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# Universal behavior of pair correlations in a strongly interacting Fermi gas 

E. D. Kuhnle, H. Hu, X.-J. Liu, P. Dyke, M. Mark, P. D. Drummond, P. Hannaford, and C. J. Vale 

ARC Centre of Excellence for Quantum-Atom Optics, Centre for Atom Optics and Ultrafast Spectroscopy, Swinburne University of Technology, Melbourne 3122, Australia 

(Dated: November 20, 2018) 

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 C /q . Bragg spectroscopy of ultracold $^{6}$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. 

Universality is a remarkable property of strongly in- teracting systems of fermions [1–3]. Universality means that all dilute Fermi systems with sufficiently strong in- teractions behave identically on a scale given by the aver- age particle separation. With the discovery of universal- ity in the Bose-Einstein condensate (BEC) to Bardeen- Cooper-Schrieffer (BCS) crossover, ultracold Fermi gases near Feshbach resonances have become a central topic in atomic physics [4–10]. Strongly interacting Fermi sys- tems arise in a wide variety of settings, from astrophys- ical to nuclear and condensed matter systems. One can therefore study universality in ultracold atomic gases to help understand other strongly interacting Fermi super- fluids, taking advantage of the ability to precisely control the atom-atom interactions. 

Understanding these strongly interacting Fermi gases, however, poses significant challenges [11]. In 2005 Shina Tan made dramatic progress by deriving several exact re- lations for Fermi gases in the BEC-BCS crossover, which relate the microscopic properties to bulk thermodynamic quantities [12–14]. These exact relations are applicable in broad circumstances: zero or finite temperatures, super- fluid or normal phases, homogeneous or trapped systems, and in few or many-body systems. 

In this letter, we experimentally verify a universal re- lation for short-range pair correlations [12] using Bragg spectroscopy. This is achieved through measurements of the static structure factor, given by the Fourier trans- form of the pair correlation function [15]. The structure factor of a unitary Fermi gas has an exact scaling with the ratio of Tan’s contact to the momentum C /q . For systems with finite scattering length, we also confirm the first order correction to the universal law. Our measure- ments are compared to new calculations for the contact based on a recently developed below-threshold Gaussian pair fluctuation theory [16]. 

The contact C in a two component Fermi gas quanti- fies the likelihood of finding two fermions with opposite spin close enough to interact with each other. In systems 

PACS numbers: 03.75.Hh, 03.75.Ss, 05.30.Fk 

where the range of the interaction potential is negligible, this single parameter encapsulates all of the information required to determine the many-body properties [17, 18]. C depends on the s -wave scattering length, density and temperature of the system. Tan showed that the inter- nal energy of a gas across the BEC-BCS crossover can be expressed as a functional of the momentum distri- bution which has a C /q 4 dependence at large momen- tum q and that the pair correlation function diverges as C /r 2 at short distance r < 1 /k$_{F}$ , where k$_{F}$ is the Fermi wavevector [12, 15]. Tan also derived the adiabatic rela- tion dE/d ( − 1 /a ) = ℏ $^{2}$C / (4 πm ) , where m is the atomic mass, giving the change in the total energy E due to an adiabatic change in the scattering length [13] and ex- tended the virial theorem to finite a and imbalanced mix- tures [14]. The contact C was first extracted [19] from the number of closed-channel molecules determined through photo-association [8] and the adiabatic and virial Tan relations were very recently verified experimentally [20]. We will generally refer to the dimensionless contact I given by C / ( Nk$_{F}$ ) where N is the number of particles. 

Short-range structure in a quantum fluid depends upon the relative wave-function of the interacting particles, in this case fermions in different spin states. In a two- component (spin-up/spin-down) Fermi gas with contact interactions this is given by ψ$_{↑↓}$ ( r ) ∝ 1 /r − 1 /a , where a is the s -wave scattering length. Starting from this wave- function, Tan showed that the spin-antiparallel pair cor- relation function is given by Eq. (1) which includes the contact as a pre-factor [12] 

Pair correlation functions are difficult to measure di- rectly in ultracold gases; however, it is possible to mea- sure macroscopic quantities which depend on correlation functions in a well defined way. A prime example is the static structure factor, S ( k ) , which is given by the Fourier transform of g $^{(2)}$( r ) ( q = $_{ℏ}$k is the probe momentum). In 

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