Evolution of the Normal State of a Strongly Interacting Fermi Gas from a Pseudogap Phase to a Molecular Bose Gas

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by  A. Perali

Evolution of the Normal State of a Strongly Interacting Fermi Gas from a Pseudogap
Phase to a Molecular Bose Gas
A. Perali 1, F. Palestini 1, P. Pieri 1, G. C. Strinati1, J. T. Stewart 2, J. P. Gaebler 2, T. E. Drake 2, D. S. Jin2
1Dipartimento di Fisica, Universita` di Camerino, I-62032 Camerino, Italy
2JILA, NIST and University of Colorado, and Department of Physics,
University of Colorado, Boulder, CO 80309-0449, USA
Wave-vector resolved radio frequency (rf) spectroscopy data for an ultracold trapped Fermi gas
1
1 are reported for several couplings at Tc, and extensively analyzed in terms of a pairing-fluctuation
theory. We map the evolution of a strongly interacting Fermi gas from the pseudogap phase into
0
2 a fully gapped molecular Bose gas as a function of the interaction strength, which is marked by a
rapiddisappearanceofaremnantFermisurfaceinthesingle-particledispersion. Wealsoshowthat
n
ourtheoryofapseudogapphaseisconsistentwitharecentexperimentalobservationaswellaswith
a
QuantumMonte Carlo data of thermodynamicquantities of a unitary Fermi gas aboveTc.
J

4 PACSnumbers: 03.75.Ss,03.75.Hh,74.40.-n,74.20.-z
1

]
While the existence of a high-temperature superfluid bodypairsinthecenterofthecrossovertothemolecular
s
a phaseintheBCS-BECcrossoverofastronglyinteracting pairs in the BEC limit [1, 2] and accordingly,in orderto
g
Fermi gas is experimentally well established, important verifythe existence ofapseudogapphase,itis criticalto
-
t questions remain as to the nature of the gas above the examine the evolution of the spectral function from the
n
superfluid transition temperature T . In particular, the center of the crossover to the molecular limit [3].
a c
u question of whether or not a pseudogap state exists and Based on two recent experiments, conflicting conclu-
q how to identify itis ofimportance [1]. This is a question sions have been reached about the existence of a pseu-
t. that may have relevance to the controversy surrounding dogap state in the strongly interacting Fermi gas. On
a
m the pseudogap state in the high-Tc cuprates. While the the one hand, thermodynamic measurements [4] have
origin of this state in the cuprates is a hotly debated beeninterpretedaswelldescribedbyFermiliquidtheory,
-
d topic, with atomic Fermi gases we can answer the sim- without the need for a pseudogap state. On the other
n pler question of whether or not strong interactions and hand, momentum-resolved rf spectroscopy [5], which
o pairingfluctuationsalonecanleadtoapseudogapphase. measures the single-particle spectral function, has been
c This,inturn, tells us whetherusing suchanapproachto interpreted as evidence for a pseudogap state above T .
[ c
explain the pseudogap phase in the cuprates is a viable In this work, we present a theoretical investigation

2 option or if other mechanisms are required. of the pseudogap regime based on the t-matrix pairing-
v
As a function ofincreasinglystrongattractive interac- fluctuation approach of Ref.[3], addressing both the
6
0 tions,aFermigasexhibitsasmoothcrossover(calledthe single-particle spectral function and the thermodynam-
4 BCS-BECcrossover),fromaweaklyattractiveFermigas ics of the gas, as a function of interaction strength in
3 with a superfluid transition explained by conventional the BCS-BEC crossover. We find that, in the pseudo-
6. BCS theory, to a Fermi gas where interparticle attrac- gapregime,thesingle-particledispersionback-bendsata
0 tionsaresostrongthatthe fermionpairsformmolecules wavevectorkL neartheFermiwavevectorkF,indicating
0 andthegasiswelldescribedasamolecularBosegaswith theexistenceofa remnant Fermi surface inthisstrongly
1 a Bose-Einstein condensation transition. In the BCS interacting gas and the importance of Fermi statistics
:
v limitthe phenomena ofCooperpairingandsuperfluidity to the pairing. As interactions are increased towards
Xi occursimultaneouslyatthephasetransition,whileinthe the BEC limit, kL disappears rapidly when entering the
BEC limit pairing and Bose condensation are decoupled regimeofmolecularpairing. Thispictureissupportedby
r
a with pairing of fermionic atoms into molecules occurring acomparisonofourtheoreticalresults,whereweinclude
well above the condensation temperature. The pseudo- the effects of the trapping potential, with new experi-
gap phase refers to the normal state of a strongly inter- mental data using momentum resolved rf spectroscopy
acting Fermi gas in the center of this crossover,where it to probe the gas for different interaction strengths. In
is proposed that pairs exist above the superfluid transi- addition, we show that the theory also reproduces the
tion in analogy with the normal state of the gas in the observed linear behavior in the thermodynamics.
