Abstract
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.
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