Fermi gas, in whichZ 2 andDLPturn out to be
sizable near the superfluid transition ( 26 ).
In Bragg spectroscopy ( 27 – 29 ), the density
response function links the density fluctuation
dn(k,w) (i.e., the response of the system) to an
applied weak potential perturbationdV(k,w)
viadnkðÞ¼;w cnnðÞk;wdVkðÞ;w. To achieve a
good signaldn, we create a high-density spin
mixture of^6 Li atoms, which are equally pop-
ulated in the lowest two hyperfine states at
832.18 G [i.e., the unitarity, where s-wave scat-
tering length diverges ( 30 )]. After using forced
evaporative cooling in a crossed dipole trap,
about 1 × 10^7 atoms close toTcare adiabatical-
ly loaded into a 151mm–by– 55 mm–by 55mm
rectangular-box trap ( 23 , 24 , 31 ). The box trap
consists of a square tube and two sheets of
532-nm laser beams, as depicted in Fig. 1A,
and has a maximal potential depth of about
2 pħ× 160 kHz. To prepare homogeneous Fermi
superfluids at variousT=Tc, whereTc≃ 0 : 17 TF,
we adiabatically lower the potential depth to
different final values and hold the trap for an
additional 500 ms to reach thermal equilibrium.
We find that the densitynand the reduced
temperatureT=Tcdecrease monotonically with
the decreasing potential depth of the box trap.
For a typical cloud atT=Tc≃ 0 :84, the realized
density isn≃ 1 : 56 1013 cm^3 , the Fermi wave
number iskF≃ 2 p 1 : 23 mm^1 ,andtheFermi
energy isEF≃ 2 pℏ 50 :1 kHz. Two important
features of our system are worth mention-
ing: (i) The densityndecreases by only about
8%, from 1.64 × 10^13 cm–^3 close toTcto 1.50 ×
1013 cm–^3 at 0:75 2ðÞTc. (ii) The 1/elifetime,
whereeis Euler’s number, of the unitary Fermi
superfluid is quite long, that is, more than 20 s,
and the heating of the system is very weak.
This preparation of a homogeneous unitary
Fermi gas with well-controlled temperature
and extremely large Fermi energy makes the
probe of the extremely weak second sound
response possible ( 32 ).
The Bragg lattice potentialdVzðÞ;tB ¼
V 0 sinðÞkzwtBQðÞtB is engineered by apply-
ing a pair of coherent 741-nm laser beams with
a frequency differencewthat intersect at the
location of the gas (see Fig. 1B). Here, 2V 0 is
the potential depth,zis the longitudinal axis
of the cloud, andQðÞtB is the Heaviside step
function. The laser beams are carefully chosen
to be far-off-resonant and to have a large beam
diameter, which is pivotal for minimizing un-
wanted heating during the perturbation and
ensuring the uniformity of the Bragg lattice
potential. It is known that the correlation
length diverges asx∼kF^1 jtjnnear the super-
fluid transition with the critical exponent
n≃ 2 =3 given by theFmodel ( 5 ). Experimentally,
a small wave numberk¼ 2 p 0 : 071 mm^1 ≃
0 : 058 kF is applied by adjusting the inter-
section angle between two lattice lasers. If we
use the criterionkx∼ 0 : 058 jtj^2 =^3 <1, the hy-
drodynamic regime could be reached over a
wide range of temperatures unless it is very
close toTc,thatis,jjt< 0 :014. Arising fromthe Bragg lattice potential, the steady-state
density response takes the form ofdnzðÞ¼;tB
jjcnnðÞk;w V 0 sin½kzwtBþfðÞk;w , where
jjcnnðÞk;w andfðÞk;w are the modulus and
argument ofcnnðÞk;w, respectively. Experimen-
tally, a carefully chosenV 0 of about 0:5%EF
(1.51 × 10–^31 J) and perturbation duration of
3 ms are implemented to satisfy the criteria
of linear steady-state response. With these
optimized parameters, the density response
dnatwcan be acquired by subtracting two
high-resolution in situ images, which are
taken at the givenwandwref¼ 2 p1 MHz,
respectively, with the latter being the ref-
erence. Figure 1, C and D, shows two distinct
density wavesdn(z)/n(z) of the superfluid
atT≃ 0 : 84 Tc,thatis,firstsoundatw=2p×
2.1kHzandsecondsoundatw=2p× 0.3 kHz,
respectively ( 32 ).
Two key technical advantages of our Bragg
spectroscopy are worth noting ( 32 ): (i) The
modulusjjcnnðÞk;w can be directly obtained
from the integration ofjjdnzðÞ=nzðÞas a func-
tion ofwso that we avoid potential errors
owing to the imperfect phase synchronization
for acquiring Im½cnnðÞk;w from out-of-phase
density response. (ii) A steady-state density
response is taken, and thus the finite pertur-
bation duration does not lead to a spectrum
broadening nor does it set a frequency reso-
lution in our experiment. These two advan-
tages, combined with the ability to prepare a
homogeneous Fermi gas with extremely large530 4FEBRUARY2022•VOL 375 ISSUE 6580 science.orgSCIENCE
0.000 0.020 0.040 0.060 0.000 0.020 0.040 0.0603.03.54.00.000 0.006 0.0123.43.94.40.000 0.006 0.0123.23.74.20.75
0.79
0.84
0.88
0.91
0.94T/Tc
0.97
0.98
0.99
1
1.01
1.0416 T/Tc116
T/Tc= 0.84^5AB
139/EF /EFT/Tc= 0.94Fig. 2. Cascade plots of density response spectra at various temperatures.(A) The spectra from 0:75 2ðÞTcto 0:94 2ðÞTc(top to bottom) are shown on the right.
The two subplots on the left give a zoomed-in view of the low-frequency second sound response at 0:84 2ðÞTcand 0:94 2ðÞTc, respectively. (B) The spectra from
0 :97 2ðÞTcto 1:04 2ðÞTc(top to bottom), with the second sound response highlighted on the left. Every data point corresponds to an average value of about 30 to
50 independent results, each obtained from a measured single-shot density wave similar to the one shown in Fig. 1, C or D ( 32 ). The error bars represent one standard
deviation. The solid lines are the fitting curves, obtained by using Eq. 1.
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