework. momentum Cooper pairs are strongly hybridized with each
Theseinclude(i) T 0breathingmodesexperiments[3, 4] other, making up the “bosonic” excitations which contribute
≈
andtheory[13,14],(ii)radiofrequency(RF)pairinggapex- tothermodynamics.
2
.
Our many-body formalism has been described below the
superfluid transition temperature Tc[11]. The parameter ∆, G0(K)=(cid:2)
(when squared), is the analogue of the total number of par-
G(K) =
ticles, in the simplest theory of BEC. Just as in BEC, there Ωb = (cid:3).
D0(Q)=
are two self-consistency conditions: (i) the effective chemi- (cid:4)
cal potential of the pairs, µpair, is zero, for T Tc (as is Uk,k′ =(cid:5)
≤
that of the closed-channel molecular bosons µ ), and, (ii)
mb . (cid:1)
the number of pairs, reflected in ∆2(T) contains two addi-
tivecontributionsrepresentingcondensed(∆˜2 )andnoncon- FIG.1:Bosoniccontributiontothethermodynamicalpotential.Here
sc
densed(∆2 )pairs.Thefirstconditionimpliesthat∆(T)sat- G0 (G) and D0 (D) are the “bare” (“full”) propagators associated
pg with the fermions and closed-channel molecular bosons, respec-
isfies a BCS-like gap equation. Then, the condensate is de-
duced,justasinBEC,bydeterminingthedifferencebetween tively, K and Q are four-momenta, and Uk,k′ is the open-channel
pairinginteraction.
∆2 and∆2 . Inthisapproachthe hybridizedpairshavedis-
pg
persionΩ =~2q2/2M∗,witheffectivepairmassM∗.
q
WenowextendthisapproachaboveTc. Ourfirstequation contributionsΩb. Thelatterisgivenbythesumofallpossible
representstheimportantdefiningconditiononµ : thatthe ringdiagramsshowninFig.1. Itcanbeeasilyshownthatthis
pair
inversepairpropagator(orT-matrix)t−1(Q)(cid:12)(cid:12)Q≡0 =Zµpair, Ωbisconsistentwiththeselfenergydiagramsforthefermions
with (inverse) “residue” Z. While in the superfluid regions andthemolecularbosons.Afterregrouping,weseethatShas
µ andµ vanish,ingeneral,wehave twocontributions,fromfullydressedfermions(S )andfrom
pair mb f
theirbosoniccounterpart(S ). Thetotalentropyinvolvesan
b
Ue−f1f(0)+Xk 1−22Efk(Ek) =Zµpair, (1) ifnotreSgrfalanodveSrbt)h,ewtrhaepr,egivenbyS =R d3rs(r)(andsimilarly
whereUeff(0) = U +g2/(2µ ν)involvesthesumofthe s = sf +sb,
−
directattractionU betweenopen-channelfermions,aswellas
sf = 2X[fklnfk+(1 fk)ln(1 fk)],
thevirtualprocessesassociatedwiththeFeshbachresonance. − − −
k
Heref(x)istheFermidistributionfunction. Thedetermina-
tion of the inter-channel coupling constant, g, and the mag- sb = −X[bqlnbq−(1+bq)ln(1+bq)], (4)
neticfielddetuning,ν,isdescribedelsewhere[25],asarethe q6=0
residues Z and Zb [11]. The contribution from hybridized wherefk f(Ek),andbq b(Ωq µmb);arelativelysmall
bosonswillleadtoanormalstateexcitationgap[9,11,26,27] ≡ ≡ −
contributionassociatedwiththeT dependenceofΩ hasbeen
q
or pseudogap(pg). Thiscan be written in termsof the Bose
dropped. The fermion contribution coincides formally with
distributionfunctionb(x)as thestandardBCSresultfornoninteractingquasiparticles[al-
though here ∆(T ) = 0]. And the bosonic contribution is
∆2pg =Z−1X b(Ωq−µpair). (2) given by the expcress6ion for non-directly-interacting bosons
q withdispersionΩ . Thesebosonsarenotfree,however;be-
q
Weusethelocaldensityapproximation(LDA)throughout causeofinteractionswiththefermions,theirpropagatorcon-
withaharmonictrappotentialV(r). Fornotationalsimplicity, tainsimportantself-energyandmassrenormalizationeffects.
