Higher-order local and non-local correlations for 1D strongly interacting Bose gas

Abstract

The correlation function is an important quantity in the physics of ultracold quantum gases because it provides information about the quantum many-body wave function beyond the simple density profile. In this paper we first study the $M$-body local correlation functions, $g_M$, of the one-dimensional (1D) strongly repulsive Bose gas within the Lieb-Liniger model using the analytical method proposed by Gangardt and Shlyapnikov [1,2]. In the strong repulsion regime the 1D Bose gas at low temperatures is equivalent to a gas of ideal particles obeying the non-mutual generalized exclusion statistics (GES) with a statistical parameter $α=1-2/γ$, i.e. the quasimomenta of $N$ strongly interacting bosons map to the momenta of $N$ free fermions via $k_i\approx αk_i^F $ with $i=1,\ldots, N$. Here $γ$ is the dimensionless interaction strength within the Lieb-Liniger model. We rigorously prove that such a statistical parameter $α$ solely determines the sub-leading order contribution to the $M$-body local correlation function of the gas at strong but finite interaction strengths. We explicitly calculate the correlation functions $g_M$ in terms of $γ$ and $α$ at zero, low, and intermediate temperatures. For $M=2$ and $3$ our results reproduce the known expressions for $g_{2}$ and $g_{3}$ with sub-leading terms (see for instance [3-5]). We also express the leading order of the short distance \emph{non-local} correlation functions $\langleΨ^\dagger(x_1)\cdotsΨ^\dagger(x_M)Ψ(y_M)\cdotsΨ(y_1)\rangle$ of the strongly repulsive Bose gas in terms of the wave function of $M$ bosons at zero collision energy and zero total momentum. Here $Ψ(x)$ is the boson annihilation operator. These general formulas of the higher-order local and non-local correlation functions of the 1D Bose gas provide new insights into the many-body physics.