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arXiv:1003.3511 · Europhysics Letters (EPL)

Static structure factor of a strongly correlated Fermi gas at large momenta

Hui Hu, Xia-Ji Liu, P. D. Drummond

Published Jul 1, 2010• 39 citations
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Abstract

We theoretically investigate the static structure factor of an interacting Fermi gas near the BEC-BCS crossover at large momenta. Due to short-range two-body interactions, we show that the structure factor of unlike spin correlations S ↑↓ ( q ) falls off as 1/ q in a universal scaling region with large momentum ℏ q and large scattering length. The scaling coefficient is determined by the celebrated Tan's contact parameter, which links the short-range behavior of many-body systems to their universal thermodynamic properties. By implementing this structure-factor Tan relation together with the random-phase approximation and the virial expansion theory in various limiting cases, we show how to calculate S ↑↓ ( q ) at zero and finite temperatures for arbitrary interaction strengths, at momentum transfer higher than the Fermi momentum. Our results provide a way to experimentally confirm one of the Tan relations and to accurately measure the value of contact parameter.

DOI
10.1209/0295-5075/91/20005

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We theoretically investigate the static structure factor of an interacting Fermi gas near the BEC-BCS crossover at large momenta. Due to short-range two-body interactions, we show that the structure factor of unlike spin correlations S ↑↓ ( q ) falls off as 1/ q in a universal scaling region with large momentum ℏ q and large scattering length. The scaling coefficient is determined by the celebrated Tan's contact parameter, which links the short-range behavior of many-body systems to their universal thermodynamic properties. By implementing this structure-factor Tan relation together with the random-phase approximation and the virial expansion theory in various limiting cases, we show how to calculate S ↑↓ ( q ) at zero and finite temperatures for arbitrary interaction strengths, at momentum transfer higher than the Fermi momentum. Our results provide a way to experimentally confirm one of the Tan relations and to accurately measure the value of contact parameter.
This paper presents studies of the universal properties of strongly interacting Fermi gases using Bragg spectroscopy. We focus on pair-correlations, their relationship to the contact C introduced by Tan, and their dependence on both the momentum and temperature. We show that short-range pair correlations obey a universal law, first derived by Tan through measurements of the static structure factor, which displays a universal scaling with the ratio of the contact to the momentum C/q . Bragg spectroscopy of ultracold 6 Li atoms is employed to measure the structure factor for a wide range of momenta and interaction strengths, providing broad confirmation of this universal law. We show that calibrating our Bragg spectra using the f -sum rule leads to a dramatic improvement in the accuracy of the structure factor measurement. We also measure the temperature dependence of the contact in a unitary gas and compare our results to calculations based on a virial expansion.
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 6Li 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.
We present a comprehensive experimental study of Tan's universal contact parameter in a two-component ultracold Fermi gas, using Bragg spectroscopy. The contact uniquely parameterizes a number of universal properties of Fermi gases in the strongly interacting regime. It is linked to the spin-antiparallel component of the static structure factor S ↑↓ ( k ) at high momenta, which can readily be obtained via Bragg scattering. Contact depends upon the relative interaction strength 1/( k F a ) and temperature T / T F , where k F is the Fermi wave vector, a is the s-wave scattering length and T F is the Fermi temperature. We present measurements of both of these dependencies in a cloud of 6 Li atoms and compare our findings to theoretical predictions. We also compare Bragg spectroscopic methods based on measuring the energy and momentum transferred to the cloud and examine the conditions under which the energy transfer method provides improved accuracy. Our measurements of the dynamic structure factor and contact are found to be in good agreement with theoretical predictions based on the quantum virial expansion.
The contact parameter in unitary Fermi gases governs the short-range correlations and high-momentum properties of the system. We perform accurate quantum Monte Carlo calculations with highly optimized trial functions to precisely determine this parameter at T=0T=0T=0, demonstrate its universal application to a variety of observables, and determine the regions of momentum and energy over which the leading short-range behavior is dominant. We derive Tan's expressions for the contact parameter using just the short-range behavior of the ground-state many-body wave function, and use this behavior to calculate the two-body distribution function, one-body density matrix, and the momentum distribution of unitary Fermi gases; providing a precise value of the contact parameter that can be compared to experiments.
Universality of strongly interacting fermions is a topic of great interest in diverse fields. Here we investigate theoretically the universal dynamic density response of resonantly interacting ultracold Fermi atoms in the limit of either high temperature or large frequency. At high temperature, we use quantum virial expansion to derive universal, nonperturbative expansion functions of the dynamic structure factor; at large momentum, we conjecture that the second-order expansion function gives the Wilson coefficient used in the operator product expansion method. The dynamic structure factor is therefore determined by its second-order expansion function with an overall normalization factor given by Tan's [S. Tan, Ann. Phys. (NY) 323, 2952 (2008); S. Tan, Ann. Phys. (NY) 323, 2971 (2008); S. Tan, Ann. Phys. (NY) 323, 2987 (2008)] contact parameter. We show that the spin-parallel and -antiparallel dynamic structure factors have, respectively, a tail of the form $\ensuremath{\sim}\ifmmode\pm\else\textpm\fi{}{\ensuremath{\omega}}^{\ensuremath{-}5/2}$ for $\ensuremath{\omega}\ensuremath{\rightarrow}\ensuremath{\infty}$, decaying slower than the total dynamic structure factor found previously ($\ensuremath{\sim}{\ensuremath{\omega}}^{\ensuremath{-}7/2}$). Our predictions for the dynamic structure factor at high temperature or large frequency are testable using Bragg spectroscopy for ultracold atomic Fermi gases.
We present a high-precision determination of the universal contact parameter in a strongly interacting Fermi gas. In a trapped gas at unitarity, we find the contact to be 3.06±0.08 at a temperature of 0.08 of the Fermi temperature in a harmonic trap. The contact governs the high-momentum (short-range) properties of these systems, and this low-temperature measurement provides a new benchmark for the zero-temperature homogeneous contact. The experimental measurement utilizes Bragg spectroscopy to obtain the dynamic and static structure factors of ultracold Fermi gases at high momentum in the unitarity and molecular Bose-Einstein condensate regimes. We have also performed quantum Monte Carlo calculations of the static properties, extending from the weakly coupled BCS regime to the strongly coupled Bose-Einstein condensate case, that show agreement with experiment at the level of a few percent.