Science - USA (2022-02-04)

EFas mentioned earlier, enable us to measure
the density response functioncnnwith high
signal-to-noise ratio.

Density response spectra

The density response spectrajjcnnðÞk;w over a
wide temperature range of 0: 42 ≤T=Tc≤ 1 : 04
were systematically measured. The spectra from
0 :75 2ðÞTcto 1:04 2ðÞTcare presented in Fig. 2,
accompanied by the fittings to Eq. 1 (solid
lines). A high-frequency first sound peak at
ℏw∼ 0 : 046 EFis clearly visible over the whole
temperature range, varying smoothly across
the superfluid transition. More importantly,
a second sound peak can be unambiguously
identified at low frequencyℏw< 0 : 01 EF. This
is particularly evident in the temperature range
of 0: 79 ≤T=Tc≤ 0 :94, as highlighted in the left
panelofFig.2A.Thereisanotablechange
in the second sound peak from 0:98 2ðÞTcto
1 :01 2ðÞTc: The line shape becomes diffusive
and shoulder-like with much reduced height
(see the left panel of Fig. 2B), indicating that
second sound is the critical mode character-
izing the superfluid transition. Two intriguing
features of the spectra are worth mention-
ing: (i)jjcnnðÞk;w is nonzero atw= 0, which
is contributed by the real part ofcnnðÞk;w
and agrees with the compressibility sum rule
( 6 ),cnnðÞ¼k;w¼ 0 n=mv^2 T



. Here,vTSðÞ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð@P=@nÞTSðÞ=m

q
is the isothermal (adiabatic)

sound speed. (ii) The coupling between first
and second sound is appreciable, and thus
the sound peaks do not show the symmetric
Lorentzian line shape expected for prop-
agating sound. To recognize the respective
contributions of the first and second sound

responses, the imaginary parts ofcnnðÞk;w

andc

ðÞi

nnðÞk;w are reconstructed using the fit-

ting results of sound speed and diffusivity. A

well-defined propagating second sound with a

Lorentzian line shape is then observed for tem-

peratures up to a threshold value of 0:98 2ðÞTc

(see fig. S10).

The threshold temperature 0:98 2ðÞTcis con-

sistent withT< 0 : 986 Tc, an estimate based

on the hydrodynamic criterionkx<1. The

confirmation of two-fluid hydrodynamics is

also supported by the excellent curve fittings in

the temperature range of 0: 75 ≤T=Tc≤ 0 :98,

as reported in Fig. 2, allowing us to accurately

determine the sound speedciand diffusivityDi.

Moreover, to independently validate the reach

of the hydrodynamic regime, a series of wave

numberskaround 0: 058 kFare implemented to

measurejjcnnðÞk;w, andGi¼Dik^2 is achieved

with nearly the same sound speed and dif-

fusivity ( 32 ).

However, for temperatures from 0:99 2ðÞTc

to 1:01 2ðÞTc, the two-fluid hydrodynamic mod-

el becomes inadequate. Although the sound

speeds and diffusivities can be still acquired

from the curve fitting to Eq. 1, a more accurate

determination requires a nonperturbative dy-

namic scaling analysis ( 3 – 5 ), in which the crit-

ical second sound response is a universal

function ( 33 ) ofw=a∞k^3 =^2



at fixed values of

kx. Here, the constanta∞sets the energy scale

in the critical regime. Finally, we mention that

the second sound cannot be resolved in the

spectra forT< 0 : 75 Tc(see fig. S6 for an ex-

ample) for two possible reasons. First, the

Landau-Placzek ratioDLPbecomes smaller

( 7 , 26 ), leading to a negligible second sound

weightZ 2 in the density response spectrum

jjcnnðÞk;w. Second, there is a transition from

the hydrodynamic to collisionless regime toward

low temperatures ( 7 ), which occurs at a typical

temperature of about 0: 4 Tcfor superfluid

helium ( 34 ). Nevertheless, the high-frequency

sound peak is consistently well-resolved in the

spectra down to the lowest achieved temper-

ature of 0:42 2ðÞTc.

Sound speeds and superfluid fraction

The normalized sound speedsc 1 =vFandvS=vF

as a function ofT=Tcare reported in Fig. 3A.

Our high-resolution spectra yield very accu-

rate first sound speedsc 1 and adiabatic sound

speedsvS, with a typical relative error of just

~1%, allowing us to determine the universal

state functions ( 16 – 19 ) of the unitary Fermi

gas through standard thermodynamic relations

( 32 ). Specifically, from the saturated first sound

speedc 1 =vF¼ 0 :350 4ðÞat the lowest achieved

temperature 0:42 2ðÞTc,wededucetheBertsch

parameterx¼ 0 :367 9ðÞby using the relation

c 1 =vF¼

ffiffiffiffiffiffiffiffi

x= 3

p

. This value is in excellent agree-
ment with the previous thermodynamic mea-
surement value ( 18 ) corrected in ( 30 ) ofx¼
0 :370 5ðÞðÞ 8 and the latest quantum Monte
Carlo result ( 35 ) ofx¼ 0 :367 7ðÞ.
For the second sound speedc 2 , an intriguing
feature is the sensitive temperature depend-
ence:c 2 =vFdecreases rapidly with increasing
temperature up to 0:98 2ðÞTc(Fig. 3B). Notably,
c 2 =vFsuddenly jumps to a saturated value of
~0.02 at 0:99 2ðÞT=Tc, implying the breakdown
of hydrodynamics near the superfluid transi-
tion. From the measured sound speedc 2 orv,
we determine a fundamental quantity for the
macroscopic description of superfluidity—
the superfluid fractionns=n—by applying the

SCIENCEscience.org 4 FEBRUARY 2022•VOL 375 ISSUE 6580 531

0.4 0.6 0.8 1.0

0.35

0.39

0.43

0.7 0.8 0.9 1.0

0.00

0.05

0.10

0.7 0.8 0.9 1.0

0.0

0.2

0.4

0.6

n

/ns

T/TC

A B

c 1 /vF

vs/vF

c^1

/v

F

(v

/vs

)F

T/TC

C

c 2 /vF

v/vF

c^2

/v

F

(v/v

)F

T/TC

Fig. 3. Sound speeds and superfluid fraction.(A) Temperature dependence of the normalized first sound speedc 1 =vF(blue circles) and adiabatic sound speedvS=vF(orange

circles). (B) Temperature dependence of the normalized second sound speedc 2 =vF(purple circles) and associated sound speedv=vF(green circles), wherev¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

c^21 þc^22 v^2 S

q

.

Here,vF¼ℏkF=mis the Fermi speed, and all the sound speeds are obtained from fitting the density response spectra (see Fig. 2). (C) Temperature dependence of
the superfluid fractionns=n, compared with that of superfluid helium (green dash-dotted line). Vertical error bars represent one standard uncertainty obtained from the
curve fitting and the measured universal thermodynamic functions; the horizontal error bars show the statistical uncertainty of the temperature determination.

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