Ground band and a generalized Gross-Pitaevskii equation for spinor Bose-Einstein condensates

Abstract

For spinor Bose-Einstein condensates both the total spin $S$ and its $Z$ component ${S}_{Z}$ should be conserved. However, in existing theories, only the conservation of ${S}_{Z}$ has been taken into account. To remedy this, here we attempt to take the conservation of both $S$ and ${S}_{Z}$ into account. For this purpose, a total spin state with the good quantum numbers $S$ and ${S}_{Z}$ is introduced in the trial wave function; thereby a generalized Gross-Pitaevskii equation has been derived. With this equation, the ground bands of the $^{23}\mathrm{Na}$ and $^{87}\mathrm{Rb}$ condensates have been studied, where the levels that are distinct in $S$ split. It was found that the level density is extremely dense in the bottom of the ground band of $^{23}\mathrm{Na}$, i.e., in the vicinity of the ground state. On the contrary, for $^{87}\mathrm{Rb}$, the levels are extremely dense in the top of the ground band.