BEC limit. However, unlike the pairs in the BEC limit, By the experimental technique introduced in Ref.[6],
thepairsinthepseudogapstatehavemany-bodycharac- excitations of the trapped gas produced by an rf pulse
ter with the underlying Fermi statistics playing a crucial are analyzed by time-of-flight imaging to determine the
role, in analogy with the Cooper pairs of the BCS limit. wave vector of the excited atoms once the trap has been
A key prediction of theories of the pseudogap phase is switched off. The new data are presented with an im-
that there should be a smooth evolutionfrom the many- proved signal-to-noise ratio at the critical temperature
2
(kFaF)-1=0.0 (kFaF)-1=0.15 (kFaF)-1=0.45 (kFaF)-1=0.57 (kFaF)-1=0.78 (a) (b)
2 6
00..48 k/kF=0.3 1 (kFaF)-1=0.0
4
0 0
0.8 k/kF=0.6 -1
2
0.4 -2
00 ..048 k/kF=0.9 - 13 (kFaF)-1=0.15 0
4
0
0 .08 k/kF=1.2 -1
2
0.4 -2
F I00G ..0048-3.k/k-1F2=.1.-51 E0x 1xp3e-3rim-2 e-1n t 0al 1x3(c-S3iinrg-cle2-lpea-r1stic)le e0naern g1xy3d /E-F3th-2e-o1r e 0ti c1x3a-l3(-2fu-l1l 0lin 1xe3s 2) C left peak position / EF ---- 013213 (kFaF)-1=0.45 0240 HM (EDC left peak) / EF
D W
EDCforthetrapatTc,forseveralcouplingsandwavevectors. E 1 (kFaF)-1=0.57 F
4
0
-1
2
-2
T , which is accurately determined as the temperature -3 0
wchere the condensate fraction disappears. We concen- 1 (kFaF)-1=0.78
4
trate in the coupling range 0.0 <∼ (kFaF)−1 <∼ 1.0, be- - 01
2
cause the evolution of interest from the pseudogap state -2
to the molecular Bose gas occurs on the positive side of -3 0
0 0.5 1 1.5 0 0.5 1 1.5 2
the resonance. Here, aF is the scattering length associ- k/kF k/kF
ated with the Fano-Feshbach resonance and k is given
F
by h¯2k2/(2m) = E = h¯ω (3N)1/3, where ¯h is Planck FIG.2. (a)Dispersionsand(b)widthsofthelow-energyEDC
F F 0 peak. Experimentaldata(circles)andtheoreticalcalculations
constant,mtheatommass,N thetotalnumberofatoms,
for the trap (full lines) are shown for the same couplings of
and ω the average trap frequency (we set ¯h=1).
0 Fig. 1, and compared with the contribution from the radial
Ultracold Fermi gases are peculiar systems, in that shell with the largest particle number (dashed lines). In the
their interparticlecoupling canbe increasedto the point leftpanelsthefree-particledispersionk2/(2m)isalsoreported
when a descriptionin terms of a gasof molecularbosons for comparison (thin full lines).
holds, for which a real gap exists in the single-particle
spectra. Thismolecular(two-body)physicsisofnointer-
estinthecontextofthepseudogap,inasimilarfashionof
molecularbinding invacuumbeingdistinctfromCooper various panels at different k. This renders quite strin-
pairing at finite density in the presence of a Fermi sur- gent the comparison with the corresponding theoretical
face(cf. footnote18ofRef.[7]). Thequestionthenarises calculations,whichinturncontainno adjustable param-
about what fermionic feature distinguishes the pseudo- eters. Good agreement results from this comparison. In
gap from the molecular phase. We shall find that the particular,thetheoreticalcalculationswellreproducethe
back-bending of the dispersion curves obtained from the asymmetry of the experimental curves between positive
single-particle spectral function A(k,ω) (with wave vec- and negative energies, in addition to the peak positions,
tor k andfrequency ω)occursata wavevectork which widthsandheights(note howthelatterchangebyabout
L
remainsclosetok overawidecouplingrangeevenwhen one orderofmagnitude fromsmallto largek). Note fur-
F
approachingthe molecularlimit. We refer tothis special thertheexcellentagreementbetweenthetheoreticaland
wave vector as k because it is reminiscent of the Lut- experimental negative energy tails, and the gradual flat-
L
tingertheorem[8],accordingtowhichinanormalFermi tening of the EDC curves for increasing coupling due to
liquid the radius k of the Fermi sphere is unaffected by the increase of intrapair correlations.
