Universal Behavior of the Energy of Trapped Few-Boson Systems with Large Scattering Length

We calculate energy levels of two and three bosons trapped in a harmonic oscillator potential with oscillator length $a_{\mathrm{o}\mathrm{s}\mathrm{c}}$. The atoms are assumed to interact through a short-range potential with a scattering length $a$, and the short-distance behavior of the three-body wave function is characterized by a parameter $\theta$. For large positive $a/a_{\mathrm{o}\mathrm{s}\mathrm{c}}$, the energies of states that, in the absence of the trap, correspond to three free atoms approach values independent of $a$ and $\theta$. For other states, the $\theta$ dependence of the energy is strong, but the energy is independent of $a$ for $|a/a_{\mathrm{o}\mathrm{s}\mathrm{c}}|\gg 1$.

Figure 1

Eigenvalues ${\lambda }_i(\rho /|a|)$ of the hyperangular Eq. (5) for $a>0$ (solid lines) and $a<0$ (dashed line).

Figure 2

Energy of states as a function of $\theta$ for $a/a_{\mathrm{o}\mathrm{s}\mathrm{c}}=1$. Dashed and dotted lines show energies of low-lying states in the hyperspherical adiabatic approximation, i.e., neglecting couplings between hyperangular functions. The dashed lines correspond to the lowest hyperangular eigenvalue, and the dotted lines to the next lowest (see Fig. 1). The full lines show the results when coupling between the two lowest hyperangular functions is included. The stars show the central energy value of an atomic state, while the crosses show its smallest and largest energies.

Figure 3

Energies of trapped three-boson systems as a function of $a/a_{\mathrm{o}\mathrm{s}\mathrm{c}}$ showing the dependence on the three-body parameter $\theta$ (see text.) The solid line shows the energy calculated neglecting three-body correlations.