Thermalization measurements on an ultracold mixture of metastable ${}^4$He and ${}^{87}$Rb atoms in a quadrupole magnetic trap

Recently we have reported (Knoop et al. [arXiv:1404.4826]) on an experimental determination of metastable triplet ${}^4$He+${}^{87}$Rb scattering length by performing thermalization measurements for an ultracold mixture in a quadrupole magnetic trap. Here we present our experimental apparatus and elaborate on these thermalization measurements. In particular we give a theoretical description of interspecies thermalization rate for a quadrupole magnetic trap, i. e. in the presence of Majorana heating, and a general procedure to extract the scattering length from the elastic cross section at finite temperature based on knowledge of the $C_6$ coefficient alone. In addition, from our thermalization data we obtain an upper limit of the total interspecies two-body loss rate coefficient of $1.5\times {10}^{-12}$ cm${}^3$s${}^{-1}$.