well-known relationc^22 ¼ðÞTs^2 ns=ðÞcPnn or
v^2 ¼ðÞTs^2 ns=ðÞmcVnn, wheresis the entropy
density andnn≡nnsis the normal fluid den-
sity ( 32 ). As shown in Fig. 3C, the superfluid
fraction of a unitary Fermi superfluid is close
to that of superfluid helium ( 36 ) in the vicinity
of the superfluid transition; for example, the
system contains about 20% superfluid compo-
nent at 0:97 2ðÞT=Tc. However, as the temper-
ature decreases further,ns=nof the unitary
Fermi superfluid notably deviates from
that of superfluid helium, indicating the dis-
tinctive nature of superfluidity in the system.
ns=nof the unitary Fermi superfluid has been
indirectly extracted from the one-dimensional
second sound speed measured in a highly elon-
gated harmonic trap ( 21 ). Our direct measure-
mentsimprovetheaccuracyonns=n, thereby
providing a benchmark for theoretical calcu-
lations, which so far remain a notoriously dif-
ficult task in quantum many-body physics.
Sound diffusivity and transport coefficients
Toaddressthemainfocusofthiswork—the
first and second sound attenuation—we pres-
ent the temperature dependence of sound dif-
fusivities and transport coefficients in Fig. 4.
For the sound diffusivityDi, two important
features are evident, as shown in Fig. 4, A and
C. One is that all theDiare in the vicinity of the
quantum Heisenberg limit, that is,Di∼ℏ=m,
which is anticipated for strongly correlated
quantum liquids owing to the absence of
well-defined quasiparticles ( 7 ). For instance,
motivated by holographic duality ( 13 , 37 ), the
diffusivityDof any diffusive mode should obey
D≳ℏc^2 =ðÞkBT, wherecis a typical speed scale
of the system. By takingT∼Tc≃ 0 : 17 TFand
c∼vS≃ 0 : 43 vFatTc(see Fig. 3) for the unitary
Fermi gas, we find the boundD~ħ/m. The
other one is that each diffusivity shows a
sudden rise very close toTc(i.e., atT∼ 0 : 95 Tc),
with an incrementDDi∼ 0 : 3 ℏ=m. We interpretthe sudden rise as a precursor of quantum
criticality near the superfluid transition, where
the sound attenuation and thermal conductiv-
ity start to exhibit critical divergence ( 3 , 5 ).
In liquid helium, the critical divergence has
been observed both in the first and second
sound attenuation in the temperature interval
ofjjTTc<1 mK orjjt< 5 10 ^4 ( 9 , 10 ).
The quantum critical region of the unitary
Fermi gas (i.e.,jjt< 0 :05) is thus about 100
times larger than that of the liquid helium.
This extremely large critical region makes the
unitary Fermi gas an ideal platform to deter-
mine the universal critical ratios [e.g.,R 2 ¼
D 2 =ðÞ 2 c 2 x] that remain elusive ( 5 ). We note
that the first sound diffusivityD 1 of the unitary
Fermi gas has been measured recently ( 22 ),
and the obtained results agree with ours. How-
ever, the sudden rise inD 1 nearTchas not been
resolved because of the relatively large uncer-
tainty of the measurement ( 22 ).
Two general damping mechanisms account
for the sound attenuation: (i) The first is the
viscous damping stemming from the diffusion
of momentum, characterized by the shear vis-
cosityhand four bulk viscositieszi(i=1,2,3,
4). For the unitary Fermi gas, most of the bulk
viscosities vanish thanks to the scale invar-
iance, and the only remainingz 3 turns out to
be negligible ( 38 ). (ii) The second is the ther-
mal damping caused by the diffusion of heat,
characterized by the thermal conductivity
k. The relative contribution of these mecha-
nisms can be quantified by the dimensionless
Prandtl number Pr≡hcP=k. From the sound dif-
fusivity, we determinek¼cV½ðD 1 þD 2 Þmn
4 hn=ðÞ 3 nn andh¼ 3 nm DðÞ 1 þD 2 Ds=4( 32 )
and present the results in Fig. 4, B and D,
respectively.
The shear viscosity in Fig. 4D exhibits a
weak temperature dependence below about
0 : 95 Tc, settling at a nearly constant value—
the quantum limith~nħ. However, a smooth,
but pronounced, increase is observed in the
vicinity of the superfluid transition. The trap-
averaged shear viscosity of a unitary Fermi gas
in a harmonic trap has been previously mea-
sured through anisotropic expansion ( 20 ), and
the local shear viscosity has also been indirect-
ly extracted ( 39 ). Thehobtained from our
direct measurement is about two times larger
than the previous result ( 39 )inthesuperfluid
phase. Moreover, as a quantitative measure,
the inset of Fig. 4D shows the ratio of shear
viscosity to entropy densityh/s, which is ex-
pressed in the unit ofℏ=ðÞ 4 pkB, the lower
bound conjectured by Kovtun, Son, and Star-
inets (KSS) for a perfect fluid ( 40 ). Around the
superfluid transition,h/sis about 18 times
larger than the KSS bound, suggesting that
the unitary superfluid is not a“perfect fluid.”
The thermal conductivityksimilarly attains
the universal quantum limit∼nℏkB=mbelow
about 0: 95 Tc, as shown in Fig. 4B. Notably, a532 4 FEBRUARY 2022•VOL 375 ISSUE 6580 science.orgSCIENCE
0.7 0.8 0.9 1.01.31.61.92.21.01.21.41.6D2D1T/TC0.70.80.91.01.21.51.8DsBCD0.7 0.8 0.9 1.00.81.01.21.41.6T/TC0.7 0.8 0.9 1.017253301234A0.7 0.8 0.9 1.01234PrkB)/m)B)Fig. 4. Temperature dependence of sound diffusivities and transport coefficients.(A) The second
sound diffusivityD 2 .(B) The thermal conductivityk. The inset shows the Prandtl number, with the line
marking Pr = 1. (C) The first sound diffusivityD 1 , together with the associated sound diffusivityDsin the inset,
whereDs¼D 1 þD 2 34 nmh.(D) The shear viscosityh. The inset shows the viscosity-to-entropy ratio, in the
units ofℏ=ðÞ 4 pkB. Away from the superfluid transition, the temperature dependence inD 1 andD 2 can be
understood from the relationsD 1 ∼h=ðÞnmandD 2 ∼hns=ðÞnmnn, which are valid at low temperatures. The
saturatedD 1 is consistent with a nearly constant shear viscosity, whereas the rapid increase ofD 2 may be
caused by the loss of the normal fluid component, that is,ns=nn→∞asT→0. A similar temperature
dependence of the second sound diffusivityD 2 has been observed in the superfluid helium ( 10 ). Vertical error
bars represent one standard error.
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