F
the interaction. In Fig. 2 the dispersion and full width at half maxi-
Figure 1 compares the experimental and theoretical mum of the peak at lower energies are reported over a
energy distribution curves (EDC) at T for five different dense set of k values for the same couplings of Fig. 1,
c
couplings in the window of interest (see Ref.[9] for de- and compared with our theoretical calculations. Note
tails). Weemphasizethatthe experimentaldatabearon that a characteristic back-bending is revealed from these
an absolute normalization, in that only the integral over dispersions [10]. This kind of back-bending is typical of
wave vector and energy of the EDC curves (and not the a BCS-like dispersion, and is associated with the pres-
separate spectra) has been normalized to unity [9]. For ence of a pseudogap in a strongly interacting Fermi sys-
this reason,there isno independent normalizationinthe tem [3, 5, 12–14]. In addition, the large values of the
3
Figure 3(a) shows that for a homogeneous system k
L
1 (a) (b) drops rapidly to zero when (kFaF)−1 ≃ 0.75, where the
pseudogapinA(k,ω)turnsintoarealgapandthemolec-
0.8
ularlimitisreached. Accordingly,weidentifythebound-
1
k/kLF 00..46 k/kF athryisbdertowpeeoncctuhres.psAeulodnoggatphiasndevmoloulteiocunlainrtpohtahseesmwohleerce-
ular regime, the disappearance of the underlying Fermi
0.2 0
0.25 0.75 1.25 surface about occurs when the molecular size becomes
0 T/TF smaller than the interparticle spacing. The existence of
-1 -0.5 0 0.5 1 1.5 -0.5 0 0.5 1 1.5
(kFaF)-1 (kFaF)-1 a remnant Fermi surface with an enclosed volume con-
sistent with Luttinger theorem was already pointed out
FIG.3. (a)CouplingdependenceoftheLuttingerwavevector byARPESexperimentsforthepseudogapphaseofhigh-
kL for a homogeneous system at Tc, according to the theory Tc superconductors [18], but its importance for delim-
ofRef.[3](fullline)[thevalueatunitarityfromtheQMCcal- iting the pseudogap region was not appreciated in that
culation of Ref.[17] is also reported (star)]. The inset shows context [19] because the interparticle interaction could
thetemperaturedependenceofkL atunitarity(fullline),and not be controlled. The inset of Fig. 3(a) shows the
comparesitwiththoseobtainedfromthetemperaturedepen-
temperature dependence of k calculated for a homo-
denceofthechemicalpotentialofthenon-interacting(dashed L
geneous system at unitarity (full line). At high temper-
line)andinteracting(dashed-dottedline)systems. (b)Theo-
retical (full line) and experimental(squares) couplingdepen- atures when the pseudogap closes up, we have identi-
denceof kL for the trap system at Tc. fied kL as the value where the dispersion of the peak
at lower energy in A(k,ω) crosses the chemical poten-
tial [9]. This does not contradict our argument that at
low temperatures the presence of a pseudogap requires
widths (which are at least of the order of EF) and their an underlying Fermi surface, since at high temperatures
asymmetric behavior between k < kF and k > kF are the underlying Fermi surface of a Fermi liquid is not re-
associated with strong deviations from the expected be- lated to a pseudogap. The plot also shows the tempera-
havior of a normal Fermi liquid (which requires instead ture dependence of kµ0 = 2mµ0(T) (dashed line) and
the quasi-particle widths to be vanishingly small at kF kµ = 2mµ(T) (dashed-dpotted line), where µ0(T) and
[15]), and confirm the fact that single-particle states in µ(T) are the chemical potentials of the non-interacting
this region constitute poor quasi-particles. Large values and inpteracting Fermi systems, in the order, at the tem-
ofthewidthsarenotsurprisinginthecontextofthepseu- peratureT. NotethatkL aboutcoincideswithkµ0,while
dogap physics that results from pairing fluctuations [3]. k is not related with k .
µ L
Large widths were also obtained by the self-consistent t-
Figure 3(b) shows the coupling dependence of k at
L
matrix approach of Ref. [16], which however masked the
T for the trapped system, for which the theoretical pre-
c
occurrence of a pseudogap near k .
F dictions can be directly comparedwith the experimental
It is relevant to discuss how trap averaging affects data(the latterareobtainedbyaBCS-likefittothe dis-
the above results, because different radial shells in the persions of Fig. 2(a), as explained in Ref.[9]). The good
trap correspond to different locations in the coupling- comparison that results between theory and experiment
vs-temperature phase diagram of the homogeneous sys- confirmsour identificationof k as the relevantquantity
L
tem. Areasonablehypothesisisthattheradialshellwith for identifying the remnant Fermi characteristics of the
the largest particle number (whose radius rmax is esti- system in the pseudogap phase.
mated to be (0.5−0.6)RF where RF =[2EF/(mω02)]1/2 However, the occurrence of a pseudogap for a unitary
istheThomas-Fermiradius)contributesmosttothetotal FermigasaboveT has recently been questioned,follow-
c
signal. The dispersions and widths contributed by this ingaresultreportedinRef.[4]wherealineardependence
shell at rmax are represented by dashed lines in Fig. 2, oftheequationofstateasafunctionof[kBT/µ(T)]2 (kB
which show good agreement with the complete calcula- beingBoltzmannconstant)wasfittedbytheFermi-liquid
tion. This indicates that both the back-bending of the equation of state and then interpreted [20] as evidence
dispersion relations and the associated large widths are that the Fermi-liquid theory with no pseudogap can de-
not an artifact of trap averaging. scribe a unitary Fermi gas above T . To compare with
c
Despite these deviations from the behavior of a nor- the data of Ref.[4] and resolve this controversy, we have
mal Fermi liquid, in the experimental data and theoreti- usedthetheoreticalapproachofRef.[3],whichcontainsa
cal calculations there yet appears a feature which is pre- robustpseudogapassociatedwithanon-Fermi-liquidbe-
served from the physics of a Fermi liquid. That is the havior consistent with the data obtained by momentum
Luttingerwavevectork wheretheback-bendingoccurs, resolved rf spectroscopy, also to calculate the thermo-
L
whichisplottedatT vs(k a )−1 inFig.3,forahomo- dynamic properties of a homogeneous system above T .
c F F c
geneous [panel (a)] and trapped [panel (b)] system. Figure 4(a) reports the pressure in the grand-canonical
4
data, to obtain the total energy in the canonical ensem-
ble as a function of (T/T )2 reported in Fig. 4(b). This
F
10
(a) showsthatinthenewvariablethelinearbehaviorislost.
