Interferometric measurement of interhyperfine scattering lengths in Rb 87

We present interferometeric measurements of the 𝑓=1 to 𝑓=2 interhyperfine scattering lengths in a single-domain spinor Bose-Einstein condensate of 87 Rb . The interhyperfine interaction leads to a strong and state-dependent modification of the spin-mixing dynamics with respect to a noninteracting 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 comagnetometer strategy to cancel laboratory magnetic noise. The method allows precise determination of interhyperfine scattering length differences, calibrated to intrahyperfine scattering length differences. We report (𝑎(12) 3−𝑎(12) 2)/(𝑎(1) 2−𝑎(1) 0)=−1.27⁢(15) and (𝑎(12) 1−𝑎(12) 2)/(𝑎(1) 2−𝑎(1) 0)=−1.31⁢(13), limited by atom number uncertainty. With achievable control of atom number, we estimate precisions of ≈0.3% should be possible with this technique.