Differential atom interferometry with ${}^{87}$Rb and ${}^{85}$Rb for testing the UFF in STE-QUEST

In this paper we discuss in detail an experimental scheme to test the universality of free fall (UFF) with a differential ${}^{87}$Rb / ${}^{85}$Rb atom interferometer applicable for extended free fall of several seconds in the frame of the STE-QUEST mission. This analysis focuses on suppression of noise and error sources which would limit the accuracy of a violation measurement. We show that the choice of atomic species and the correctly matched parameters of the interferometer sequence are of utmost importance to suppress leading order phase shifts. In conclusion we will show the expected performance of $2$ parts in ${10}^{15}$ of such an interferometer for a test of the UFF.