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 = 2\sqrt{N} and N_{k=2πj/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 (β=\frac{ν_{0}m}{π^{2}\hbar^{2}n} \ll 1), we obtain N_{0}/N = (\frac{2}{N\sqrtβ})^{\sqrtβ/2} and N_{k=2πj/L} = \frac{N_0\sqrtβ}{4|j|^{1-\sqrtβ/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 $β$ is small (e.g., for N = 10^{10}, β= 0.01). A method of measurement of such fragmented quasicondensate is proposed.