Observation of Universal Efimov's Ratios across an Intermediate-Strength Feshbach Resonance in $^{39}\mathrm{K}$

Efimov's original scenario is featured by an infinite number of three-body bound states (trimers) accumulating at unitarity where $E=1/a=0$. The binding energies of these trimers have a self-similar structure with a fixed scaling factor between adjacent branches. This scheme is valid in the zero-range limit and in real systems only applies to highly-excited trimers with finite-range interactions. In this work, we unambiguously measured the benchmarks associated with the Efimov spectrum in $^{39}\mathrm{K}$, denoted as $a_{-}^{(n=0)}$, $a_{*}^{(n=1)}$ and $a_{+}^{(n=0)}$, with $n$ indexing the parentage of trimer. $a_{-}^{(n)}$ are tri-atomic resonances at $a<0$, $a_{*}^{(n)}$ are scattering resonances between atoms and Feshbach molecules at $a>0$, $a_{+}^{(n)}$ are interference minima in three-atom recombination at $a>0$. We report a universal ratio $a_{*}^{(1)}/a_{-}^{(0)}$ on the two lowest-lying trimers. The within-ten-percent consistency between this ratio and zero-range result implies that finite range perturbations are suppressed as expected for Feshbach resonances with intermediate strength. We introduce multi-channel van der Waals three-body model that can reproduce all three benchmarks.