µµP(,T)/P(,0)0 2468 2µ[kT/(T)]B 000 ...1468 0.07 (0T.0/T9F 0).211 0.13 talianhuteYtinroheminett,,aosdirdityneyrsgnFeplameeitrm-eapmiiatcnihrsgtqeiaducfsilaaffienactdbctiuetothnlyvtasettitthToyaecapoeppsfvpreeserutnsedaecbtnoieaygcsteatepohodbefiistratace-ilcmpnetsealaedyrtulrfybdirxyoopgmrctaeahsptleechnuitinst--
0.1 0.3 [k B0T.5/µ(T)]2 0.7 0.9 matrixasshownintheinsetofFig.4(b)wheredeviations
1.6 fromthe non-interactingbehavior (ω+µ(T ))/E are
c F
1.4
(b) evident. Accordingly, by suitable numerical differentia-
1.2 p
0) tion of the energy data we have obtained in Fig. 4(c)
E(n,T)/E(n,0 000 ...1468 DOS 1 .125-3 -2 -111n1....026i.ω1/ 0EF 1 2 3 tmahboeodvsyepneTacimfiwicchheqreueaattnhtveistpyT,seb/uTedgFoi.ngnaAipnssgheatastrpiant,uerpmetsupurelntrsaotcuflertaehriTlsy∗tfrhwoeemrl-l
0.2 0.5 c
the t-matrix calculation, and it is also visible from the
0
0
0 0.1 0.2 0.3 0.4 QMC data at the corresponding value of T .
3.5 (T/TF)2 The experimental data in Fig. 4(c) appcear too scat-
3 (c) tered to draw definite conclusions about the presence of
2.5 the upturn and thus of a pseudogap above T . It should
T)/kB 2 be mentioned, however,thata similarupturncofthe spe-
c(V 1.5 cific heatatatemperature T∗ aboveTc wasmeasuredin
1
underdopedhigh-T cupratesandinterpretedasrevealing
c
0.5
the onset of the pseudogap regime, whereby a “residual
0
0.2 0.3 0.4 0.5 0.6 superconductivity” remains far above T [24].
c
T/TF
Inconclusion,wehaveprovidedclearexperimentaland
theoreticalevidence for non-Fermi-liquidbehaviorin the
FIG.4. ThermodynamicsofahomogeneousFermigasatuni-
tarity. (a)Pressurevs[kBT/µ(T)]2: Experimentaldatafrom normal phase of a strongly interacting Fermi gas, which
Ref.[4] (circles) are compared with QMC data from Refs.[21] we have qualified in terms of a pseudogap picture. We
(squares) and [22] (triangles), and with the t-matrix (full have further shown that this picture, that appears ev-
2
line). In the inset, the variable [kBT/µ(T)] is transformed ident in the single-particle dynamics, is also consistent
to(T/TF)2 accordingtothet-matrix. (b)Energyvs(T/TF)2 with the thermodynamic behavior of the system.
atfixeddensity: ExperimentaldatafromRef.[23](circles)are
We acknowledge financial support from the NSF and
compared with QMC data from Refs.[21] (squares) and [22]
(triangles),andwiththet-matrix(fullline). Theinsetshows from the Italian MIUR under contract PRIN-2007 “Ul-
thedensityofstatesperspin(inunitsofmkF/(2π)2)forsev- tracold Atoms and Novel Quantum Phases”.
eraltemperaturesinunitsofTcaccordingtothet-matrix,and
contrastsitwiththenon-interacting(n.i.) result. (c)Specific
heat per particle vs T/TF obtained from the t-matrix (full
line),theexperimentaldataofRef.[23](circles),andtheQMC
[1] See, Q. Chen, Y. He, C. -C. Chien, and K. Levin, Rep.
data of Ref.[21] (squares) - the dotted line is a guide to the
Prog. Phys. 72, 122501 (2009), and references therein.
eye for the QMC data. The behavior of the non-interacting
[2] M.Randeria,inProc.oftheIntern.SchoolofPhysics“En-
Fermi gas (brokenline) is reported for reference [9].
ricoFermi”CourseCXXXVIonModelsandPhenomenol-
ogy for Conventional and High-temperature Superconduc-
tivity, G. Iadonisi, J. R. Schrieffer, and M. L. Chiafalo,
Eds. (IOS Press, Amsterdam, 1998), p. 53.
[3] A.Perali,P.Pieri,G.C.Strinati,andC.Castellani,Phys.
ensemble vs [k T/µ(T)]2 as in Ref.[4], and shows that
B Rev.B 66, 024510 (2002).
the linear behavior seen in the experimental data and
[4] S. Nascimb`ene, N. Navon, K. J. Jiang, F. Chevy, and C.
