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arXiv:1002.1987 · Physical Review Letters

Verification of Universal Relations in a Strongly Interacting Fermi Gas

John T. Stewart, John Gaebler, Tara E. Drake, D. S. Jin

Published Jun 7, 2010• 270 citations
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Abstract

Many-body fermion systems are important in many branches of physics, including condensed matter, nuclear, and now cold atom physics. In many cases, the interactions between fermions can be approximated by a contact interaction. A recent theoretical advance in the study of these systems is the derivation of a number of exact universal relations that are predicted to be valid for all interaction strengths, temperatures, and spin compositions. These equations, referred to as the Tan relations, relate a microscopic quantity, namely, the amplitude of the high-momentum tail of the fermion momentum distribution, to the thermodynamics of the many-body system. In this work, we provide experimental verification of the Tan relations in a strongly interacting gas of fermionic atoms by measuring both the microscopic and macroscopic quantities in the same system.

DOI
10.1103/physrevlett.104.235301

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Many-body fermion systems are important in many branches of physics, including condensed matter, nuclear, and now cold atom physics. In many cases, the interactions between fermions can be approximated by a contact interaction. A recent theoretical advance in the study of these systems is the derivation of a number of exact universal relations that are predicted to be valid for all interaction strengths, temperatures, and spin compositions. These equations, referred to as the Tan relations, relate a microscopic quantity, namely, the amplitude of the high-momentum tail of the fermion momentum distribution, to the thermodynamics of the many-body system. In this work, we provide experimental verification of the Tan relations in a strongly interacting gas of fermionic atoms by measuring both the microscopic and macroscopic quantities in the same system.
Contact interactions play a fundamental role for understanding correlated quantum systems from microscopic models. For example, the 1/k4 momentum tail, the weight of which is known as Tan's contact, reveals a set of universal relations in interacting atomic systems. Nevertheless, the experimental measurement of contact has still remained unexplored for multi-component fermions with high spins. Here, we report on the measurement of s-wave contact parameter in a cold spin-balanced gas of 173Yb atoms with SU(N) symmetry. The high-momentum tail is recorded after time-of-flight expansion, and Tan's contact is directly quantified with a tunable number of spin component N without changing the number of atoms per component. We experimentally verify the linear increase in the contact with N providing experimental confirmation of the SU(N) interaction. Furthermore, we investigate the momentum distribution of SU(N) fermions at the low momentum regime showing that multi-component fermions with large N exhibit a more bosonic behavior. Our measurement directly reveals the interaction effect in a multi-component Fermi gas with SU(N) symmetry, and paves the way for study of many-body physics with high spins.
Ultracold fermions with SU(N >2) symmetry offer a unique opportunity to study quantum dynamics and interaction effects in the large spin systems that have no analogue in condensed matter physics. In this poster, we present a set of experiments on the measurement of contact parameters and collective excitations with a SU(N) Fermi gas of 173Yb atoms. First, we explore the short-range interaction effect via Tan’s contact parameters in a multi-component Fermi gas with SU(N >2) symmetry. The s-wave contact parameter is experimentally measured by recording the high-momentum tail of weakly interacting fermions. For a tunable number of spin component N with a fixed number of atoms per component, we verify the linear increase in the contact with N providing experimental confirmation of SU(N) interactions. Furthermore, we explore the momentum distribution of SU(N) fermions at the low momentum regime. Next, we measure collective excitations of a harmonically trapped twodimensional SU(N) Fermi gas. Various collective modes are investigated with a tunable number of spin component N showing a decrease in the ratio of quadruple and dipole mode with N. Our work will pave the way for the experimental study of interacting SU(N) Fermi gases with large spin.
Ultracold quantum gases are an ideal toolbox for simulating complex condensed or nuclear matter systems and to investigate fundamental quantum properties of matter. In this thesis, we will investigate universal properties connecting the high-momentum tail of the momentum distribution, the short-range correlations and the thermodynamics, encapsulated in Tan's relations. These relations are especially useful in the strongly interacting case, where perturbative approaches usually fail. With the aid of Tan's universal relations, we can still come to general conclusions about strongly interacting quantum gases. In particular, the momentum distribution exhibits a characteristic algebraic decay, unlike the exponential decay of the non-interacting case. The main focus in this thesis is on the one-dimensional, fermionic case, where we study the highly polarized case (the one-dimensional Fermi polaron), verifying Tan's relations using a variety of theoretical tools. In addition, we show that localized systems exhibit a universal, dynamical instability to delocalization when a short-range interaction between particles is switched off rapidly. This delocalization process relies on the algebraic decay of the momentum distribution, which guarantees that at least some of the delocalized single-particle states are occupied with a finite probability. Finally, we investigate the Bardeen-Cooper-Schrieffer (BCS) to Bose-Einstein condensation (BEC) crossover for the three-dimensional Fermi gas and develop a novel method to describe the breakdown of the Fermi liquid description in the vicinity of the critical temperature for superfluidity, in good agreement with a recent experiment.
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 Tan contact is a pivotal quantity for characterizing many-body quantum systems, bridging microscopic correlations to macroscopic thermodynamic behavior. It is defined as the weight of universal [Formula: see text] momentum tails, but, so far, its direct measurement has been hindered in Bose gases due to interactions strongly altering time-of-flight imaging. Here, we report the first direct measurement of the Tan contact in a strongly correlated Lieb-Liniger gas. Leveraging the one-dimensional geometry of our system, we implement a two-stage expansion scheme, yielding interaction-immune imaging. Our results show good agreement with theoretical predictions and are consistent with a predicted universal scaling law. Our work paves the way for further characterization of the Lieb-Liniger gas across broad interaction regimes and holds promise for extension to other correlated quantum gases in confined geometries.
Shina Tan has derived some universal relations that hold for any state of a system consisting of fermions with two spin states that have a large scattering length. These relations involve an intensive quantity called the contact that measures the number of pairs of atoms that are very close together. We show how these relations can be derived in the framework of quantum field theory using standard renormalization methods and the operator product expansion. They allow the contact density to be identified as the expectation value of a local operator constructed out of quantum fields.