Measurement of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>s</mml:mi></mml:math>-wave scattering lengths in a two-component Bose-Einstein condensate

We use collective oscillations of a two-component Bose-Einstein condensate (2CBEC) of ${}^{87}$Rb atoms prepared in the internal states $|1\ensuremath{\rangle}\ensuremath{\equiv}|F=1,{m}_{F}=\ensuremath{-}1\ensuremath{\rangle}$ and $|2\ensuremath{\rangle}\ensuremath{\equiv}|F=2,{m}_{F}=1\ensuremath{\rangle}$ for the precision measurement of the interspecies scattering length ${a}_{12}$ with a relative uncertainty of $1.6\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$. 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 ${a}_{12}/{a}_{11}$, where ${a}_{11}$ is the intraspecies 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 ${a}_{12}=98.006\phantom{\rule{0.16em}{0ex}}(16)\phantom{\rule{0.16em}{0ex}}{a}_{0}$, where ${a}_{0}$ is the Bohr radius. Our reported value is in reasonable agreement with the theoretical prediction ${a}_{12}=98.13\phantom{\rule{0.16em}{0ex}}(10)\phantom{\rule{0.16em}{0ex}}{a}_{0}$ but deviates significantly from the previously measured value ${a}_{12}=97.66{a}_{0}$ [Phys. Rev. Lett. 99, 190402 (2007)] which is commonly used in the characterization of spin dynamics in degenerate ${}^{87}$Rb atoms. Using Ramsey interferometry of the 2CBEC we measure the scattering length ${a}_{22}=95.44\phantom{\rule{0.16em}{0ex}}(7)\phantom{\rule{0.16em}{0ex}}{a}_{0}$ which also deviates from the previously reported value ${a}_{22}=95.0{a}_{0}$ [Phys. Rev. Lett. 99, 190402 (2007)]. We characterize two-body losses for component 2 and obtain the loss coefficients ${\ensuremath{\gamma}}_{12}=1.51\phantom{\rule{0.16em}{0ex}}(18)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}14}\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{3}/\mathrm{s}$ and ${\ensuremath{\gamma}}_{22}=8.1\phantom{\rule{0.16em}{0ex}}(3)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}14}\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{3}/\mathrm{s}$.

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