QMC calculations also results from our t-matrix ap- Salomon, Nature463, 1057 (2010).
proach,both above and below the temperature at which [5] J. P. Gaebler, J. T. Stewart, T. E. Drake, D. S. Jin, A.
the pseudogapappears(indicatedbythe verticalarrow). Perali, P. Pieri, and G. C. Strinati, Nature Phys. 6, 569
TheinsetofFig.4(a)showsthatthislinearbehaviorcan (2010).
[6] J. T. Stewart, J. P. Gaebler, and D. S. Jin, Nature 454,
be ascribed to the pronounced temperature dependence
744 (2008).
of the chemical potential, because a non-linear behavior
[7] J.Bardeen,L.N.Cooper,andJ.R.Schrieffer,Phys.Rev.
results when transforming [kBT/µ(T)]2 to (T/TF)2 over 108, 1175 (1957).
the relevant range. The same change of variables can [8] J. M. Luttinger, Phys.Rev.119, 1153 (1960).
be performedin the experimental [23] and QMC [21, 22] [9] For more details, see the“supplemental material”.
5
[10] W. Schneider and M. Randeria [Phys. Rev. 81, 021601 Supplemental material: “Evolution of the
(2010)]pointedoutthattheuniversalbehaviorofaFermi Normal State of a Strongly Interacting Fermi
gaswithacontactinteraction[11]yieldsaweaknegatively Gas from a Pseudogap Phase to a Molecular
dispersing spectral feature at k≫kF even for arepulsive Bose Gas”
Fermigas.Inthatcase,however,thissecondarypeakcan-
not betraced down to k≃kF.
[11] S.Tan, Ann.Phys. 323, 2971 (2008). We provide details of the theoretical calculations of
[12] Q. Chen and K. Levin, Phys. Rev. Lett. 102, 190402 the wave-vector resolved rf signal and a description of
(2009). the experimental procedures. We also add information
[13] P.Magierski, G. Wlazlowski, A.Bulgac, andJ. E. Drut, about the theoretical analysis of the experimental data.
Phys. Rev.Lett. 103, 210403 (2009).
[14] S.Tsuchiya,R.Watanabe,andY.Ohashi,Phys.Rev.A Pairing-fluctuation theory
80, 033613 (2009); ibid. 82, 033629 (2010).
[15] P.Nozi`eres, Theory of interacting Fermi systems (Read- The theoretical approach of Ref. [1] is based on
ing, MA,1964). a diagrammatic t-matrix approximation, whereby the
[16] R.Haussmann,M.Punk,W.Zwerger, Phys.Rev.A80, fermionicsingle-particleself-energyΣ(k,ω)includespair-
063612 (2009). ing fluctuations. We have used that approach here to
[17] J. Carlson and S. Reddy, Phys. Rev. Lett. 95, 060401
calculate the single-particlespectral function A(k,ω) for
(2005).
[18] F. Ronninget al.,Science 282, 2067 (1998). the homogeneous case:
[19] C.KuskoandR.S.Markiewicz,Phys.Rev.Lett.84,963
1 ImΣ(k,ω)
(2000). A(k,ω) = −
[20] Y-ilShin,Nature 463, 1029 (2010). π [ω−ξk−ReΣ(k,ω)]2 + [ImΣ(k,ω)]2
[21] A. Bulgac, J. Drut, and P. Magierski, Phys. Rev. Lett. (1)
96, 90404 (2006). where ξk = k2/(2m) − µ. For given wave vector k,
[22] E. Burovski, N. Prokofev, B. Svistunov, and M. Troyer, the frequency structure of the real and imaginary parts
Phys. Rev.Lett. 96, 160402 (2006). of Σ(k,ω) determines the positions and widths of the
[23] S. Nascimb`ene, N. Navon, F. Chevy, and C. Salomon,
peaksin A(k,ω), andis thus responsiblefor the nontriv-
arXiv:1006.4052v1.
[24] H.-H.Wen et al., Phys.Rev.Lett. 103, 067002 (2009). ial shape of the dispersions of these peaks vs k =|k|.
Wave-vector resolved rf spectroscopy
When final-state effects in the rf transition [2, 3] can be
neglected (like for the case of 40K used in the experi-
ment), the rf signal in the normal phase is given by [3]:
1 dk
RF(ω˜)= dr A(k,ξ(k;r)−ω˜)f(ξ(k;r)−ω˜).
πN (2π)3
Z Z
(2)
Here, ω˜ = ω −ω is the detuning frequency where ω
rf a rf
is the frequency of the rf photon and ω the atomic hy-
a
perfine frequency, r the position in the trap, ξ(k;r) =
k2/(2m) − µ + V(r) a local energy with trapping po-
tential V(r) = m ω x2+ω y2+ω z2 /2, and f(ǫ) =
x y z
eǫ/(kBT)+1 −1 the Fermi function. [The prefactor in
(cid:0) (cid:1)
Eq.(2) is chosen to make the total area of the rf signal
(cid:0) (cid:1)
equalunity.] Equation(2)isbasedonalocal-densityap-
proximationwherecontributionsofadjacentshellsinthe
trap are separately considered.
