Measurement of the Homogeneous Contact of a Unitary Fermi Gas

By selectively probing the center of a trapped gas, we measure the local, or homogeneous, contact of a unitary Fermi gas as a function of temperature. Tan’s contact, $C$, is proportional to the derivative of the energy with respect to the interaction strength and is thus an essential thermodynamic quantity for a gas with short-range correlations. Theoretical predictions for the temperature dependence of $C$ differ substantially, especially near the superfluid transition, $T_c$, where $C$ is predicted to either sharply decrease, sharply increase, or change very little. For $T/T_F>0.4$, our measurements of the homogeneous gas contact show a gradual decrease of $C$ with increasing temperature, as predicted by theory. We observe a sharp decrease in $C$ at $T/T_F=0.16$, which may be due to the superfluid phase transition. While a sharp decrease in $C$ below $T_c$ is predicted by some many-body theories, we find that none of the predictions fully account for the data.

Figure 1

Schematics of the experiment. (a) We probe the center of the gas by optically pumping the outer parts of the cigar-shaped cloud to a dark state using two intersecting second-order Laguerre-Gaussian beams [22]. By changing the beams’ power, we control the fraction of atoms probed. The power of the two beams is set such that the number of atoms optically pumped by each beam is about the same. (b) The magnetic field is ramped from 203.4 G, where the atoms are initially prepared, to the Feshbach resonance. The hollow light beams are turned on $280\mu \mathrm{s}$ before trap release; initially, the beam that propagates perpendicular to the long axis of the cloud is pulsed on for $10\mu \mathrm{s}$, followed by $40\mu \mathrm{s}$ of the second beam. The line shape is measured using an rf pulse with a total duration of $100\mu \mathrm{s}$ and a Gaussian field envelope with $\sigma =17\mu \mathrm{s}$, centered $180\mu \mathrm{s}$ before the trap release. The cloud expands for 3 ms before being detected by absorption imaging. To improve the signal-to-noise ratio, we remove the remaining atoms from the $|9/2,-9/2⟩$ and $|9/2,-7/2⟩$ states and then transfer the outcoupled atoms in the $|9/2,-5/2⟩$ state to the $|9/2,-9/2⟩$ state, where we image on the cycling transition [28].

Figure 2

An rf line shape for the unitary Fermi gas at $T/T_F=0.25$ with 30% of the atoms probed. The solid red line is a fit to Eq. (1) with the normalization ${\int }_{-\infty }^{\infty }\Gamma (\nu )d\nu =1/2$, due to the 50%–50% spin mixture. The inset shows the same data multiplied by $2^{3/2}{\pi }^2{\nu }^{3/2}$. We make sure that the rf pulse induces only a small perturbation by setting its power to well below the value where we see the onset of saturation of the number of outcoupled atoms [23]. The measurement at different frequencies is done with different rf powers, and, when analyzing the data, we linearly scale the measured number of atoms outcoupled at each frequency to correspond to a common rf power.

Figure 3

The contact of a nearly homogeneous sample (about 30% of the trapped atoms probed) versus $T/T_F$ at unitarity (black circles). The shaded area marks the superfluid phase transition, with some uncertainty in its exact position ($T_c/T_F=0.16–0.23$) [29]. As a comparison, we plot the Gaussian pair-fluctuation (GPF) Noziéres and Schmitt-Rink model [18], the self-consistent $t$-matrix model (GG) [29], the non-self-consistent $t$-matrix model ($G_0{G}_0$) [5], the second- and third-order virial expansion [18], a QMC calculation [19], and the contact extracted from a thermodynamic measurement done at École Normale Supérieure [25]. The error bars represent one standard deviation. The inset shows the high temperature behavior of the contact, where we find good agreement with the virial expansion.

Figure 4

Contact versus the fraction of atoms probed for a gas with $T/T_F=0.46$ at the center of the cloud. In the main plot, the measured contact (squares) is normalized in respect to the trap $k_F$ and is compared to the predictions of several theoretical models (lines) using the local density approximation. The measured contact increases as we probe fewer atoms at the cloud center, where the local density is largest. The inset shows the contact normalized by the average $k_F$ of the probed atoms (squares), compared to theoretical predictions of the homogeneous contact at the average $T/T_F$ (lines).