Bose-Einstein condensates with attractive 1/r interaction: The case of self-trapping

TL;DR

This paper investigates self-trapped Bose-Einstein condensates with attractive 1/r interactions, revealing scaling laws, calculating properties, and discovering a second solution indicating an unstable excited state.

Contribution

It introduces a scaling framework for self-trapped condensates, computes detailed properties, and uncovers a new unstable excited state solution for negative scattering lengths.

Findings

Scaling depends on two key parameters $N^2 a/a_u$ and $\gamma/N^2$.

Accurate numerical wave functions and energies were obtained.

A second solution indicates an unstable excited state for negative scattering lengths.

Abstract

Amplifying on a proposal by O'Dell et al. for the realization of Bose-Einstein condensates of neutral atoms with attractive $1/r$ interaction, we point out that the instance of self-trapping of the condensate, without external trap potential, is physically best understood by introducing appropriate "atomic" units. This reveals a remarkable scaling property: the physics of the condensate depends only on the two parameters $N^2 a/a_u$ and $\gamma/N^2$, where $N$ is the particle number, $a$ the scattering length, $a_u$ the "Bohr" radius and $\gamma$ the trap frequency in atomic units. We calculate accurate numerical results for self-trapping wave functions and potentials, for energies, sizes and peak densities, and compare with previous variational results. As a novel feature we point out the existence of a second solution of the extended Gross-Pitaevskii equation for negative scattering lengths, with and without trapping potential, which is born together with the ground state in a tangent bifurcation. This indicates the existence of an unstable collectively excited state of the condensate for negative scattering lengths.