Finite one-dimensional impenetrable Bose systems: Occupation numbers

Abstract

Bosons in the form of ultracold alkali-metal atoms can be confined to a one-dimensional (1D) domain by the use of harmonic traps. This motivates the study of the ground-state occupations ${\ensuremath{\lambda}}_{i}$ of effective single-particle states ${\ensuremath{\varphi}}_{i},$ in the theoretical 1D impenetrable Bose gas. Both the system on a circle and the harmonically trapped system are considered. The ${\ensuremath{\lambda}}_{i}$ and ${\ensuremath{\varphi}}_{i}$ are the eigenvalues and eigenfunctions, respectively, of the one-body density matrix. We present a detailed numerical and analytic study of this problem. Our main results are the explicit scaled forms of the density matrices, from which it is deduced that for fixed i the occupations ${\ensuremath{\lambda}}_{i}$ are asymptotically proportional to $\sqrt{N}$ in both the circular and harmonically trapped cases.