Precision measurement of spin-dependent inter-hyperfine scattering lengths in $^{87}$Rb.

We present precision measurements of the $f=1$ to $f=2$ inter-hyperfine scattering lengths in a single-domain $^{87}$Rb spinor Bose-Einstein condensate. The inter-hyperfine interaction leads to a strong and state-dependent modification of the spin-mixing dynamics with respect to a non-interacting description. We employ hyperfine-specific Faraday-rotation probing to reveal the evolution of the transverse magnetization in each hyperfine manifold for different state preparations, and a co-magnetometer strategy to cancel laboratory magnetic noise. We find the scattering length ratios $a_{32}^{(12)}/a_{20}^{(1)}=-1.27(15)$ and $a_{12}^{(12)}/a_{20}^{(1)}=-1.31(13)$, limited by atom number fluctuations. Using better control of atom number, we estimate precisions of $\sim 0.3\%$ should be possible with this technique.