density, caused by interatomic collisions and the
inhomogeneous dephasing along the imaging
line of sight.
In Fig. 2B, we show the density dependence of
W (assuming for now W = –d0) for weak and nearly
unitary interactions. n is the ↑ density experienced
by an atom, averaged over the imaging line of sight
and a small radial bin in the image plane (30). For
weak interactions, W º n, consistent with the
expected dominance of two-body correlations for
n|a|3 ≪ 1. However, close to unitarity, WðnÞis
nonlinear and even changes sign, which cannot
be explained by two-body physics.
For a quantitative analysis, we first focused
on very low densities. In this limit, W is dominated by two-body correlations at all interaction
strengths. From the measurements of WðnÞ, we
extracted the initial slope a ¼ @W=@njn¼0 (Fig.
2B), which gives the behavior of W at vanishing
In Fig. 3A, we plot a across the Feshbach resonance for T = 125 ms. The solid red line shows
a ¼ 8pℏℜðf Þ=m, where ℜð f Þ is the real part
of the scattering amplitude f (34), averaged
over the thermal momentum distribution, and
ℏ is the reduced Planck constant; the dashed
orange line is the weakly interacting limit a ¼
8pℏa=m. Using Eq. 1, from our measurements,
we extracted C2/n2 = a4pma=ℏ. This is plotted
in Fig. 3B, along with an analytic prediction for
C2 (35). Over two orders of magnitude in C2, we
find excellent agreement between theory and
In our search for C3, a key prediction of Eq. 1 is
that the C2 contribution to W vanishes exactly
at B0. In the inset of Fig. 3A, we show mea-
surements focused on the resonance region and
verify that this is indeed the case. Here, we
measured d0 for two evolution times, T1 = 40 ms
and T2 = 125 ms, to assess the instantaneous W at
t = 82.5 ms according to Eq. 2. We also varied T1
and T2 and found that a is always consistent
with the equilibrium theory curve (red shad-
ing). This is in agreement with our simulations
of the two-particle dynamics after an interac-
tion quench (30). We theoretically find that C2
equilibrates on a time scale t2, which is ≈ma2=ℏ
away from the Feshbach resonance and ≈ml2=ℏ
at unitarity; for our experimental parameters, t2
is shorter than the first RF pulse.
We next turned to higher densities and strong
interactions, where the effect of C3 should be
prominent. We always reconstructed the instan-
taneous W(t), and in Fig. 4A, we show it for t =
90 ms and two different densities. At high density, a
nonzero W at unitarity is evident, which, per
Eq. 1, cannot arise from a C2 contribution [see
also (36, 37)]. Additionally, away from unitarity,
at B < B0, an intriguing suppression of W is ap-
parent, which coincides with the previously ob-
served strong suppression of three-body losses
(at a ≈ 5600a0) (23), qualitatively associated
with atom-dimer physics (38).
We focused on the nonzero W at unitarity and
verified that it arises from three-particle corre-
lations by looking at its scaling with density. Ac-
cording to Eq. 1, a C3 contribution to W should
scale linearly with n2l4. In Fig. 4B, we show that
log[W(B0)] is linearly dependent on logðn2l4Þ,
confirming a power-law scaling (30). To determine
the exponent itself, we fitted a line to the data and
extracted the slope g = dðlogWÞ=d½logðn2l4 Þ;.
We find g = 1.0(1), which is in excellent agree-
ment with the predicted three-body scaling.
Last, we studied the value of the unitary C3. In
contrast to C2, we observed a gradual develop-
ment of C3 over the time scale of our experiment
(Fig. 4C), which means that after the interac-
tion quench, the three-body correlations develop
slower than the two-body ones. For t ≲ 50 ms, the
three-body contact is consistent with zero (within
our error bars), whereas at our longest times, t ≈
100 ms, it approaches the theoretical expecta-
tion for the equilibrium unitary gas, C3/(n3l4) ≈
Our measurements provide a conclusive obser-
vation of the effects of three-body correlations on
the coherent behavior of an atomic Bose gas. The
nonequilibrium dynamics of the three-body con-
tact are an interesting open problem for future
study. It would be exciting to extend our technique
to a deeply degenerate gas, for which C3 is not
even theoretically known (20). In our harmonic
trap setup, starting with a noninteracting Bose
condensate would result in prohibitively short
lifetimes after the quench to unitarity, but this
problem could be mitigated by using a uniform
trapping potential (39).
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SCIENCE sciencemag.org 27 JANUARY 2017 • VOL 355 ISSUE 6323 379
Fig. 4. Three-body contact. (A) W at t = 90 ms, for densities nl3= 0.13 (light blue) and nl3 = 0.54 (purple). For reference, we also show the two-body
prediction for the lower-density data (blue line). The red band indicates the position of the Feshbach resonance, and the gray band shows the location of
the previously observed minimum in the three-body loss rate (23). (B) Log-log plot of the density dependence of W(B0); t = 90 ms data are averaged
within the red band shown in (A). The linear fit (blue line) gives a slope g = 1.0(1), in excellent agreement with the three-body scaling law. (C) The three-body contact density C3 as a function of time after the interaction quench. The horizontal purple line shows the theoretical prediction for the equilibrium
unitary Bose gas (20).