The rf signal can be analyzed into its individual k-
componentstocomparewiththeexperimentaltechnique
ofRef.[4]. Theresultingwave-vector resolved rf signal is
obtained by dropping the k-integration and considering
onek-componentatatime. Moreprecisely,the selection
is made over the magnitude k = |k| while kˆ = k/k is
integrated over the solid angle, yielding:
48k2 ∞
RF(k,ω˜)= drr2A(k,ξ(k;r)−ω˜)f(ξ(k;r)−ω˜)
π2
Z0
(3)
6
where the factor k2 is from the sphericalintegrationand 2
r = |r| is the radial position in the trap. The prefactor
1
here results by expressing wave vectors in units of k ,
F
energies in units of E , and radial positions in units of 0
F
the Thomas-Fermi radius RF = [2EF/(mω02)]1/2 where -1
ω0 =(ωxωyωz)1/3 is the averagetrap frequency. -2 (k a )-1=0
Finally, to obtain an expression that can be directly F F
comparedwiththe experimentalEDCspectra,itis suffi- -3
cienttoexpressthefrequencyω˜ inEq.(3)intermsofthe 1
single-particle energy E =k2/(2m)−ω˜ via the relation
s 0
ξ(k;r)−ω˜ = E −µ(r), where µ(r) = µ−mω2r2/2 is
s 0
the local chemical potential in the harmonic trap. This -1
yields eventually: EF -2 (kFaF)-1=0.15
EDC(k,Es)= 4π8k22Z0∞drr2A(k,Es−µ(r))f(Es−µ(r)). sition / - 13
(4) o
p 0
ThenumericalresultsobtainedfromEq.(4)arethencon- k
a -1
voluted by a Gaussian broadening with a rms of about e
0.25EF, corresponding to the experimental resolution. eft p -2 (kFaF)-1=0.45
When the interparticle interaction is switched off, C l -3
A(k,Es − µ(r);r) is given by δ(Es − k2/(2m)). This D 1
defines the zero of the single-particle energy in the EDC E
0
curves as the energy of an isolated atom at rest. The
chemical potential has thus disappeared from the free- -1
particlebranchk2/(2m),whichremainspositiveforallk -2
-1
and can be used to reckon the value of the pseudogap. -3 (kFaF) =0.57
-4
Experimental procedures
1
We refer to Ref. [5] for a detailed description of the ex- 0
perimental techniques and procedures. Here, we add a -1
few comments that are specifically relevant to the data
-2
presented in the main paper. -1
-3 (k a ) =0.78
These data are taken at T/T = 1.0±0.1, where T F F
c c -4
is determined in the trapped system by the vanishing of
0 0.5 1 1.5 2
the measured condensate fraction. Note, however, that,
k/k
because the density of the trapped gas is spatially inho- F
mogeneous,thelocalcriticaltemperaturedecreasesaway
FIG. 5. χ 2−fit to theexperimental data. The experimental
from the cloud center.
data of Fig. 2(a) of the main paper (circles) are fitted over
Itwasalreadyremarkedinthe mainpaperthatanab- the interval 0.0 ≤ k/kF ≤ 2.0 by the BCS-like dispersion of
solute comparison can be made between the experimen- Eq.(5)(fulllines). Thefree-particledispersionk2/(2m)isalso
tal and theoretical EDC curves at given coupling and shown for comparison (thin full lines).
wave vector, in such a way that only their overall inte-
gral over k and E (and not the individual EDC curves)
s
are normalizedto unity. This is possible because the ex-
perimental radio frequency data (from which the EDC where k ≈ k is the special wave vector about which
L F
curves are obtained) were taken for the first time over a the back-bending occurs and µ˜ accounts for an overall
wide range of ω˜, such that their long high-frequency tail (upward) displacement of the dispersion curves. Note
could be determined and accurately compared with the that, contrary to the homogeneous case, in a trap µ˜ is
ω˜−3/2 behavior predicted theoretically [2]. not related to the value of the thermodynamic chemical
The experimental data reported in Fig. 2(a) of the potential close to T .
c
main paper have been analyzedin terms of the BCS-like Aχ2−analysisofthedataintheinterval0.0≤k/k ≤
F
dispersion: 2.0 yields the fits shown in Fig. 5. The five values of k
L
thusobtainedhavebeenreported(togetherwiththecor-
k2 k2 2 responding error bars) in Fig. 3(b) of the main paper
E (k) = µ˜ − − L + ∆˜2 (5)
s s(cid:18)2m 2m(cid:19) (there, the additionalvalue for the coupling(kFaF)−1 =
7
1.1 has been inferred from the experimental data re-
0.5
ported in Ref.[4]). F
E
The sameanalysisalsoshowsthat pairsof(µ˜,∆˜)with n / 0
µ˜−∆˜ = constant produce comparable χ2 tests. From o
Eq.(5) we note that µ˜ − ∆˜ = Es(kL) ≡ Emax corre- positi -0.5
spondstothemaximumvalueofEs(k). Forthefivecou- ak
plings here considered we obtain the values Emax/EF = pe -1
(0.40,0.24,−0.5,−1.1,−1.8),in the order. eft
L
Using these values, one can extract a rough estimate -1.5
of a (trap averaged)pseudogap energy, by relating them 0 0.25 0.5 0.75 1 1.25 1.5
with the free-particle dispersion k2/(2m) at kL. The k2- k/kF
dispersion can, in fact, be considered as a lower bound
tothedispersionoftheupperbranchintheEDCcurves,
FIG. 6. Evolution in temperature of the dispersion ω(k) ob-
which results from the two-peak structure of A(k,ω) in
tained by following the peak at lower energy in A(k,ω) for a
thepresenceofapseudogap[1]andbehaveslikek2/(2m)
homogeneoussystematunitarity. Thefourcurvescorrespond
for kF ≪ k. [In the analysis of the experimental data to temperatures T/Tc = (1.0,1.2,1.65,4.0) from bottom to
the visibility of this upper branch is suppressed by the top.
