Determination of the scattering length of erbium atoms

An accurate knowledge of the scattering length is fundamental in ultracold quantum gas experiments and essential for the characterization of the system as well as for a meaningful comparison to theoretical models. Here, we perform a careful characterization of the $s$-wave scattering length $a_{\mathrm{s}}$ for the four highest-abundance isotopes of erbium, in the magnetic field range from 0 to 5 G. We report on cross-dimensional thermalization measurements and apply the Enskog equations of change to numerically simulate the thermalization process and to analytically extract an expression for the so-called number of collisions per rethermalization (NCPR) to obtain $a_{\mathrm{s}}$ from our experimental data. We benchmark the applied cross-dimensional thermalization technique with the experimentally more demanding lattice modulation spectroscopy and find good agreement for our parameter regime. Our experiments are compatible with a dependence of the NCPR with $a_{\mathrm{s}}$, as theoretically expected in the case of strongly dipolar gases. Surprisingly, we experimentally observe a dependency of the NCPR on the density, which might arise due to deviations from an ideal harmonic trapping configuration. Finally, we apply a model for the dependency of the background scattering length with the isotope mass, allowing us to estimate the number of bound states of erbium.

Figure 1

Effective temperatures $T_z$ (blue circles) and $T_y$ (red diamonds) after the increase of the trapping potential along the weakest trapping direction $y$. The measurement was performed at 1 G and $\theta =0°$ for the ${}^{166}\mathrm{Er}$ isotope. The red dashed line represents a guide to the eye. The black solid line denotes the results of the Enskog simulations for this specific data set. The error bars denote the standard error for three repetitions. The inset shows a schematic representation of our experimental system.

Figure 2

(a) Dependency of $\alpha$ on $\theta$ and $a_{\mathrm{s}}$ for $a_{\mathrm{s}}=0a_0, 5a_0, 10a_0, 36.5a_0$, and $68.3a_0$. These values are chosen such that the angle dependence at small $a_{\mathrm{s}}$ becomes visible. Note that, at $68.3a_0$ ($a_{\mathrm{s}}$ at 1 G, see later measurements) the variation of $\alpha$ with $\theta$ is strongly suppressed. (b) $\alpha$ vs $a_{\mathrm{s}}$ for $\theta =0°$. The inset shows an enlargement of the region for $a_{\mathrm{s}}$ between $0a_0$ and ${40}_0$. The gray dashed lines show the values of $\alpha$ for $s$-wave and $p$-wave scattering, respectively.

Figure 3

Atom-loss spectroscopy (orange circles) as a function of $B$ for a fixed holding time of 250 ms. For each $B$ value, the data point is an average of three repetitions and it is normalized to the maximum averaged atom number recorded in the explored magnetic-field range. Further, $a_{\mathrm{s}}$ extracted from cross-dimensional thermalization measurements using both the Enskog equations (red squares) and the analytic formula of Eq. (8) (blue diamonds) is shown for ${}^{166}\mathrm{Er}$. Additionally $a_{\mathrm{s}}^{\mathrm{LMS}}$ (black triangles) values obtained from lattice modulation spectroscopy measurements are given. The solid black lines represent a fit of Eq. (10) to $a_{\mathrm{s}}^{\mathrm{LMS}}$. Error bars and the shaded area of the fitting results denote the standard error.

Figure 4

Measurements of $\alpha (a_{\mathrm{s}},\theta )$ as a function of $\overline{n}$. The blue circles correspond to the data sets at 1 G, shown in Fig. 3. The black solid line marks the value given by the analytic formula in Eq. (9). All measurements were performed with $\theta =0°$. Error bars denote the standard error.

Figure 5

Atom-loss spectroscopy (orange circles) as a function of $B$ for a fixed holding time of (a) 250 ms for ${}^{164}\mathrm{Er}$ and (b) 500 ms for ${}^{170}\mathrm{Er}$. For each $B$ value, the data point is an average of four to five repetitions and it is normalized to the maximum averaged atom number recorded in the explored magnetic-field range. Further, $a_{\mathrm{s}}$ values extracted from cross-dimensional thermalization measurements using both the Enskog equations (red squares) and the analytic formula of Eq. (8) (blue diamonds) are shown. The solid black lines represent a fit of Eq. (10) to $a_{\mathrm{s}}$ obtained using the Enskog equations. Error bars and the shaded area of the fitting results denote the standard error.

Figure 6

Background scattering length $a_{\mathrm{s}}^{\mathrm{bg}}$ for four bosonic isotopes (red circles). The solid line represents the best fit with $\varphi /\pi =144\left(1\right)$ (see text). The shaded area, enclosed by the dotted lines, represents the fitting function for $\varphi =143$ and $\varphi =145$. The error bars denote the standard error of the fit of Eq. (10) to the experimental data.

Figure 7

Benchmarking of the Enskog simulation results for $T_z$ (red solid line) with Monte Carlo simulations (black dashed line). The data set is the same as that in Fig. 1.

Figure 8

Atom-loss spectroscopy (orange circles) as a function of $B$ for a fixed holding time of 500 ms. For each $B$ value, the data point is an average of six to seven repetitions and it is normalized to the maximum averaged atom number recorded in the explored magnetic-field range. Further, the measured scattering lengths $a_{\mathrm{s}}$ obtained for ${}^{168}\mathrm{Er}$ from lattice modulation spectroscopy measurements are shown. The solid black line represents a fit to $a_{\mathrm{s}}^{\mathrm{LMS}}$. The shaded area and the error bars denote the standard error.