$s$-wave scattering lengths of the strongly dipolar bosons ${}^{162}\mathrm{Dy}$ and ${}^{164}\mathrm{Dy}$

We report the measurement of the deca-heptuplet $s$-partial-wave scattering length $a$ of two bosonic isotopes of the highly magnetic element dysprosium: $a=112\left(10\right)a_0$ for ${}^{162}\mathrm{Dy}$ and $a=92\left(8\right)a_0$ for ${}^{164}\mathrm{Dy}$, where $a_0$ is the Bohr radius. The scattering lengths are determined by the cross-dimensional relaxation of ultracold gases of these Dy isotopes at temperatures above quantum degeneracy. In this temperature regime, the measured rethermalization dynamics can be compared to simulations of the Boltzmann equation using a direct-simulation Monte Carlo method employing the anisotropic differential scattering cross section of dipolar particles.

Figure 1

A qualitative comparison between the experimentally measured rethermalization of ${}^{162}\mathrm{Dy}$ versus results from the DSMC simulation. We show the rethermalization dynamics for (a, b) $\beta =0^{\circ }$ and (c, d) $\beta ={90}^{\circ }$. In each plot the data points with error bars correspond to experimental measurements. In addition, there are multiple solid lines (each with a different color). These solid lines correspond to simulation results, and the color corresponds to the value of the scattering length used for simulation. The phase offset between the data and simulation is likely due to experimental uncertainty in the trap parameters. We employ a two-step fitting method to extract estimates of the scattering length in a manner immune to these phase offsets; see Sec. 3b. Uncertainty in these data and in those of Fig. 2 are given as $1\sigma$ standard errors. Statistical fluctuations dominate systematic uncertainties in these data.

Figure 2

Fits to ${}^{162}\mathrm{Dy}$ data with $\beta =0^{\circ }$. (a, b) Data points show the experimental measurements, and the solid line shows the best fit using Eq. (5). (c, d) Plot showing ${\chi }^2$ versus fit parameter. We vary either ${\tau }_{1z}$ [in (c)] or ${\tau }_{1x}$ [in (d)] while allowing all other parameters to be reoptimized. The blue bar along the bottom axes of (c) and (d) show the $1\sigma$ uncertainties (where ${\chi }^2$ increases by 1 [34]) in ${\tau }_{1z}$ and ${\tau }_{1x}$, respectively. Results for other $\beta$ and for ${}^{164}\mathrm{Dy}$ are qualitatively similar.

Figure 3

Analyses of ${}^{164}\mathrm{Dy}$ data. The dots show the value of ${\tau }_{1x,1z}$ extracted by fitting the functional form Eq. (5) to the simulation results. The dark grey band denotes a $1\sigma$ uncertainty on the simulated ${\tau }_{1x,1z}$, and the larger grey band includes experimental uncertainty. See text for details. The horizontal dashed lines show the upper and lower bounds at $1\sigma$ uncertainty of ${\tau }_{1x,1z}$ found by fitting the same functional form Eq. (5) to the experimental data. The blue bar along the bottom axis of each figure shows the $1\sigma$ estimation of $a/a_d$, i.e., where the grey area lies between the $1\sigma$ experimental bounds. (a, b) $\beta =0^{\circ }$, (c, d) $\beta ={45}^{\circ }$, and (e, f) $\beta ={90}^{\circ }$.

Figure 4

Analyses of ${}^{162}\mathrm{Dy}$ data. The dots show the value of ${\tau }_{1x,1z}$ extracted by fitting the functional form Eq. (5) to the simulation results. The dark grey band denotes a $1\sigma$ uncertainty on the simulated ${\tau }_{1x,1z}$, and the larger grey band includes experimental uncertainty. See text for details. The horizontal dashed lines show the upper and lower bounds at $1\sigma$ uncertainty of ${\tau }_{1x,1z}$ found by fitting the same functional form Eq. (5) to the experimental data. The blue bar along the bottom axis of each figure shows the $1\sigma$ estimation of $a/a_d$, i.e., where the grey area lies between the $1\sigma$ experimental bounds. (a, b) $\beta =0^{\circ }$, (c, d) $\beta ={45}^{\circ }$, and (e, f) $\beta ={90}^{\circ }$. Data for $\beta ={90}^{\circ }$ fail to constrain ${\tau }_{1x}$ due to the fast thermalization time scale for $T_x$ and hence do not yield an estimate of $a/a_d$.

Figure 5

Summaries of the measured scattering lengths extracted from each individual experiment along with the weighted average (dashed line) and its $1\sigma$ error (grey band). Results are shown for (a) ${}^{162}\mathrm{Dy}$ and (b) ${}^{164}\mathrm{Dy}$. The weighted averages and $1\sigma$ standard errors are $a/a_d=0.63\left(5\right)$ for ${}^{162}\mathrm{Dy}$ and $a/a_d=0.47\left(4\right)$ for ${}^{164}\mathrm{Dy}$.