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    Finite one dimensional impenetrable Bose systems: Occupation numbers

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    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 λi\lambda_i of effective single particle states ϕi\phi_i, in the theoretical 1d impenetrable Bose gas. Both the system on a circle and the harmonically trapped system are considered. The λi\lambda_i and ϕi\phi_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 ii the occupations λi\lambda_i are asymptotically proportional to N\sqrt{N} in both the circular and harmonically trapped cases.Comment: 22 pages, 8 figures (.eps), uses REVTeX
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