Finite one dimensional impenetrable Bose systems: Occupation numbers

TL;DR

This paper investigates the ground state occupation numbers of one-dimensional impenetrable Bose gases confined in circular and harmonic traps, providing explicit forms and asymptotic behavior of the density matrices.

Contribution

It offers a detailed numerical and analytical analysis of occupation numbers and density matrices in 1D impenetrable Bose systems, including explicit scaled forms and asymptotic relations.

Findings

Occupation numbers scale as √N for fixed i

Explicit scaled forms of density matrices derived

Asymptotic proportionality of λ_i to √N

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 $\phi_i$, 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 $\sqrt{N}$ in both the circular and harmonically trapped cases.