Prediction of a weakly bound excited state of Efimov character in a ${}^7\mathrm{Li}{}^4\mathrm{He}_2$ system

We carry out calculations on the van der Waals trimer ${}^7\mathrm{Li}{}^4\mathrm{He}_2$ using the mapping method within the frame of hyperspherical coordinates, which allows us to give accurate binding energies and wave functions for both the ground and excited state of the system. When the realistic two-body potentials are adopted, the system presents an excited state which shows Efimov character. We study the range of the interaction strength in which the excited state could exist and find that the state persists within the experiment error band for binding energy of LiHe molecule. We also study the three-body parameter (3BP) of ${}^7\mathrm{Li}{}^4\mathrm{He}_2$ system and its relationship with the background scattering length $a_{\text{HeHe}}$. Our calculations demonstrate that the 3BP of ${}^7\mathrm{Li}{}^4\mathrm{He}_2$ system is dependent on the value of the scattering length $a_{\text{HeHe}}$, independent of the short-range details of the He-He interaction. The results confirm the prediction of Wang et al. [Phys. Rev. Lett. 109, 243201 (2012)] that the 3BP for a heteronuclear atomic system is universally determined from the van der Waals lengths and the homonuclear scattering length.

Figure 1

He-He and Li-He potentials used in this paper.

Figure 2

${}^7{\mathrm{Li}}^4{\mathrm{He}}_2$ effective hyperradial potential curve (solid line) $W\left(R\right)=U_0\left(R\right)-\frac{1}{2\mu }\langle {\Phi }_0\left(R\right)|\frac{d^2}{d{R}^2}|{\Phi }_0\left(R\right)\rangle$, and the ground-state wave function (dotted line), and the first excited-state wave function (dash line). Notice that the hyperradius has a logarithmic scale.

Figure 3

Evolution of the bound-state energies of ${}^7{\mathrm{Li}}^4{\mathrm{He}}_2,{}^4{\mathrm{He}}_2$, and ${}^7{\mathrm{Li}}^4\mathrm{He}$ as the function of the parameter $\lambda$. (a) The three-body ground state; (b) the three-body excited state.

Figure 4

To clarify the figure, these quantities are normalized by $r_{\text{vdW,HeLi}}$ and $E_{\text{vdW,HeLi}}$. (a) The 3BP $a_{-}$ as the function of the scattering length $a_{\text{HeHe}}$. (b) The ground Efimov state energy $E_{\text{Efimov}}^0$ with respect to the scattering length $a_{\text{HeHe}}$ for ${}^7{\mathrm{Li}}^4{\mathrm{He}}_2$ system. The figure is obtained at the heteronuclear unitarity $\left(a_{\text{LiHe}}=\infty \right)$.