Generalized Efimov Scenario for Heavy-Light Mixtures

Motivated by recent experimental investigations of Cs-Cs-Li Efimov resonances, this work theoretically investigates the few-body properties of $N-1$ noninteracting identical heavy bosons, which interact with a light impurity through a large $s$-wave scattering length. For Cs-Cs-Cs-Li, we predict the existence of universal four-body states with energies $E_4^{(n,1)}$ and $E_4^{(n,2)}$, which are universally linked to the energy $E_3^{(n)}$ of the $n$th Efimov trimer. For infinitely large ${}^{133}\mathrm{Cs}\text{−}{}^6\mathrm{Li}$ and vanishing ${}^{133}\mathrm{Cs}\text{−}{}^{133}\mathrm{Cs}$ scattering lengths, we find $(E_4^{(1,1)}/E_3^{(1)}{)}^{1/2}\approx 1.51$ and $(E_4^{(1,2)}/E_3^{(1)}{)}^{1/2}\approx 1.01$. The ${}^{133}\mathrm{Cs}\text{−}{}^6\mathrm{Li}$ scattering lengths at which these states merge with the four-atom threshold, the dependence of these energy ratios on the mass ratio between the heavy and light atoms, and selected aspects of the generalized Efimov scenario for $N>4$ are also discussed. Possible implications of our results for ongoing cold atom experiments are presented.

Figure 1

Efimov plot for the $B_{N-1}X$ system ($N=3$ and 4) with $\kappa =133/6$. The dashed line shows the energy of the $BX$ system. The solid lines show the three lowest energies of the $BBX$ system. The dotted lines show the energies of the two bound states of the $BBBX$ system that are tied to the lowest Efimov trimer. The excited tetramer becomes unbound at $r_0/a_s\approx 0.02$. The calculations are performed for $V_0=3.2E_{\text{sr}}$, where $E_{\text{sr}}$ is equal to ${\hslash }^2/(2\mu r_0^2)$ and $\mu$ denotes the reduced mass, $\mu =m_B{m}_X/(m_B+m_X)$.

Figure 2

Pair distribution functions for infinitely large $BX$ and vanishing $BB$ scattering lengths for (a) and (b) the lowest trimer, (c) and (d) the lowest tetramer, and (e) and (f) the first excited tetramer. Panels (a), (c), and (e) show the scaled pair distribution functions $r^2{P}_{BB}(r)$ while panels (b), (d), and (f) show the scaled pair distribution functions $r^2{P}_{BX}(r)$. Dotted, solid, and dashed lines are for $\kappa =8$, $133/6$, and 40, respectively. The calculations are performed for $V_{3b}=3.2E_{\text{sr}}$.