We use collective oscillations of a two-component Bose-Einstein condensate (2CBEC) of Rb atoms prepared in the internal states $ket{1}equivket{F=1, m_F=-1}$ and $ket{2}equivket{F=2, m_F=1}$ for the precision measurement of the interspecies scattering
length $a_{12}$ with a relative uncertainty of $1.6times 10^{-4}$. 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{o}dinger equation for an effective harmonic oscillator. The frequency of the collective oscillations is defined by the axial trap frequency and the ratio $a_{12}/a_{11}$, where $a_{11}$ is the intra-species scattering length of a highly populated component 1, and is largely decoupled from the scattering length $a_{22}$, 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 $a_{12}=98.006(16),a_0$, where $a_0$ is the Bohr radius. Our reported value is in a reasonable agreement with the theoretical prediction $a_{12}=98.13(10),a_0$ but deviates significantly from the previously measured value $a_{12}=97.66,a_0$ cite{Mertes07} which is commonly used in the characterisation of spin dynamics in degenerate Rb atoms. Using Ramsey interferometry of the 2CBEC we measure the scattering length $a_{22}=95.44(7),a_0$ which also deviates from the previously reported value $a_{22}=95.0,a_0$ cite{Mertes07}. We characterise two-body losses for the component 2 and obtain the loss coefficients ${gamma_{12}=1.51(18)times10^{-14} textrm{cm}^3/textrm{s}}$ and ${gamma_{22}=8.1(3)times10^{-14} textrm{cm}^3/textrm{s}}$.
We observe the coherence of an interacting two-component Bose-Einstein condensate (BEC) surviving for seconds in a trapped Ramsey interferometer. Mean-field driven collective oscillations of two components lead to periodic dephasing and rephasing of
condensate wave functions with a slow decay of the interference fringe visibility. We apply spin echo synchronous with the self-rephasing of the condensate to reduce the influence of state-dependent atom losses, significantly enhancing the visibility up to 0.75 at the evolution time of 1.5s. Mean-field theory consistently predicts higher visibility than experimentally observed values. We quantify the effects of classical and quantum noise and infer a coherence time of 2.8 s for a trapped condensate of 5.5e4 interacting atoms.
