Precision Test of the Limits to Universality in Few-Body Physics

We perform precise studies of two- and three-body interactions near an intermediate-strength Feshbach resonance in ${}^{39}\mathrm{K}$ at 33.5820(14) G. Precise measurement of dimer binding energies, spanning three orders of magnitude, enables the construction of a complete two-body coupled-channel model for determination of the scattering lengths with an unprecedented low uncertainty. Utilizing an accurate scattering length map, we measure the precise location of the Efimov ground state to test van der Waals universality. Precise control of the sample’s temperature and density ensures that systematic effects on the Efimov trimer state are well understood. We measure the ground Efimov resonance location to be at $-14.05(17)$ times the van der Waals length $r_{\mathrm{vdW}}$, significantly deviating from the value of $-9.7r_{\mathrm{vdW}}$ predicted by van der Waals universality. We find that a refined multichannel three-body model, built on our measurement of two-body physics, can account for this difference and even successfully predict the Efimov inelasticity parameter $\eta$.

Figure 1

Survey of experimental $a_{-}$ values in homonuclear systems, inspired by [15]. Previous results (blue circles) [3, 17, 37, 41, 42, 43, 44, 45] show a tentative dependence of the $a_{-}$ value on the Feshbach resonance strength parameter $s_{\mathrm{res}}$. Our measurement (red star; red band in the inset) is the strongest evidence of departure from the $-9.7\pm 15\%r_{\mathrm{vdW}}$ value (dashed line and gray area) predicted by van der Waals universality [4, 5, 7, 9]. The inset shows calculations for $a_{-}$ based on a single van der Waals potential [7] with $N=1-7$ $s$-wave two-body bound states (green squares) and results from our multichannel model [19] with $N=2-5$ (black triangles).

Figure 2

Precise measurement of Feshbach dimer binding energies $E_b$ as a function of magnetic field $B$. Small experimental uncertainties on $E_b$, spanning from 56 Hz at $E_b/h=2.103\mathrm{kHz}$ to 1.0 kHz at $E_b/h=1167.2\mathrm{kHz}$, are not resolvable in the figure. A coupled-channel (cc) model is required to describe our data [19]. The solid curve shows the resulting fit, and the inset shows remarkably small fractional residuals. Contrary to applicability near broad Feshbach resonances [65], universal expressions (dashed and dotted curves) are insufficient for describing $E_b$ near our intermediate strength resonance.

Figure 3

Temperature dependence of the three-body loss coefficient $L_3$, scaling as $a^4$ scaling (dashed) [73, 74, 75], enhanced near an Efimov ground state located at $a_{-}$. For each temperature, we fit our data using a zero-temperature zero-range model [Eq. (1)], limiting fits to data points for which $|a|<\lambda /10$ (short vertical lines), to extract the $L_3/a^4$ peak location and a finite-temperature zero-range model [Eq. (2), solid] to extract the true $a_{-}$ value. The inset shows the extracted peak locations (circles) and $a_{-}$ values (squares), where both coincide at the lowest temperature. The observed $a_{-}$ value significantly deviates from the $a_{-}=-630a_0$ value (inset dashed line) predicted by van der Waals universality [7].

Figure 4

Suppression of the Efimov resonance in a high-density gas. Measurements of high- and intermediate-density samples performed with the same experimental conditions, contrasting only in the initial atom number. As a result, differential comparison of $L_3$ values between those two measurements is of greatest interest. Small $L_3$ deviations at low $|a|$ between the lowest-density data and other data are attributed to differing trap conditions that result in evaporation. However, for our highest-density data, we observed a strong suppression of $L_3$ near $a=a_{-}$.