Bose-Einstein condensation in a one-dimensional system of interacting bosons

TL;DR

This paper investigates Bose-Einstein condensation in a one-dimensional system of interacting bosons, revealing the absence of condensate in impenetrable bosons and the presence of a quasicondensate in weakly interacting cases, with implications for measurement methods.

Contribution

It provides analytical calculations of the momentum distribution and condensate fractions in 1D interacting bosons, distinguishing between impenetrable and weakly interacting regimes, and proposes a measurement method for quasicondensates.

Findings

Impenetrable bosons have no condensate at large N.

Weakly interacting bosons exhibit a quasicondensate on low momentum levels.

A measurement method for fragmented quasicondensates is proposed.

Abstract

Using the Vakarchuk formulae for the density matrix, we calculate the number N_k of atoms with momentum \hbar k for the ground state of a uniform one-dimensional periodic system of interacting bosons. We obtain for impenetrable point bosons N_0 = 2\sqrt{N} and N_{k=2\pi j/L} = 0.31N_{0}/\sqrt{|j|}. That is, there is no condensate or quasicondensate on low levels at large N. For almost point bosons with weak coupling (\beta=\frac{\nu_{0}m}{\pi^{2}\hbar^{2}n} \ll 1), we obtain N_{0}/N = (\frac{2}{N\sqrt{\beta}})^{\sqrt{\beta}/2} and N_{k=2\pi j/L} = \frac{N_0\sqrt{\beta}}{4|j|^{1-\sqrt{\beta}/2}}. In this case, the quasicondensate exists on the level with k=0 and on low levels with k\neq 0, if N is large and $\beta$ is small (e.g., for N = 10^{10}, \beta = 0.01). A method of measurement of such fragmented quasicondensate is proposed.