understanding and quantitative account of crit-
ical scaling functions remain elusive. Related
issuesariseinawiderangeofstronglycorre-
lated materials such as high-temperature super-
conductors, where the anomalous charge and
energy transport near quantum criticality is
not well understood ( 12 , 13 ).
Ultracold fermionic atoms in the strongly
interacting limit, that is, the unitary Fermi gas
( 14 ), offer great promise for studying the sec-
ond sound attenuation and elucidating the
critical phenomena. First, as a consequence of
scale invariance, the thermodynamic and dy-
namic properties of the unitary Fermi gas are
universal functions of the reduced tempera-
tureT=TF( 15 – 20 ). Here, the Fermi tempera-
tureTF≡ℏ^2 k^2 F=ðÞ 2 mkB is determined by the
atomic massm, the densityn, and the Fermi
wave numberkF¼ð 3 p^2 nÞ^1 =^3 ;kBandħdenote
the Boltzmann and reduced Planck constants,
respectively. Thus, the second sound diffusiv-
ityD 2 and the thermal conductivitykat tem-
peratureTare similarly universal functions of
T=TF. Second, thanks to the unprecedented
controllability ( 14 ), the critical region of the
unitary Fermi gas can be precisely probed to
investigate the critical transport behaviors.
Over the past decades, great efforts have been
devoted to probing the sound propagation and
attenuation in the unitary Fermi gas. The sec-
ond sound propagation has been observed in a
highly elongated harmonic trap ( 21 ), but the
attenuation remains undetermined because of
the density inhomogeneity. The first sound at-
tenuation has been recently measured by con-
fining the unitary Fermi gas into a box potential,
eliminating the inhomogeneity problem ( 22 ).
However, observing the second sound attenu-
ation is challenging because the signal is too
weak to be resolved from noise.
Here,wemeasuredsecondsoundattenua-
tion in a homogeneous unitary Fermi gas of(^6) Li atoms ( 23 , 24 )withextremelylargeFermi
energy by developing a Bragg spectroscopy
technique with small wave numberkand high
energy resolution. We successfully determined
the second sound diffusivity and the thermal
conductivity of the unitary Fermi gas; the super-
fluidfractionandtheshearviscosityarealso
obtained with improved accuracy. In the super-
fluid phase,D 2 andkattain the universal
quantum values ofħ/mandnℏkB=m, respec-
tively. Near the superfluid transition, a sudden
rise inD 2 andkis observed, consistent with
the critical divergence phenomena predicted
by the dynamic scaling theory ( 3 – 5 ). We find a
surprisingly large quantum critical region char-
acterized byjjt≲ 0 :05, where the dimension-
less temperaturet≡ 1 T=Tcmeasures the
proximity to the superfluid transition tem-
peratureTc. Our measurements accomplish a
quantitative experimental examination of the
dissipative two-fluid hydrodynamic theory for
the strongly interacting Fermi gas. Furthermore,
the observed universal transport coefficients can
provide insight into the anomalous transport
of strongly correlated materials such as the
cuprates ( 13 ) and provide a benchmark for
many-body theories ( 25 ).
Experimental scheme and setup
The measurement of first and second sound
rests on the dissipative two-fluid hydrodynam-
ic theory for the density response function at
wave numberkand frequencyw( 6 – 8 , 26 ):
cnnðÞ¼k;w
nk^2
m
w^2 v^2 k^2 þiDsk^2 w
ðÞw^2 c^21 k^2 þiD 1 k^2 wðÞw^2 c 22 k^2 þiD 2 k^2 w
ð 1 Þ
which is deduced from the conservation laws
for momentum and energy. Here,c 1 (c 2 ) andD 1
(D 2 ) are, respectively, the speed and diffusivity
of first (second) sound, whereasvandDsare,
respectively, the speed and diffusivity associ-
ated with thermodynamic and transport prop-
erties of the system ( 6 – 8 , 26 ). In the superfluid
phase, two propagating sound waves with at-
tenuation or damping rateGi≡Dik^2 (i= 1, 2)
can be clearly identified near the two poles
wi¼cikin the response function, which can
be expressed asc
ðÞi
nn∼Zi=w^2 c^2 ik^2 þiGiw
with weightZi; above the superfluid transition,
the critical second sound becomes a thermally
diffusive modecðÞnn^2 ∼ 1 =ðÞwþiD 2 k^2 , and the
diffusivityD 2 ¼k=ðÞmncP is fully character-
ized by the thermal conductivitykand the
specific heat at constant pressurecP.
It is notable that the simple form of Eq. 1 is
applicable for quantum liquids with strong cor-
relations ( 6 ), provided thatkandware small
in comparison with the inverse correlation
lengthx ^1 and inverse collision timet ^1. How-
ever, a careful experimental validation of this
theory is difficult in liquid helium for two
reasons: (i) The narrow critical region is diffi-
cult to reach by Brillouin scattering ( 9 ), and (ii)
the second sound weightZ 2 determined by the
Landau-Placzek ratioDLPis very small; here,
DLP≡cP=cV 1, withcVbeing the specific heat
at constant volume. In this work, we accom-
plish this by developing a high-resolution Bragg
spectroscopy technique with smallkto probe
the density response of a homogeneous unitary
SCIENCEscience.org 4 FEBRUARY 2022•VOL 375 ISSUE 6580 529
Square Tub
e
DM
Moving
lattice
k^2
k 112
Sheet
Lens
Dipo
le Trap
Dipole Tr
ap
A
C
z (
Y Z
X
0 20 40 60 80
-8
-4
0
4
8
D
B
k 1 k 2
z (
0 20 40 60 80
-4
-2
0
2
4
Fig. 1. Creation of sound waves.(A) Sketch of the experimental setup. The rectangular-box trap consists
of a square tube and two sheets of 532-nm laser beams, generated by two spatial light modulators. DM,
dichroic mirror; CCD, charge-coupled device. (B) A pair of 741-nm laser beams with wave vectors (k 1 ,k 2 ) and
frequencies (w 1 ,w 2 ) exactly intersect on the homogeneous unitary Fermi gas with a small angle, producing
a one-dimensional moving-lattice potential. The wave numberk=|k 1 −k 2 | of the Bragg lattice can be accurately
determined by the horizontal CCD camera in (A), whereas another vertical CCD camera is used to probe the
in situ density profile of the cloud. (CandD) First and second sound waves in a unitary Fermi superfluid at
0 :84 2ðÞT=Tc. The top panels show typical single-shot difference images of density responsedn, with the lattice
frequencyw¼w 1 w 2 of 2p× 2.1 kHz for first sound (C) and 2p× 0.3 kHz for second sound (D) (see text).n(z)
anddn(z) are obtained, respectively, by further integrating the reference and difference images along the
transverse direction in the dashed box (i.e., region of interest 84.40mm by 41.39mm). The bottom panels show
the normalized density wavesdn(z)/n(z) along the longitudinalzaxis. The solid line is a guide to the eye.
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