Universal behavior of pair correlations in a strongly interacting Fermi gas 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) E. D. Kuhnle  H. Hu  X.-J. Liu  P. Dyke  M. Mark  P. D. Drummond  P. Hannaford  and C. J. Vale 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. PACS numbers: 03.75.Hh  03.75.Ss  05.30.Fk 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 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 g (2) ↑↓ ( r ) → I 16 π 2 $^{(}$1 r 2 − 2 ar$^{)}$ . (1) arXiv:1001.3200v2 [cond-mat.quant-gas] 28 Jun 2010