Determining $s$-wave Scattering Length Ratios from Binary Condensate Dynamics

We characterize experimentally and computationally the component separation dynamics of a binary Bose-Einstein condensate (BEC) created suddenly by a two-photon transition from a single BEC. In addition to inelastic losses, the condensate dynamics depend crucially on the three $s$-wave scattering lengths (inter- and intraspecies), which are nearly identical in ${}^{87}$Rb. The experimentally observed density distributions exhibit striking transient ring patterns that closely correspond to the results of numerical simulations. The simulations exhibit a sensitive dependence on the ratios of the scattering lengths; we can therefore extract these ratios by comparing simulation to experiment.