Finite one dimensional impenetrable Bose systems: Occupation numbers

Abstract

Bosons in the form of ultra cold alkali 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 ${\lambda }_i$ of effective single particle states ${\varphi }_i$, in the theoretical 1d impenetrable Bose gas. Both the system on a circle and the harmonically trapped system are considered. The ${\lambda }_i$ and ${\varphi }_i$ 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 ${\lambda }_i$ are asymptotically proportional to $\sqrt{N}$ in both the circular and harmonically trapped cases.

22 pages, 8 figures (.eps), uses REVTeX4

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