Occupation numbers of the harmonically trapped few-boson system

We consider a harmonically trapped dilute $N$-boson system with pairwise interactions, which are characterized by the two-body $s$-wave scattering length $a_{s}$ and the effective range $r_{e}$. We construct the one-body density matrix of the weakly-interacting $N$-boson system and calculate the condensate fraction, defined as the largest occupation number, by employing a perturbative treatment within the framework of second quantization. The condensate fraction for the harmonically trapped $N$-boson system, calculated within first order perturbation theory, is $1-(N-1)0.420004a_{s}^{2}$. Corrections of order $a_s^{3}$ and $a_s^{3}r_{e}$ are also considered. The condensate depletion induced by effective three-body interactions is identified to occur at order $a_s^{3}$. Our expression for $N=2$ is confirmed by comparing with the expansion of the exact solution [1]. Our results for $N=3$ and $4$ are compared with high precision $ab$ $initio$ calculations for Bose gases that interact through finite-range two-body model potentials. \\[4pt] [1] T. Busch, B.-G. Englert, K. Rzazewski, and M. Wilkens, Foundations of Phys. \textbf{28}, 549 (1998).