weomitwritingV(r) infavorofµ(r)accordingtotheLDA Figure 2 illustrates the behavior of S as a function of T
prescription: µ µ(r) µ V(r), whereµ µ(0). The obtained from our self-consistent equations, over the entire
totalatomicnum→berN ≡ d3−rn(r),where ≡ experimentallyrelevantcrossoverregime. Themagneticfield
R
≡ is contained in the parameter 1/k a, which increases with
F
n = 2nb0+2Zb−1Xb(Ωq−µmb) decreasing field. Here a is the s-wave fermionic scattering
q length, kF is the Fermi wavevector at the trap center, and
+2X(cid:2)vk2(1−f(Ek))+u2kf(Ek)(cid:3) . (3) tkuBreT.FTw=oim~2pkoF2rt/a2nmtasipsecthtseonfothneinfteerrmacitoinngicFceornmtriibtuetmiopneSra-
f
k
shouldbenoted.Generally,thefermionshaveagap∆intheir
Hdeenrseednbc0lo=sedg-2c∆ha2sncn/e[(lνm−ole2cµu(lre)s),Uan]2dius2t,hve2d=ens[i1ty of(ǫcon- aexncditmatoiorenovsperecthtriusmT (dwepheicnhdeinntcgreaapseissiwnhitohmdoegcerneeaosiunsgsofiethlda)t
k k ± k −
µ(r))/E ]/2. Thetotalorderparameter[23,24,26]isgiven thefermionsnearthetrapedgeoftenbehaveasfreeparticles
k
by∆˜sc =∆sc+ g √nb0. atT > ∆. Thesequasi-“free”fermionschangetheT depen-
| |
Tomakeprogress,wenumericallysolveEqs.(1)-(2)ateach dence of S fromexponentialto power law. Theyhave also
f
r forgivenµandthenself-consistentlyadjustµviathetotal beenseeninRFexperiments[9,15],asafreefermionpeakin
numberconstraint.NextweobtaintheentropySdirectlyfrom thespectra.
the thermodynamicalpotential [28]. This potential contains We refer to Fig. 2, starting from the high field or BCS
fermioniccontributionsfrombarefermions,Ω ,andbosonic regime where S is linear in T. As the field is lowered to-
f
3
4 1 generalgroundstate, their contribution, at unitarity, will not
F1-1r/keeF aB =os -e2 gas 0 1/2 1/kaF = 3/2 beTaosismhepdoratadndtitaisonthaaltlifgrohmt otnhetheedgceomfeprmonioennst.fermionic and
-1/2 2 bosoniccontributions,intheinsettoFig.2wedecomposethe
0
1/2 3 various terms in the entropy to reveal their spatial distribu-
1 A tions, for the unitary case at T = T /4. It can be seen that
3/2 c
NkB2 2 B thefermioniccontributionsf (redcurve)islimitedtothetrap
S/ 3 edge,where∆issmall. Bycontrast,thebosoniccontribution
2
1/2 B s sb(bluecurve)isevenlydistributedoverthesuperfluidregion
- k andrapidlydecaysatlargerradii.
N s
-2 Bose gas s(r)/0 sf b swFeeigpuerxep2eripmroevnitds.eTshaevbearstiiscaflolrintehseirlmluosmtraetteryhoiwn taodciahboaotsice
0 r/R 1 aninitialtemperature(T =0.5T atpoint“A”)withaninitial
0 TF i F
0 0.2 0.4 0.6 0.8 1 valueof1/k a(=1)anduseanisentropicsweep(represented
F
T/TF bythehorizontallinethrough“A”and“B”)toobtainthefinal
temperature(T = 0.28T atpoint“B”)withthefinalvalue
f F
FIG.2: (coloronline)EntropyperatomasafunctionofT fordif-
of1/k a(=0). Itismostconvenienttobeginwitheitherthe
ferentvaluesof1/kFafromBCStoBECinaharmonictrap. The BCSreFgimeorBECregimes,sincehereT can,inprinciple,
dottedlinesshowanisentropicsweepbetween1/kFa=1anduni- i
bedeterminedbyfittingthedensityprofiles.
tarity. Forcomparison,wealsoplotSforanidealBosegas(dashed
line).The1/kFa=3curveliesbelowthedashedlineatT >0.8TF Figure 3presentsaplotofthesuperfluidfraction[10,11]
reflectsthatM∗ 6=2m.Theinsetplotsthespatialprofileoftotalen- N /N in the intermediate regime as a function of an effec-
s
tropy s (black curve) and its fermionic (sf, red) and bosonic (sb, tive temperature Teff/TF for different values of initial fields
blue)componentcontributionsatunitarityforT =Tc/4.HereRTF or 1/k a. Here T is the temperaturereachedafter an adi-
F eff
istheThomas-Fermiradius,andTc =0.27TF. abatic sweep to a BCS-like state. Based on experiment, we
take thefinalstate as1/k a = 0.59at1025G for6Li[2]
F
andanoninteractingFermigasfo−r40K[8].