presence of the Fermi function.] The values we obtain
for[k2/(2m)−E ]/2are(0.38,0.34,0.58,0.82,1.08)E
L max F
for the five couplings of Fig. 5, which are in line with
F 1
the expected trend for the pseudogap of a homogeneous E
system (cf. Fig. 17 of Ref.[1]). n /
o
siti 0
Determination of kL for a homogeneous system k po
A comment is in order about the procedure for iden- ea
p -1
tifying the Luttinger wave vector kL for a homogeneous eft
system, as reported in Fig. 3(a) of the main paper. C l
Quite generally, A(k,ω) given by Eq.(1) has a pro- D
E -2
nounced peak when the following condition is satisfied 0 0.5 1 1.5 2
k/k
F
ω−ξk−ReΣ(k,ω) = 0 (6)
and provided ImΣ(k,ω) is sufficiently small. In a BCS- FIG. 7. The dispersion (two arcs drawn by full lines) of
like situation we write: thelow-frequency peak of A(k,ω) for a homogeneous system
with the density of the shell at r = rmax for the coupling
∆2 −1
Σ(k,ω) ≈ pg + δµ (7) (kFaF) = 0.15, is compared with the corresponding dis-
ω+iη+ξk+δµ persion (dashed line) obtained multiplying A(k,ω) by f(ω).
A BCS-like fit to thetwo arcs is also shown (dotted line).
withη =0+ andδµ=µ−µ whereµ =k2/(2m). Note
L L L
that the shift δµ is the part of the self-energy which is
responsibleforthepersistenceofaremnantFermisurface
aboutthe(temperaturedependent)Fermiwavevectorof energy in A(k,ω), from which the value of k = kL is
theunderlyingnon-interactingsystem. Acombinationof extractedaccordingtotheabovecriterion. Atsufficiently
Eqs.(6) and (7) then yields: hightemperatureswhenthedispersionω(k)crosseszero,
k is seen to quickly converge to the value associated
L
k2 k2 2 with the temperature dependent chemical potential of
ω ≈− (ξk+δµ)2+∆2pg =−s 2m − 2mL +∆2pg the non-interacting Fermi system.
q (cid:18) (cid:19)
(8)
Additional theoretical analysis of the
for the lower branch (which has the largest spectral
experimental data
<
weight for k ∼k ), where ω is measured with respect to
L
the chemical potential. The maximum value ω ≈ −∆ In the expression (4) the presence of the Fermi func-
pg
is for k = k where the back-bending occurs. For in- tion f(ω) may be of considerable help for the analysis
L
creasingtemperaturesuchthat∆ closesupeventually, of the dispersion of the low-ω peak, in situations when
pg
Eq.(6) yields accordingly k =k for ω =0, which corre- two broad non-Lorentzian structures in A(k,ω) merge
L
sponds to the familiar condition for a Fermi liquid [6]. togetheroveralimitedrangeofk. Thisisbecausemulti-
Figure 6 shows a typical temperature evolution of the plicationofA(k,ω)byf(ω)inthatexpressionactseffec-
dispersion ω(k) obtained by following the peak at lower tively as a “filter” for the low-ω structures of A(k,ω), in
8
(kFaF)-1=0.45 (kFaF)-1=0.57 1.5
0.5
0.4 k/k =0.3
F
0.3
0.2 B 1
k
0.1 T)/
0 (V
0.4 k/k =0.6 c 0.5
F
0.3
0.2
0
0.1
0 0.2 0.4 0.6 0.8 1
0
T/T
0.4 k/k =0.9 F
F
C 0.3
D
E 0.2
FIG. 9. Specific heat per particle of a non-interacting Fermi
0.1
gas (full line) reported over an extended temperature range.
0
Thedashedlinecorrespondstothelinearbehaviorthatholds
0.4 k/kF=1.2 when T/TF ≪1.
0.3
0.2
0.1
0 gap in integrated quantities, like the single-particle den-
0.4 k/kF=1.5 sity of states (DOS) [8], where a spectraldepressionsur-
0.3 x3 x3
0.2 vives at much higher temperature than in A(k ≈ kF,ω)
[cf. the inset of Fig.4(b) of the main paper, where a cal-
0.1
culation of the DOS is explicitly reported].
