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

We demonstrate how simultaneously-operated $^{39}$K -- $^{87}$Rb interferometers exhibiting a high level of correlation can be used to make competitive tests of the university of free fall. This work provides an overview of our experimental apparatus and data analysis procedure, including a detailed study of systematic effects. With a total interrogation time of $2T = 40$ ms in a compact apparatus, we reach a statistical uncertainty on the measurement 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 arise from a combination of wavefront aberrations, the quadratic Zeeman effect in $^{39}$K, parasitic interferometers in $^{87}$Rb, and the velocity sensitivity of our detection system. These systematic errors limit the accuracy of our measurement to $\eta = 0.9(1.6) \times 10^{-6}$. We discuss prospects for improvements using ultracold atoms at extended interrogation times.