wards unitarity, Sf will vary as a low-T power law which The same vertical axis appears in the inset but with the
is higher than linear. Simultaneously, the bosonic degrees physical temperature scale T, so that this figure provides a
of freedom emerge. Here one sees a T3/2 power law from meansofdirectlycalibratingthetemperaturescaleT which
eff
these excited bosons. At unitarity, bosonic effects dominate hasbeenusedintheimportantphasediagramsof6Liand40K.
forT/TF . 0.05orT/Tc . 0.2. For an extendedrangeof Figure 3alsoprovidesameansofcomparingthecondensate
T < Tc,thefermionsandbosonscombinetoyield,S T2, fractionswiththoseinthephasediagrams. For6Liat900G,
∝
whichcanbecomparedwiththeexperimentalpowerlaw[18] withT /T =0.2,0.1and0.05,theexperimentalcondensate
eff F
T2.73. Finallyinthenear-BECregimeoneseesanessentially fractionsare0.0,0.1and0.6. Thisshouldbecomparedwith
purebosonicT3/2 powerlawinS atlowT inthesuperfluid ourcalculatedvalues,0.0060.36,and0.73,respectively. For
phase. The relative contribution of the bosonic excitations, 6Liat770G,thecondensatefirstappearsatT /T = 0.18,
eff F
Sb/S evolves continuously from 0 to 1 as 1/kFa increases consistentwiththeory.FromthevaluesofkFaandTcatboth
from to + . S becomes dominantly bosonic once µ ends(importantly,the latter can be read fromthe inset), one
−∞ ∞
becomesnegative. can easily see that the sweep from 770 G to 1025 G is still
ThebosonicT3/2 powerlawfoundinthetrapisthesame veryfarfromafullBEC-BCSsweep.
asfoundforthehomogeneoussituation.Inhomogeneityeffec- There are two reasons for the larger condensate fractions
tivelydisappearsherebecausethefermion-bosoninteractions found theoretically for 6Li. A calculation of the BEC-like
lead to the self-consistent constraint that µ = 0 for the densityprofilesshowsthatthenoncondensedpairsinsidethe
pair
entiresuperfluidregion.Thissamedisappearanceofinhomo- superfluid region have a flat density distribution, which re-
geneityisfoundinRef.[20]. Thisisdifferentfromastrictly flects the vanishing of µ [17]. The superfluid fraction
pair
non-interactingBosesystem[22](dashedlineinFig.2)where extracted experimentally (assuming a Gaussian form for the
onedoesnothaveavanishingbosonchemicalpotentialbelow noncondensed particles inside the condensate core) is, thus,
T exceptatthetrapcenter.ThepreviousworkofRef.[20]is underestimated,mostnotablyaroundT /2. In addition, ear-
c c
basedoninteractingbuttruebosons. Thepresentsituationis lierwork[18]showsthatwhenthesystemistreatedasanon-
morecomplexsinceCooperpairoperatorsdonotobeyBose interactingFermigas,T willbeunderestimatedwhenevera
eff
commutation relations, (nor does the linear combination of condensateispresent(atT < T 0.17T at1025G).This
c F
≈
Cooper pair and closed-channelboson operators). This sug- suggeststhattheoryandexperimentcanbebroughtintorather
gests that a theory based on a true Bose liquid may not be goodagreementforthe case of 6Li. For40K, one hasto ap-
appropriateforthefieldsthathavebeenaccessedexperimen- pealtonon-adiabaticityandothercomplicationsofthesweep
tally.Moreover,ifoneweretocontemplatecontributionsfrom processtounderstandthesmallmeasuredfractions.
thelinearlydispersingGoldstonebosons,albeitwithinamore Forthiscase, weemphasizetemperaturescales. Forafull
4
1 fieldcannotbedirectlycomparedtoanytheory. Thepresent
1
6Li Li: 1/kFa=3 work presents a theory for the entropy S of a Fermi gas, at
1025G general accessible magnetic fields, which thereby calibrates
0.8 N T invariousexisting[2,8,9]andfutureexperiments.
/
S
N WeareextremelygratefultoJ.E.Thomas,J.KinastandA.
40K Turlapov for many helpful discussions, and to N. Nygaard,
0.6
C. Chin, M. Greiner, C. Regal, D.S. Jin, and M. Zwierlein
N 0
/S 0 T/T 0.4 as well. This work was supported by NSF-MRSEC Grant
N F
0.4 Li: 900G↔ 1025G No. DMR-0213745 and by the Institute for Theoretical Sci-
Li: 820G↔ 1025G encesandDOE,No. W-31-109-ENG-38(QC).
