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arXiv:cond-mat/0506609 · Physical Review A

Exact results for three-body correlations in a degenerate one-dimensional Bose gas

Vadim Cheianov, H. Smith, Mikhail B. Zvonarev

Published May 26, 2006• 61 citations
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Abstract

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.

DOI
10.1103/physreva.73.051604

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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 \ensuremathγ\ensuremath{\gamma}\ensuremathγ 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.
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.
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 use the coordinate Bethe ansatz to exactly calculate matrix elements between eigenstates of the Lieb–Liniger model of one-dimensional bosons interacting via a two-body delta-potential. We investigate the static correlation functions of the zero-temperature ground state and their dependence on interaction strength, and analyze the effects of system size in the crossover from few-body to mesoscopic regimes for up to seven particles. We also obtain time-dependent nonequilibrium correlation functions for five particles following quenches of the interaction strength from two distinct initial states. One quench is from the noninteracting ground state and the other from a correlated ground state near the strongly interacting Tonks–Girardeau regime. The final interaction strength and conserved energy are chosen to be the same for both quenches. The integrability of the model highly constrains its dynamics, and we demonstrate that the time-averaged correlation functions following quenches from these two distinct initial conditions are both nonthermal and moreover distinct from one another.
We derive an exact analytic expression for the three-body local correlations in the Lieb-Liniger model of 1D Bose gas with contact repulsion. The local three-body correlations control the thermalization and particle loss rates in the presence of terms which break integrability, as is realized in the case of 1D ultracold bosons. Our result is valid not only at finite temperature but also for a large class of nonthermal excited states in the thermodynamic limit. We present finite temperature calculations in the presence of external harmonic confinement within local density approximation, and for a highly excited state that resembles an experimentally realized configuration.
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 LLL 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 NNN 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.