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

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 an 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 N²a∕a_u and γ∕N², where N is the particle number, a the scattering length, a_u the “Bohr” radius, and γ the trap frequency in atomic units. We calculate accurate numerical results for self-trapping wave functions and potentials, and for energies, sizes, and peak densities, and compare with previous variational results. 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.