Li: 770G↔ 1025G
K: unitarity ↔ FG
0.2 K: k a=1.67↔ FG
K: kFa=0.99↔ FG
F
∗ Present address: Los Alamos Nat’l Lab, Los Alamos, NM
0 87545
0 0.1 0.2
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T /T
eff F 040403(2004).
[2] M.W.Zwierleinetal.,Phys.Rev.Lett.92,120403(2004).
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eff
measuredinanear-BCS(at1025Gfor6Li)ornoninteractingFermi [5] M.W.Zwierlein,J.R.Abo-Shaeer,A.Schirotzek,andW.Ket-
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icaltemperatureateachfieldvalue. Thesystemisnotfarfromthe [7] A. J. Leggett, in Modern Trends in the Theory of Condensed
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chosen based onRefs. [2] and [8]. For 6Li, TF = 3.6µK, sothat [9] C.Chinetal.,Science305,1128(2004).
1/kFa = −0.59,−0.26,0.07,0.38for1025G,900G,820Gand [10] Definingthecondensatefractioninthefermionicregimeisnot
770G,respectively. unambiguous.Ns/N whichweuseheregivesthelargestpos-
siblefraction.
[11] Q.J.Chen,J.Stajic,S.N.Tan,andK.Levin,Phys.Rep.412,1
BCStonear-BEC[(k a) =1.7]adiabaticsweep[8]with (2005).
F final
[12] I. Kosztin, Q.J.Chen, Y.-J.Kao, andK.Levin, Phys.Rev. B
initialT T = 0.19T ,thefinalreportedtemperaturesin
eff i F
≡ 61,11662(2000).
experiment[8] and in theory are T = 0.47T and 0.33T ,
f F F [13] H.Huetal.,Phys.Rev.Lett.93,190403(2004).
respectively. For the same (k a) butwith T = 0.17T ,
F final i F [14] H.Heiselberg,Phys.Rev.Lett.93,040402(2004).
we find Tf Tc, in agreement with the observed sudden [15] J.Kinnunen, M.Rodriguez, andP.To¨rma¨, Science305, 1131
≈
onset of a bimodal distribution in the density profile. Simi- (2004).
larlyforasweepfromaFermigasdownto(k a) = 0.99 [16] Y.He,Q.J.Chen,andK.Levin,Phys.Rev.A72,011602(R)
F final
with T = 0.06T , the experimentally quoted and theoreti- (2005).
i F
[17] J.Stajic,Q.J.Chen,andK.Levin,Phys.Rev.Lett.94,060401
callycalculatedT are0.25T and0.18T ,respectively.The
f F F
(2005).
experimentalsweepswerenotstrictlyadiabatic[8],sothatthe
[18] J.Kinast,A.Turlapov,J.E.Thomas,Q.J.Chen,J.Stajic,and
experimentalT shouldserveasupperbounds. Ourcalcula-
f K.Levin,Science307,1296(2005).
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sumedaT3 powerlawforS intheBECregime,fromwhich Lett.93,090404(2004).
onewouldinferT = 0.52T and 0.37T , respectively,ex- [20] L.D.Carr,G.V.Shlyapnikov, andY.Castin,Phys.Rev.Lett.
f F F
ceedingtheupperbounds. 92,150404(2004).
Experimentally, 40K gases [8] are prepared in the nonin- [21] L.D. Carr, R. Chiaramonte, and M. J.Holland, Phys.Rev. A
70,043609(2004).
teracting limit, where, as a result of heating associated with
[22] J.E.Williams,N.Nygaard, andC.W.Clark,NewJ.Phys.6,
an adiabatic sweep, low T is difficult to reach. By contrast
123(2004).
6Li gases [2, 4] are prepared in the BEC regime, so that [23] Y.OhashiandA.Griffin,Phys.Rev.Lett.89,130402(2002).
higher condensate fractions of 80% and 95% have been re- [24] J. N. Milstein, S.J. J. M. F. Kokkelmans, and M. J. Holland,
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we notethat the ratherlarge T 0.17T at1025G makes [25] Q.J.ChenandK.Levin,Phys.Rev.Lett.95,260406(2005).
c F
it hard to access the Fermi gas r≈egime in 6Li. This may be [26] J.Stajicetal.,Phys.Rev.A69,063610(2004).
[27] M. Greiner, C. A. Regal, and D. S. Jin, Phys. Rev. Lett. 94,
circumvented either by reduction in the size of N or T or,
F 070403(2005).
possibly, by sweeps to 528 G [29]. In 40K, one avoids this
[28] Q.J.Chen,K.Levin,andI.Kosztin,Phys.Rev.B63,184519
problemaltogether. (2001).
Without knowing the temperature, measurements in this [29] K.M.O’Haraetal.,Science298,2179(2002).