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arXiv:cond-mat/0304159 · New Journal of Physics

Local correlations in a strongly interacting one-dimensional Bose gas

Dimitri M. Gangardt, Gora V. Shlyapnikov

Published Jun 18, 2003• 86 citations
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Abstract

We develop an analytical method for calculating local correlations in strongly interacting 1D Bose gases, based on the exactly solvable Lieb-Liniger model. The results are obtained at zero and finite temperatures. They describe the interaction-induced reduction of local many-body correlation functions and can be used for achieving and identifying the strong-coupling Tonks-Girardeau regime in experiments with cold Bose gases in the 1D regime.

DOI
10.1088/1367-2630/5/1/379

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We develop an analytical method for calculating local correlations in strongly interacting 1D Bose gases, based on the exactly solvable Lieb-Liniger model. The results are obtained at zero and finite temperatures. They describe the interaction-induced reduction of local many-body correlation functions and can be used for achieving and identifying the strong-coupling Tonks-Girardeau regime in experiments with cold Bose gases in the 1D regime.
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.
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We address the problem of calculating finite-temperature response functions of an experimentally relevant low-dimensional, strongly correlated system: the integrable one-dimensional Bose gas with a repulsive $\ensuremath{\delta}$-function interaction (the Lieb-Liniger model). Focusing on the dynamical density-density function, we present a Bethe ansatz-based method allowing for its accurate evaluation in finite but large systems, over broad ranges of momenta, frequencies, temperatures, and interaction parameters which are difficult to access using other methods. This allows us to quantify the reshaping of the zero-temperature critical behavior by thermal fluctuations, in experimentally accessible regimes.
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.
Phase correlations, density fluctuations and three-body loss rates are relevant for many experiments in quasi one-dimensional geometries. Extended mean-field theory is used to evaluate correlation functions up to third order for a quasi one-dimensional trapped Bose gas at zero and finite temperature. At zero temperature and in the homogeneous limit, we also study the transition from the weakly correlated Gross–Pitaevskii regime to the strongly correlated Tonks–Girardeau regime analytically. We compare our results with the exact Lieb–Liniger solution for the homogeneous case and find good agreement up to the cross-over regime.