One-Dimensional Behavior of Dilute, Trapped Bose Gases

TL;DR

This paper rigorously proves that dilute, trapped Bose gases can behave as one-dimensional systems under certain conditions, providing a mathematical foundation for the observed quantum phenomena in low-dimensional Bose gases.

Contribution

It offers a rigorous proof of one-dimensional behavior in dilute Bose gases, extending beyond previous variational and numerical approaches.

Findings

Ground state energy and density match one-dimensional models

Behavior occurs when trap width is small compared to longitudinal size

Energy can be derived from a one-dimensional density functional

Abstract

Recent experimental and theoretical work has shown that there are conditions in which a trapped, low-density Bose gas behaves like the one-dimensional delta-function Bose gas solved years ago by Lieb and Liniger. This is an intrinsically quantum-mechanical phenomenon because it is not necessary to have a trap width that is the size of an atom -- as might have been supposed -- but it suffices merely to have a trap width such that the energy gap for motion in the transverse direction is large compared to the energy associated with the motion along the trap. Up to now the theoretical arguments have been based on variational - perturbative ideas or numerical investigations. In contrast, this paper gives a rigorous proof of the one-dimensional behavior as far as the ground state energy and particle density are concerned. There are four parameters involved: the particle number, $N$, transverse and longitudinal dimensions of the trap, $r$ and $L$, and the scattering length $a$ of the interaction potential. Our main result is that if $r/L\to 0$ and $N\to\infty$ the ground state energy and density can be obtained by minimizing a one-dimensional density functional involving the Lieb-Liniger energy density with coupling constant $\sim a/r^2$.