Universal Behavior of Pair Correlations in a Strongly Interacting Fermi Gas

We show that short-range pair correlations in a strongly interacting Fermi gas follow a simple universal law described by Tan’s relations. This is achieved through measurements of the static structure factor which displays a universal scaling proportional to the ratio of Tan’s contact to the momentum $\mathcal{C}/q$. Bragg spectroscopy of ultracold ${}^6\mathrm{Li}$ atoms from a periodic optical potential is used to measure the structure factor for a wide range of momenta and interaction strengths, providing broad confirmation of this universal law. We calibrate our Bragg spectra using the $f$-sum rule, which is found to improve the accuracy of the structure factor measurement.

Figure 1

Theoretical prediction for Tan’s contact in the BEC-BCS crossover. (a) Evolution of contact for a trapped Fermi gas with $1/(k_{F}a)$ at different temperatures. Points are experimental data from Refs. [8, 19]. The $T=0$ curve is calculated using the contact for a homogeneous Fermi gas [shown in (b)] and a local density approximation. Finite temperature contact is determined from a cluster expansion theory [24]. (b) Zero temperature contact in free space, obtained from a Gaussian pair fluctuation theory [16]. Red circle shows the contact from QMC for pair correlation functions at unitarity [15]. (c) Temperature dependence of the trapped contact at unitarity. With decreasing $T$, contact increases rapidly towards the $T=0$ result (purple star).

Figure 2

Static structure factor in the BEC-BCS crossover. Experimental points were obtained by integrating Bragg spectra normalized via the $f$-sum rule for $k/k_F=4.8$. Error bars are due to shot-to-shot atom number fluctuations and uncertainties in measuring the center of mass. The theoretical line is a zero temperature result, calculated by interpolating the near-resonance structure factor Eq. (2) with the asymptotic results in the BCS and BEC limits. Inset: A Bragg spectrum obtained at $1/(k_{F}a)=0$ and $k/k_F=8.5$ showing center-of-mass displacement $\Delta X(k,\delta )$ versus Bragg frequency $\delta$. Points are single shots and the line is a guide to the eye. The pair ($\delta \sim 150\mathrm{kHz}$) and free-atom peaks ($\delta \sim 300\mathrm{kHz}$) are clearly distinguished.

Figure 3

Universal dependence of the static structure factor of a strongly interacting Fermi superfluid. Measured and calculated static structure factor versus $k_F/k$ for $1/(k_{F}a)=+0.3$, 0.0, and $-0.2$. Bragg momentum $k$ is fixed while $k_F$ is varied by changing the mean trapping frequency $\overline{\omega }$. Vertical error bars are due to atom number fluctuations and uncertainties in measuring the center of mass and horizontal error bars are due to atom number fluctuations and uncertainties in $\overline{\omega }$. Solid lines are the zero temperature theory and the dashed line is a straight line fit to the $1/(k_{F}a)=0$ data yielding a slope of $0.75\pm 0.03$.