Five- and Six-Body Resonances Tied to an Efimov Trimer

We explore the properties of weakly bound bosonic states in the strongly interacting regime. Combining a correlated-Gaussian basis set expansion with a complex-scaling method, we extract the energies and structural properties of bosonic cluster states with $N\le 6$ for different two-body potentials. The identification of five- and six-body resonances attached to the first-excited-Efimov trimer provides strong support to the premise of Efimov universality in bosonic systems. Our study also reveals a rich structure of bosonic cluster states. Besides the lowest cluster states that behave as bosonic droplets, we identify cluster states weakly bound to one or two atoms forming effective cluster-atom dimers and cluster-atom-atom “trimers.” The experimental signatures of these cluster states are discussed.

Figure 1

Energies of the lowest universal cluster state as a function of inverse scattering length. From top to bottom, the solid curves represent the $N=3$, 4, 5, 6 energies, respectively. The energies for $N=3$ and $N=4$ were obtained in previous studies [9]. Note that, in this figure, the energy of the second $N=4$ state would lie almost on top of the trimer energy. The $N=5$ solid curve and symbols correspond to predictions from an excited resonance for $V_r$ and $V_a$, respectively. The circles and squares are the $N=6$ predictions for $V_r$ and $V_a$, respectively, and the solid curve is a guide to the eye. Dashed curves correspond to the $N=5$, 6 predictions from Ref. 16. Inset: Width of the five-body resonances for $V_r$ (red) and $V_a$ (blue).

Figure 2

Pair-distribution function of the universal five-body (main figure) and six-body (inset) states. Here ${\kappa }_N=\sqrt{m|E_N^u|/{\hslash }^2}$, where $E_N^u$ is the binding energy of the $N$-body droplet at unitarity. Solid and long-dashed curves are the prediction from the $V_a$ and $V_r$ potential, while the short-dashed curves are predictions from Ref. 16.

Figure 3

Spectrum of the lowest Efimov family as a function scattering length for $3\le N\le 6$. The thin solid curve corresponds to the trimer state energy, the short-dashed curves correspond to the four-body states, the long-dashed curves correspond to the five-body states, and the thick solid curves correspond to the six-body states. Inset: States formed below the trimer threshold.

Figure 4

Pair-distribution functions of the lowest $N$-body cluster states ($N=3,\dots ,6$) at unitarity. Inset: Distribution functions for the trimer (“3”), the trimer-atom four-body state (“$3+1$”), and the trimer-atom-atom five-body state (“$3+1+1$”).