0
-3 -2 -1 0 1 -3 -2 -1 0 1 2 Thepresenceofthe factorf(ω)inEq.(4)obviouslyaf-
Single-particle energy / E F fects more the large-ω than the low-ω peak of the EDC
curves. The discrepancies that are evident in the large-
FIG. 8. The experimental EDC (circles) for the two cou-
ω peak from Fig. 1 of the main paper, when comparing
plings 0.45 and 0.57 are reproduced from Fig. 1 of the main
experimental and theoretical EDC curves at T for the
paper,andcompared withtheoreticalcalculations (fulllines) c
couplings0.45and0.57,canaccordinglybeattributedto
in which the temperature in the Fermi function has been de-
creased to0.7Tc. Thetheoretical curvesreported in Fig. 1of the larger absolute values of Tc at which the theoretical
themain paper are also reproduced here (dashed lines). spectraarecalculated[7],withrespecttotheexperimen-
tal values of T at which the data are taken.
c
InFig.8wereproducethe experimentalEDC(circles)
from Fig. 1 of the main paper for the two couplings 0.45
particular for those values of ω through which the back- and 0.57, and compare them with the theoretical calcu-
bending occurs in the dispersion. lations(full lines)inwhichthe temperature inthe Fermi
ThisisshownexplicitlyinFig.7,wherethedispersion function has been decreasedto 0.7Tc while the tempera-
of the low-ω peak of A(k,ω) (corresponding to a homo- tureinA(k,ω)iskeptatTc. Thisprocedureisconsistent
geneous system with the density of the shell at r for with the fact that in this coupling range the theoreti-
max
the trap coupling (kFaF)−1 = 0.15) is drawn (full line) cal approach overestimates the absolute value of Tc by
only for those values of k for which two peaks in A(k,ω) about 30%, while close to Tc the spectral function de-
appear clearly distinguishable. This procedure results in pendsessentiallyonthe relativetemperatureT/Tc. This
two arcs separated by an empty window. A single BCS- procedure, albeit empirical, goes in the right direction
like fit (dotted line) to these two disconnected arcs via of reducing the height of the high-ω peak of the EDC
Eq.(5)providesthevalue∆˜(r )/E =0.77,inreason- curves making it closer to the experimental value, while
max F
able agreement with the value determined for the whole affecting only slightly the low-ω partof the EDC curves.
trap. Figure 7 shows also the dispersion (dashed line) The numerical difference between the theoretical and
obtained by multiplying A(k,ω) by f(ω), in such a way experimental values of Tc for a homogeneous Fermi gas
that the low-ω peak can be smoothly followed even in at unitarity is also evident from Fig. 4(c) of the main
the k-window that had to be excluded before. This pro- paper, although this difference is immaterial to the sake
cedure does not appreciablyalter the values obtained by of the argument that was there raised.
the BCS-like fit.
Specific heat of a non-interacting Fermi gas
Thisconclusionisconsistentwiththefactthatthelack
ofaspectraldepressioninA(k,ω)inalimited rangeofk InFig.4(c)ofthe mainpaper the behaviorofthe spe-
doesnotnecessarilyleadtodisappearanceofthepseudo- cificheatofanon-interactingFermigaswasreportedfor
9
comparison over the temperature interval 0.2<∼T/TF <∼
0.6, which was relevant to the experimental data shown
in the same figure. [1] A.Perali,P.Pieri,G.C.Strinati,andC.Castellani,Phys.
Rev.B 66, 024510 (2002).
[2] A. Perali, P. Pieri, and G. C. Strinati, Phys. Rev. Lett.
100, 010402 (2008).
[3] P. Pieri, A. Perali, and G. C. Strinati, Nature Phys. 5,
Itisinterestingtoshowthesamequantityoveramore
736 (2009).
extended temperature range which reaches T = 0. This [4] J. T. Stewart, J. P. Gaebler, and D. S. Jin, Nature 454,
is done for 0 ≤ T/TF ≤ 1 in Fig. 9, where the specific 744 (2008).
heat per particle of a non-interacting Fermi gas is re- [5] J. P. Gaebler, J. T. Stewart, T. E. Drake, D. S. Jin, A.
ported vs T/T (full line) and compared with its linear Perali, P. Pieri, and G. C. Strinati, Nature Phys. 6, 569
F
approximation (k π2/2)T/T (dashed line) that holds (2010).
B F
[6] E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics:
when T/T ≪ 1. Note that for T/T = 0.2 this linear
F F
Theory of the Condensed State (Butterworth-Heinemann,
approximationdeviates from the full calculation already
Oxford, 1980), Section 14.
by about 20%.
[7] A. Perali, P. Pieri, L. Pisani, and G. C. Strinati, Phys.
Rev.Lett. 92, 220404 (2004).
[8] S. Tsuchiya, R. Watanabe, and Y. Ohashi, Phys. Rev. A
80, 033613 (2009).
An analogous linear behavior is known to result for a [9] See, e.g., D. Pines and P. Nozi`eres, The Theory of Quan-
Fermi liquid when T/T ≪ 1, although with a different tum Liquids: Normal Fermi Liquids (Addison-Wesley,
F
slope reflecting the renormalization of the mass [9]. Reading, 1996), Section 1.3.