Probing Efimov discrete scaling in an atom-molecule collision

The discrete Efimov scaling behavior, well known in the low-energy spectrum of three-body bound systems for large scattering lengths (unitary limit), is identified in the energy dependence of an atom-molecule elastic cross section in mass-imbalanced systems. That happens in the collision of a heavy atom with mass $m_H$ with a weakly bound dimer formed by the heavy atom and a lighter one with mass $m_L\ll m_H$. Approaching the heavy-light unitary limit, the $s$-wave elastic cross section $\sigma$ will present a sequence of zeros or minima at collision energies following closely the Efimov geometrical law. Our results, obtained with Faddeev calculations and supplemented by a Born-Oppenheimer analysis, open a perspective to detecting the discrete scaling behavior from low-energy scattering data, which is timely in view of the ongoing experiments with ultracold binary mixtures having strong mass asymmetries, such as lithium and cesium or lithium and ytterbium.

Figure 1

The $s\text{−}$ wave cross section is shown as a function of the energy collision $E$ for zero-ranged (ZR) (blue-solid lines) and finite-ranged Gaussian (G) (red-dashed lines) potentials, for fixed mass ratio $A=0.01$ and given two-body energies ($B_2^G$ is a factor smaller than $B_2^{ZR}$ to keep both results close to the unitary limit). Results are in units of $B_3$.

Figure 2

The zero-ranged results for $\sigma$ (in arbitrary units) as functions of $E/B_3$ are given in eight panels, with mass ratios $A\equiv m_L/m_H$ as shown inside the frames. In all the panels the two-body energies are fixed such that $B_2/B_3=0.01$ (solid-blue lines), 0.03 (dot-dashed-red lines), and 0.05 (dashed-black lines).

Figure 3

Ratio between the energy positions of the successive zeros ($n+1,n$) of the cross section $\sigma$ plotted versus $1/\left(E_{n+1}^{1/2}{a}_{HL}\right)$ for mass ratio $A=0.01$. The results obtained from Faddeev calculations with the renormalized zero-range (ZR) model for given two-body energies are indicated inside the frame. The straight dashed line indicates the Efimov limit for $A=0.01$. The solid line with dots shows the results obtained from Faddeev calculations with the Gaussian potential (2). The Born-Oppenheimer (BO) results (connected by a dashed-blue line) are shown for three boundary conditions.