Asymptotic expressions for the hyperfine populations in the ground state of spin-1 condensates against a magnetic field

Based on the perturbation theory up to the second order, analytical asymptotic expressions for the variation of the population of hyperfine component $\mu =0$ particles in the ground state of spin-1 condensates against a magnetic field $B$ has been derived. The ranges of $B$ in which the asymptotic expressions are applicable have been clarified via a comparison of the numerical results from the analytical expressions and from a diagonalization of the Hamiltonian in a complete spin-space. It was found that, For ${}^{87}$Rb, the two analytical expressions, one for a weak and the other one for a strong field, together cover the whole range of $B$ from 0 to infinite. For Na, the analytical expressions are valid only if $B$ is very weak or sufficiently strong.