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We demonstrate how simultaneously-operated $^{39}$K -- $^{87}$Rbinterferometers exhibiting a high level of correlation can be used to makecompetitive tests of the university of free fall. This work provides anoverview of our experimental apparatus and data analysis procedure, including adetailed study of systematic effects. With a total interrogation time of $2T =40$ ms in a compact apparatus, we reach a statistical uncertainty on themeasurement of the E\"{o}tv\"{o}s parameter of $7.8 \times 10^{-8}$ after $2.4\times 10^4$ s of integration. The main limitations to our measurement arisefrom a combination of wavefront aberrations, the quadratic Zeeman effect in$^{39}$K, parasitic interferometers in $^{87}$Rb, and the velocity sensitivityof our detection system. These systematic errors limit the accuracy of ourmeasurement to $\eta = 0.9(1.6) \times 10^{-6}$. We discuss prospects forimprovements using ultracold atoms at extended interrogation times.

Testing the Universality of Free Fall using correlated $^{39}$K -- $^{87}$Rb interferometers