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Finite one dimensional impenetrable Bose systems: Occupation numbers

P. J. Forrester·N. E. Frankel·T. M. Garoni·N. S. Witte
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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 $λ_i$ of effective single particle states $ϕ_i$, in the theoretical 1d impenetrable Bose gas. Both the system on a circle and the harmonically trapped system are considered. The $λ_i$ and $ϕ_i$ 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 $λ_i$ are asymptotically proportional to $\sqrt{N}$ in both the circular and harmonically trapped cases.

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P. J. Forrester, N. E. Frankel, T. M. Garoni, N. S. Witte. 2002-11-07. Finite one dimensional impenetrable Bose systems: Occupation numbers. https://doi.org/10.1103/physreva.67.043607

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