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arXiv:1705.02100 · Physical Review A

Connection between nonlocal one-body and local three-body correlations of the Lieb-Liniger model

Maxim Olshanii, Vanja Dunjko, Anna Minguzzi, Guillaume Lang

Published Sep 19, 2017• 22 citations

Abstract

We derive a connection between the fourth coefficient of the short-distance Taylor expansion of the one-body correlation function, and the local three-body correlation function of the Lieb-Liniger model of $\ensuremath{\delta}$-interacting spinless bosons in one dimension. This connection, valid at arbitrary interaction strength, involves the fourth moment of the density of quasimomenta. Generalizing recent conjectures, we propose approximate analytical expressions for the fourth coefficient covering the whole range of repulsive interactions, validated by comparison with accurate numerics. In particular, we find that the fourth coefficient changes sign at interaction strength ${\ensuremath{\gamma}}_{c}\ensuremath{\simeq}3.816$, while the first three coefficients of the Taylor expansion of the one-body correlation function retain the same sign throughout the whole range of interaction strengths.

DOI: 10.1103/physreva.96.033624

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We derive a connection between the fourth coefficient of the short-distance Taylor expansion of the one-body correlation function, and the local three-body correlation function of the Lieb-Liniger model of

\ensuremath{\delta}

-interacting spinless bosons in one dimension. This connection, valid at arbitrary interaction strength, involves the fourth moment of the density of quasimomenta. Generalizing recent conjectures, we propose approximate analytical expressions for the fourth coefficient covering the whole range of repulsive interactions, validated by comparison with accurate numerics. In particular, we find that the fourth coefficient changes sign at interaction strength

{\ensuremath{\gamma}}_{c}\ensuremath{\simeq}3.816

, while the first three coefficients of the Taylor expansion of the one-body correlation function retain the same sign throughout the whole range of interaction strengths.

We present analytical results for the nonlocal pair correlations in one-dimensional bosonic systems with repulsive contact interactions that are uniformly valid from the classical regime of high temperatures down to weak quantum degeneracy entering the regime of ultralow temperatures. By using the information contained in the short-time approximations of the full many-body propagator, we derive results that are nonperturbative in the interaction parameter while covering a wide range of temperatures and densities. For the case of three particles we give a simple formula for arbitrary couplings that is exact in the dilute limit while remaining valid up to the regime where the thermal de Broglie wavelength ${\ensuremath{\lambda}}_{T}$ is of the order of the characteristic length

L

of the system. We then show how to use this result to find analytical expressions for the nonlocal correlations for arbitrary but fixed particle numbers

N

including finite-size corrections. Neglecting the latter in the thermodynamic limit provides an expansion in the quantum degeneracy parameter $N{\ensuremath{\lambda}}_{T}/L$. We compare our analytical results with numerical Bethe ansatz calculations, finding excellent agreement.

We investigate local three-body correlations for bosonic particles in three dimensions and one dimension as a function of the interaction strength. The three-body correlation function g(3) is determined by measuring the three-body recombination rate in an ultracold gas of Cs atoms. In three dimensions, we measure the dependence of g(3) on the gas parameter in a BEC, finding good agreement with the theoretical prediction accounting for beyond-mean-field effects. In one dimension, we observe a reduction of g(3) by several orders of magnitude upon increasing interactions from the weakly interacting BEC to the strongly interacting Tonks-Girardeau regime, in good agreement with predictions from the Lieb-Liniger model for all strengths of interaction.

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 functions of a one-dimensional homogeneous Bose system described by the Lieb-Liniger Hamiltonian are investigated by using exact quantum Monte Carlo techniques. This article is an extension of a previous study published in Phys. Rev. A {\bf 68}, 031602 (2003). New results on the local three-body correlator as a function of the interaction strength are included and compared with the measured value from three-body loss experiments. We also carry out a thorough study of the short- and long-range behavior of the one-body density matrix.