For spinor Bose-Einstein condensates both the total spin
S and its
Z component
SZ should be conserved. However, in existing theories, only the conservation of
SZ has been taken into account. To remedy this, here we attempt to take the conservation of both
S and
SZ into account. For this purpose, a total spin state with the good quantum numbers
S and
SZ is introduced in the trial wave function; thereby a generalized Gross-Pitaevskii equation has been derived. With this equation, the ground bands of the
23Na and
87Rb 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
23Na, i.e., in the vicinity of the ground state. On the contrary, for
87Rb, the levels are extremely dense in the top of the ground band.
We examine the use of Laguerre-Gaussian (LG) laser beams in the tripod scheme to create the coupling between spin and orbital angular momentum in a Bose–Einstein condensate (BEC) confined by a harmonic trap. We derive and solve the Gross-Pitaeskii equation numerically to obtain the density distribution of the BEC. Orbital angular momentum from the LG beam is transferred to the BEC, inducing the formation of a giant vortex at the BEC’s centre. The macroscopic flow of spinor BEC around the vortex leads to a persistent spin current in the BEC.
The interaction between two species of atoms in the condensate has been deduced and the trial wave functions that conserve the total spin have been constructed for the states of the ground band. A couple of generalized Gross-Pitaevskii equations have been derived for the spatial wave functions and the energies. The mixture of
23Na and
87Rb has been calculated as an example. The density of states is found to exhibit a ladder shape.
The analytical expressions of the fractional parentage coefficients for the total spin states of a spinor
N-boson system have been derived. Thereby an
S-conserved theory for the spinor Bose-Einstein condensation has been proposed. A set of equations has been established to describe the first excited band of the condensates. Numerical solution for
23Na has been given as an example.
By numerically solving the single-particle stationary Schrödinger equation and the Gross-Pitaevskii equation with mean-field interactions at zero temperature, the ground state properties of the rotating spin-orbital-angular-momentum coupled Bose-Einstein condensates in a harmonic trapping potential are investigated in this work. The results show that the rotation lifts the double degeneracy of the single-particle energy spectrum in the angular momentum space, and leads to the vortex state. The angular momentum of the vortex depends on the rotating frequency, the intensity of the laser beam, and the spin-orbital-angular-momentum coupling. In particular, if the rotating frequency is below a critical value, the angular momentum of the ground state vortex remains unaffected by the rotating frequency. When the rotating frequency exceeds the critical value, the angular momentum of the ground state vortex will increase with the rotating frequency increasing. By assuming that the system is confined in a ring trap, the expression of the single-particle energy spectrum in the angular momentum space can be obtained, which clarifies how the rotation frequency affects the angular momentum of the ground state. In the presence of atomic interactions, similar phenomena can also be observed in the mean-field ground state at zero temperature.