Lower bound for the hyperfine populations of spin-2 condensates against a magnetic field under the single-spatial-mode approximation

Abstract

The variation of the hyperfine populations $N{\ensuremath{\rho}}_{\ensuremath{\mu}}$ of the ground state (g.s.) of spin-2 condensates against a magnetic field $B$ is studied. A $N$-body theory which is rigorous in dealing with the spin degrees of freedom has been adopted. The conservation of the total magnetization $M$ is assumed. A combined hyperfine population $N{\ensuremath{\rho}}_{\text{comb}}=N({\ensuremath{\rho}}_{0}\ensuremath{-}3{\ensuremath{\rho}}_{2}\ensuremath{-}3{\ensuremath{\rho}}_{\ensuremath{-}2})$, which is an observable, is defined. The lower bound for this population has been derived analytically. It turns out that the lower bound is identical to the actual population when $B=0$ and $\ensuremath{\infty}$, and the population is monotonically uprising in between. Thus, via the analytical form of the lower bound, the variation of the population against $B$ can be roughly known in advance. Numerical examples are given to demonstrate the applicability of the bound. When $M=0$, the g.s. in the polar phase was found to be highly unstable against $B$ because the neighboring level density is extremely high. When $M=N$, the g.s. in the cyclic phase was also found to be highly unstable. This is because the appearance of $B$, even if it is very weak, will break the high degeneracy of the g.s. and result in having numerous levels emerging in the neighborhood of the g.s.