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 $N^{2}a∕a_u$ 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, 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.

Figure 1

(Color online) Phase diagram $N$ vs $a∕a_u$ for the self-binding ground state of monopolar degenerate quantum gases (for the explanation, see text).

Figure 2

(Color online) Numerically accurate self-binding ground-state $s$-wave solutions for different values of the scaling parameter $N^{2}a∕a_u$: (a) wave functions and (b) self-binding potentials. Both in (a) the value of the wave function at the origin and in (b) the absolute value of the self-binding potential at the origin decrease monotonically with the scaling parameter from their maximum values at $N^{2}a∕a_u=-1.02$ to their smallest values at $N^{2}a∕a_u=10$. Thus, as the scaling parameter grows the binding becomes weaker. In (b) the asymptotic $1∕r$ potential is also shown for comparison.

Figure 3

(Color online) Total energy of the condensate as a function of $N^{2}a∕a_u$: (a) on a logarithmic and (b) on a linear scale. Variational results obtained by minimizing the total energy for a Gaussian-type orbital [12] are shown by dashed lines.

Figure 4

(Color online) Root-mean-square radius of the condensate as a function of $N^{2}a∕a_u$: (a) on a logarithmic and (b) on a linear scale. Variational results [12] are shown by dashed lines.

Figure 5

(Color online) Peak density $\varrho$ of the condensate as a function of $N^{2}a∕a_u$: (a) on a logarithmic and (b) on a linear scale. Variational results [12] are shown by dashed lines.

Figure 6

(Color online) (a) Bifurcation of the chemical potential and (b) bifurcation of the total mean-field energy at the critical point $N^{2}a∕a_u=-1.0251$, for self-binding—i.e., vanishing trap potential.

Figure 7

(Color online) (a) Bifurcations of the chemical potential for nonvanishing values of the trapping potential. The case $\gamma =0$ is shown for comparison. (b) Dependence of the critical scattering length (the bifurcation point) on the frequency of the trapping potential. Numerically accurate results are given by solid lines, variational results by dashed lines. To elucidate the behavior for small values of $\gamma ∕N^2$, this region is shown in the inset on a logarithmic scale.