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arXiv:cond-mat/0210629 · Physical Review Letters

Short-Distance Correlation Properties of the Lieb-Liniger System and Momentum Distributions of Trapped One-Dimensional Atomic Gases

Maxim Olshanii, Vanja Dunjko

Published Aug 26, 2003• 239 citations
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Abstract

We derive exact closed-form expressions for the first few terms of the short-distance Taylor expansion of the one-body correlation function of the Lieb-Liniger gas. As an intermediate result, we obtain the high-p asymptotics of the momentum distribution of both free and harmonically trapped atoms and show that it obeys a universal 1/p(4) law for all values of the interaction strength. We discuss the ways to observe the predicted momentum distributions experimentally, regarding them as a sensitive identifier for the Tonks-Girardeau regime of strong correlations.

DOI
10.1103/physrevlett.91.090401

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We derive exact closed-form expressions for the first few terms of the short-distance Taylor expansion of the one-body correlation function of the Lieb-Liniger gas. As an intermediate result, we obtain the high-p asymptotics of the momentum distribution of both free and harmonically trapped atoms and show that it obeys a universal 1/p(4) law for all values of the interaction strength. We discuss the ways to observe the predicted momentum distributions experimentally, regarding them as a sensitive identifier for the Tonks-Girardeau regime of strong correlations.
The ground-state correlation properties of a one-dimensional Bose system described by the Lieb-Liniger Hamiltonian are investigated by using exact quantum Monte Carlo techniques. The pair distribution function, static structure factor, one-body density matrix, and momentum distribution of a homogeneous system are calculated for different values of the gas parameter ranging from the Tonks-Girardeau to the mean-field regime. Results for the momentum distribution of a harmonically trapped gas in configurations relevant to experiments are also presented.
We present an exact numerical study of the scaling of density and momentum distribution functions of harmonically trapped one-dimensional bosons with repulsive contact interactions at zero and finite temperatures. We use path integral quantum Monte Carlo with worm updates in our calculations at finite interaction strengths, and the Bose-Fermi mapping in the Tonks-Girardeau regime. We discuss the homogeneous case and, within the local density approximation, use it to motivate the scaling in the presence of a harmonic trap. For the momentum distribution function, we pay special attention to the high momentum tails and their ${k}^{\ensuremath{-}4}$ asymptotic behavior.
We introduce a general approximate method for calculating the one-body correlations and the momentum distributions of one-dimensional Bose gases at finite interaction strengths and temperatures trapped in smooth confining potentials. Our method combines asymptotic techniques for the long-distance behavior of the gas (similar to Luttinger liquid theory) with known short-distance expansions. We derive analytical results for the limiting cases of strong and weak interactions and provide a general procedure for calculating one-body correlations at any interaction strength. A step-by-step explanation of the numerical method used to compute Green's functions (needed as input to our theory) is included. We benchmark our method against exact numerical calculations and compare its predictions to recent experimental results.
Motivated by recent experiments we derive an exact expression for the correlation function entering the three-body recombination rate for a one-dimensional gas of interacting bosons. The answer, given in terms of two thermodynamic parameters of the Lieb-Liniger model, is valid for all values of the dimensionless coupling $\ensuremath{\gamma}$ and contains the previously known results for the Bogoliubov and Tonks-Girardeau regimes as limiting cases. We also investigate finite-size effects by calculating the correlation function for small systems of three, four, five, and six particles.