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arXiv:1801.05675 · Physical Review A

Tan's contact in a cigar-shaped dilute Bose gas

Jean Decamp, Mathias Albert, Patrizia Vignolo

Published Mar 16, 2018• 13 citations
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Abstract

We compute the Tan's contact of a weakly interacting Bose gas at zero temperature in a cigar-shaped configuration. Using an effective one-dimensional Gross-Pitaevskii equation and Bogoliubov theory, we derive an analytical formula that interpolates between the three-dimensional and the one-dimensional mean-field regimes. In the strictly one-dimensional limit, we compare our results with Lieb-Liniger theory. Our study can be a guide for actual experiments interested in the study of Tan's contact in the dimensional crossover.

DOI
10.1103/physreva.97.033611

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We compute the Tan's contact of a weakly interacting Bose gas at zero temperature in a cigar-shaped configuration. Using an effective one-dimensional Gross-Pitaevskii equation and Bogoliubov theory, we derive an analytical formula that interpolates between the three-dimensional and the one-dimensional mean-field regimes. In the strictly one-dimensional limit, we compare our results with Lieb-Liniger theory. Our study can be a guide for actual experiments interested in the study of Tan's contact in the dimensional crossover.
It is well known that Tan's contact could be calculated by using any of the following three methods: by the asymptotic behavior of momentum distribution, by Tan's adiabatic sweep theorem, or by the operator product expansion as an expectation value of the interaction term. We argue that if a theory describing a Bose (or Fermi) system with the only contact interaction is self-consistent, then it should lead to the same result in all three cases. As an example, we considered mean-field theory (MFT)-based approaches and established that among existing approximations of MFT, the Hartree-Fock-Bogoliubov (HFB) approach is the most self-consistent. Actually, HFB is able to describe existing experimental data on Tan's contact for dilute Bose gas but fails to predict its expected behavior at large gas parameters $(\ensuremath{\gamma}>0.015)$. So, for appropriate description of properties of a Bose gas even at zero temperature, this approximation needs to be expanded by taking into account fluctuations in higher order then the second one.
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.
We investigate the properties of the one-dimensional Bose gas at zero temperature, for which exact results exist for some model systems. We treat the interactions between particles in the gas with an approximate form of the many-body T-matrix, and find a form of Gross-Pitaevskii equation valid in 1D. The results presented agree with the exact models in both the weakly and strongly interacting limits, and interpolate smoothly between them. We also investigate the use of mean-field treatments of trapped BECs to describe the 1D system in the strongly interacting limit. We find that the use of the many-body T-matrix to describe interactions leads to qualitative agreement with the exact models for some physical quantities. We indicate how the standard mean-field treatments need to be modified to extend and improve the agreement.
We study elementary excitations of a system of one-dimensional bosons with weak contact repulsion. We show that the Gross-Pitaevskii regime, in which the excitations are the well-known Bogoliubov quasiparticles and dark solitons, does not extend to the low-energy limit. Instead, the spectra of both excitations have finite curvatures at zero momentum, in agreement with the phenomenological picture of fermionic quasiparticles. We describe analytically the crossover between the Gross-Pitaevskii and the low-energy regimes, and discuss implications of our results for the behavior of the dynamic structure factor.
The universal Tan relations connect a variety of microscopic features of many-body quantum systems with two-body contact interactions to a single quantity, called the contact. The latter has become pivotal in the description of quantum gases. We provide a complete characterization of the Tan contact of the harmonically trapped Lieb-Liniger gas for arbitrary interactions and temperature. Combining thermal Bethe ansatz, local-density approximation, and exact quantum Monte Carlo calculations, we show that the contact is a universal function of only two scaling parameters, and determine the scaling function. We find that the temperature dependence of the contact, or equivalently the interaction dependence of the entropy, displays a maximum. The presence of this maximum provides an unequivocal signature of the crossover to the fermionized regime and it is accessible in current experiments.