Finite one-dimensional impenetrable Bose systems: Occupation numbers

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 lambda(i) of effective single-particle states phi(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 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 i the occupations lambda(i) are asymptotically proportional to rootN in both the circular and harmonically trapped cases.