Role of collisions in creation of overlapping Bose condensates

We study the elastic scattering length and inelastic decay rate associated with collisions of ${}_{}{}^{23}{}_{}{}^{}\mathrm{Na}$ and ${}_{}{}^{87}{}_{}{}^{}\mathrm{Rb}$ atoms in different hyperfine states: one with atoms in the |f=2,${\mathrm{m}}_{\mathrm{f}}$=2〉 and one with atoms in the |f=1,${\mathrm{m}}_{\mathrm{f}}$=-1〉 state. For Na the real part of the ${\mathrm{a}}_{\mathrm{}\left(22\right)+\left(1\mathrm{}\mathrm{-}1\right)}$ scattering length for |22〉+|1-1〉 collisions is predicted to be positive and equal to 65±5 Bohr length units ${\mathrm{a}}_0$ for small B. The zero-temperature low-field decay rate is of the order of a typical exchange rate: ${\mathrm{G}}_{\mathrm{}\left(22\right)+\left(1\mathrm{}\mathrm{-}1\right)}$=(1.5±0.7)${10}^{\mathrm{-}11}$ ${\mathrm{cm}}^3$ ${\mathrm{s}}^{\mathrm{-}1}$, showing that a two-condensate experiment is not feasible for a Na gas sample in a static magnetic trap. For the case of Rb atoms the present knowledge of the singlet interaction does not allow a similar calculation. The extreme suppression of ${\mathrm{G}}_{\mathrm{}\left(22\right)+\left(1\mathrm{}\mathrm{-}1\right)}$ demonstrated by a recent experiment is shown to be very restrictive for the value of the singlet accumulated phase.