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.
Lower bound for the population of hyperfine component
μ=0 particles in the ground state of spin-1 condensates
An analytical expression for the lower bound of the average number of hyperfine component
μ=0 particles in the ground state of spin-1 condensates (denoted as
ρ0__) under a magnetic field has been derived. In the derivation the total magnetization
M is kept rigorously conserved. Numerical examples are given to show the applicability of the analytical expression. It was found that, in a broad domain of parameters specified in the paper, the lower bound is very close to the actual
ρ0__. Thereby, in this domain, $\overset{\_\_}{% ρ_{0}}$ can be directly evaluated simply by using the analytical expression.
The possible ground states of spin-2 Bose-Einstein condensates in an external magnetic field are obtained analytically and classified systematically according to the population of the condensed atoms at the hyperfine sublevels. It is shown that the atoms can populate simultaneously at four hyperfine sublevels in a weak magnetic field with only the linear Zeeman energy, in contrast to that in a stronger magnetic field with the quadratic Zeeman energy, where condensed atoms can at most populate at three hyperfine sublevels in the ground states. Any spin configuration we obtained will give a closed subspace in the order parameter space of the condensates.