Universal Three-Body Parameter in Heteronuclear Atomic Systems

In Efimov physics, a three-body parameter (3BP), previously regarded as nonuniversal, uniquely defines bound and scattering properties of three particles. A universal 3BP, however, has been recently shown in experiments and theory in ultracold homonuclear gases. Our present study further predicts a universal 3BP for heteronuclear atomic systems near broad Feshbach resonances and provides physical interpretations for its origin. We show that for a system composed of two light and one heavy atom, the physical origin of the universal 3BP is similar to the homonuclear case, while for systems composed by one light and two heavy atoms, the universality of the 3BP is now mostly controlled by the heavy-heavy interatomic properties. This new universality is explained by the universal properties of the van der Waals interactions in a simple Born-Oppenheimer picture. Finally, we show the numerically determined 3BPs for some combinations of alkali atoms used in ultracold experiments.

Figure 1

The wave functions for the first excited Efimov state at $a_{AA}=a_{AX}=\infty$. (a) The BO wave function $F_{\nu }^{\mathrm{BO}}(r)$ with increasing number of $A_2$ bound states. The short-range cutoff is tuned at $r_{c,AA}/r_{\mathrm{vdW},AA}\approx 0.92$, 0.57, 0.38, 0.31, 0.26 to give 1, 2, 4, 6, 8 $A_2$ $s$-wave bound states, respectively. (b) A comparison of the BO probability density $|F_{\nu }^{\mathrm{BO}}(r){|}^2$ and hyperspherical probability densities $|F_{\nu }(R){|}^2$ for the first excited Efimov state at unitarity. Here we use $A={}^{174}\mathrm{Yb}$ and $X={}^6\mathrm{Li}$, with $r_{\mathrm{vdW},AA}=78.7\mathrm{a}.\mathrm{u}.$ and $r_{\mathrm{vdW},AX}=38.1\mathrm{a}.\mathrm{u}.$

Figure 2

The Efimov energy spectrum $E_n(a_{AA})$ for ${}^{174}\mathrm{Yb}_2{}^6\mathrm{Li}$ at the heteronuclear unitarity ($a_{AX}=\infty$). The “1BS” and “2BS” hyperspherical calculations have different $r_{c,AA}$ and $r_{c,AX}$ to give 1 and 2 $s$-wave bound states for both $A+A$ and $A+X$ pairs, respectively. The error bar on the hyperspherical data points indicates the width of the corresponding Efimov state. The “Analytic” curve is calculated from Eq. (11) (see the discussion in the main text).

Figure 3

The adiabatic hyperspherical potentials $W_{\nu }(R)$ for the $AX+A$ channel (a) and the $A_2+X$ channel (b) for an Efimov-unfavored system with $A={}^{133}\mathrm{Cs}$, $X={}^{87}\mathrm{Rb}$. Here, $r_{\mathrm{vdW},AA}=101\mathrm{a}.\mathrm{u}.$ and $r_{\mathrm{vdW},AX}=90.9\mathrm{a}.\mathrm{u}.$ The $a_{AA}$ labels in the plots are also in a.u.