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arXiv:2007.12385 · Nature Communications

Tan’s two-body contact across the superfluid transition of a planar Bose gas

Y.-Q. Zou, Brice Bakkali-Hassani, C. Maury, É. Le Cerf, Sylvain Nascimbène, Jean Dalibard, J. Beugnon

Published Feb 3, 2021• 23 citations
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Abstract

Abstract Tan’s contact is a quantity that unifies many different properties of a low-temperature gas with short-range interactions, from its momentum distribution to its spatial two-body correlation function. Here, we use a Ramsey interferometric method to realize experimentally the thermodynamic definition of the two-body contact, i.e., the change of the internal energy in a small modification of the scattering length. Our measurements are performed on a uniform two-dimensional Bose gas of 87Rb atoms across the Berezinskii–Kosterlitz–Thouless superfluid transition. They connect well to the theoretical predictions in the limiting cases of a strongly degenerate fluid and of a normal gas. They also provide the variation of this key quantity in the critical region, where further theoretical efforts are needed to account for our findings.

DOI
10.1038/s41467-020-20647-6

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Abstract Tan’s contact is a quantity that unifies many different properties of a low-temperature gas with short-range interactions, from its momentum distribution to its spatial two-body correlation function. Here, we use a Ramsey interferometric method to realize experimentally the thermodynamic definition of the two-body contact, i.e., the change of the internal energy in a small modification of the scattering length. Our measurements are performed on a uniform two-dimensional Bose gas of 87Rb atoms across the Berezinskii–Kosterlitz–Thouless superfluid transition. They connect well to the theoretical predictions in the limiting cases of a strongly degenerate fluid and of a normal gas. They also provide the variation of this key quantity in the critical region, where further theoretical efforts are needed to account for our findings.
We determine the two-body contact in a planar Bose gas confined by a transverse harmonic potential, using the nonperturbative functional renormalization group. We use the three-dimensional thermodynamic definition of the contact where the latter is related to the derivation of the pressure of the quasi-two-dimensional system with respect to the three-dimensional scattering length of the bosons. Without any free parameter, we find a remarkable agreement with the experimental data of Zou et al. [Tan's two-body contact across the superfluid transition of a planar Bose gas, Nat. Commun. 12, 760 (2021).NCAOBW2041-172310.1038/s41467-020-20647-6] from low to high temperatures, including the vicinity of the Berezinskii-Kosterlitz-Thouless transition. We also show that the short-distance behavior of the pair distribution function and the high-momentum behavior of the momentum distribution are determined by two contacts: the three-dimensional contact for length scales smaller than the characteristic length ℓ_{z}=sqrt[ℏ/mω_{z}] of the harmonic potential and, for length scales larger than ℓ_{z}, an effective two-dimensional contact, related to the three-dimensional one by a geometric factor depending on ℓ_{z}.
We prepare and study strongly interacting two-dimensional Bose gases in the superfluid, the classical Berezinskii-Kosterlitz-Thouless (BKT) transition, and the vacuum-to-superfluid quantum critical regimes. A wide range of the two-body interaction strength 0.05 < g < 3 is covered by tuning the scattering length and by loading the sample into an optical lattice. Based on the equations of state measurements, we extract the coupling constants as well as critical thermodynamic quantities in different regimes. In the superfluid and the BKT transition regimes, the extracted coupling constants show significant down-shifts from the mean-field and perturbation calculations when g approaches or exceeds one. In the BKT and the quantum critical regimes, all measured thermodynamic quantities show logarithmic dependence on the interaction strength, a tendency confirmed by the extended classical-field and renormalization calculations.
By selectively probing the center of a trapped gas, we measure the local, or homogeneous, contact of a unitary Fermi gas as a function of temperature. Tan's contact, C, is proportional to the derivative of the energy with respect to the interaction strength and is thus an essential thermodynamic quantity for a gas with short-range correlations. Theoretical predictions for the temperature dependence of C differ substantially, especially near the superfluid transition, T(c), where C is predicted to either sharply decrease, sharply increase, or change very little. For T/T(F)>0.4, our measurements of the homogeneous gas contact show a gradual decrease of C with increasing temperature, as predicted by theory. We observe a sharp decrease in C at T/T(F)=0.16, which may be due to the superfluid phase transition. While a sharp decrease in C below T(c) is predicted by some many-body theories, we find that none of the predictions fully account for the data.
Abstract Using a formalism based on the two-body S-matrix we study two-dimensional Bose and Fermi gases with both attractive and repulsive interactions. Approximate analytic expressions, valid at weak coupling and beyond, are developed and applied to the Berezinskii–Kosterlitz–Thouless (BKT) transition. We successfully recover the correct logarithmic functional form of the critical chemical potential and density for the Bose gas. For fermions, the BKT critical temperature is calculated in BCS and BEC regimes through consideration of Tan’s contact.
We theoretically investigate the equation of state and Tan's contact of a nondegenerate three-dimensional Bose gas near a broad Feshbach resonance, within the framework of large-NNN expansion. Our results agree with the path-integral Monte Carlo simulations in the weak-coupling limit and recover the second-order virial expansion predictions at strong interactions and high temperatures. At resonance, we find that the chemical potential and energy are significantly enhanced by the strong repulsion, while the entropy does not change significantly. With increasing temperature, the two-body contact initially increases and then decreases like ${T}^{\ensuremath{-}1}$ at large temperature, and therefore exhibits a peak structure at about 4Tc04{T}_{c0}4Tc0​, where Tc0{T}_{c0}Tc0​ is the Bose-Einstein condensation temperature of an ideal, noninteracting Bose gas. These results may be experimentally examined with a nondegenerate unitary Bose gas, where the three-body recombination rate is substantially reduced. In particular, the nonmonotonic temperature dependence of the two-body contact could be inferred from the momentum distribution measurement.
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.