Bose–Einstein Condensation in a One-Dimensional System of Interacting Bosons

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 $N_{0} \approx 2\sqrt{N}$ and $N_{k=2\pi j/L} \simeq 0.31~N_{0}/\sqrt{|j|}$. 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 $\frac{N_{0}}{N} \approx \left( \frac{2}{N\sqrt{\beta }}\right) ^{\sqrt{\beta }/2}$ and $N_{k=2\pi j/L} \approx \frac{N_{0}\sqrt{\beta }}{4|j|^{1-\sqrt{\beta }/2}}$. In this case, the quasicondensate exists on the level with $k=0$ and on low levels with $k\ne 0$, if N is large and $\beta$ is small (e.g., for $N \sim 10^{10}$, $\beta \sim 0.01$). A method of measurement of such fragmented quasicondensate is proposed.