Measurement ofs-wave scattering lengths in a two-component Bose-Einstein condensate

We use collective oscillations of a two-component Bose-Einstein condensate (2CBEC) of ⁸⁷Rb atoms prepared in the internal states |1≡|F=1,mF=-1 and |2≡|F=2,mF=1 for the precision measurement of the interspecies scattering length a₁₂ with a relative uncertainty of 1.6×10^-4. We show that in a cigar-shaped trap the three-dimensional (3D) dynamics of a component with a small relative population can be conveniently described by a one-dimensional (1D) Schr\"odinger equation for an effective harmonic oscillator. The frequency of the collective oscillations is defined by the axial trap frequency and the ratio a₁₂/a₁₁, where a₁₁ is the intraspecies scattering length of a highly populated component 1 and is largely decoupled from the scattering length a₂₂, the total atom number and loss terms. By fitting numerical simulations of the coupled Gross-Pitaevskii equations to the recorded temporal evolution of the axial width we obtain the value a₁₂=98.0060.16em0ex(16)0.16em0exa₀, where a₀ is the Bohr radius. Our reported value is in reasonable agreement with the theoretical prediction a₁₂=98.130.16em0ex(10)0.16em0exa₀ but deviates significantly from the previously measured value a₁₂=97.66a₀ [Phys. Rev. Lett. 99, 190402 (2007)] which is commonly used in the characterization of spin dynamics in degenerate ⁸⁷Rb atoms. Using Ramsey interferometry of the 2CBEC we measure the scattering length a₂₂=95.440.16em0ex(7)0.16em0exa₀ which also deviates from the previously reported value a₂₂=95.0a₀ [Phys. Rev. Lett. 99, 190402 (2007)]. We characterize two-body losses for component 2 and obtain the loss coefficients γ₁₂=1.510.16em0ex(18)×10^-144pt0excm³/s and γ₂₂=8.10.16em0ex(3)×10^-144pt0excm³/s.