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

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.

Figure 1

Dimensionless coefficient ${\Omega }_{\mathrm{H}\mathrm{O}}({\gamma }^0)$ in front of the leading term of the high-$p$ asymptotics of the momentum distribution of harmonically trapped atoms, as a function of the interaction strength ${\gamma }^0$ in the center of the cloud. Also shown is the directly related to the experimental parameters Thomas-Fermi estimate ${\gamma }_{\mathrm{T}\mathrm{F}}^0=(8/3^{2/3})(Nm{a}_{\mathrm{1}\mathrm{D}}^2\omega /\hslash {)}^{-2/3}$ for the interaction strength in the center, as a function of ${\gamma }^0$. Both momentum and momentum distribution are measured in units related to the density in the center $n^0$, that can be expressed through ${\gamma }^0$ using ${\gamma }^0=2/n^0{a}_{\mathrm{1}\mathrm{D}}$.