Scaling Properties of Universal Tetramers

We evidence the existence of a universal correlation between the binding energies of successive four-boson bound states (tetramers), for large two-body scattering lengths ($a$), related to an additional scale not constrained by three-body Efimov physics. Relevant to ultracold atom experiments, the atom-trimer relaxation peaks for $|a|\to \infty$ when the ratio between the tetramer and trimer energies is $\simeq 4.6$ and a new tetramer is formed. The new scale is also revealed for $a<0$ by the prediction of a correlation between the positions of two successive peaks in the four-atom recombination process.

Figure 1

Universal scaling function for successive tetramer binding energies. Our model results, with corresponding two-body energies, are indicated inside the inset, where the red-solid line is for the unitary limit. The other model calculations are indicated inside the main frame, where 18, 19, 21, 23 report results in the unitary limit; with the others near this limit. By decreasing the ratio $B_3/B_4^{(N)}$, successive excited tetramers bind at the threshold. The shadowed region excludes tetramers for excited trimers.

Figure 2

Positions of four-atom recombination peaks ($a<0$) where two successive tetramers become unbound (blue solid line with boxes). For comparison, we show results from calculations given in [19] (green triangles) and from experiments reported in 6 (red bullets with error bars) and [5] (brown $\times$). (Our first point from the left corresponds to $B_4\simeq 64B_3$ at the unitary limit.)