The spectra of $N$-boson systems with arbitrary nonzero spin $\mathfrak{f}$ have been studied. Firstly, only the singlet pairing interaction is considered, a set of exact eigenstates together with the eigenenergies are analytically obtained. The completeness of this set is proved. The analytical expression allows us to see clearly the spin textures of various states different in $N$ and/or $\mathfrak{f}$, and to find out the similarity and relationship lying among them. Secondly, the effect of other interactions is evaluated via exact numerical calculations on the systems with a smaller $N$. Some features and notable phenomena that might emerge in high-$\mathfrak{f}$ systems, say, the ground band might have extremely high level density, have been discussed.
A generalized Gross-Pitaevskii equation adapted to the U(5) D SO(5) 3 SO(3) symmetry has been derived and solved for the spin-2 condensates. The spin-textile and the degeneracy of the ground state (g.s.) together with the factors affecting the stability of the g.s., such as the gap and the level density in the neighborhood of the g.s., have been studied. Based on a rigorous treatment of the spin-degrees of freedom, the spin-textiles can be understood in an V-body language. In addition to the ferro, polar, and cyclic phases, the g.s. might in a mixture of them when \M\ is not equal to 0 and 2N (M is the total magnetization). The great difference in the stability and degeneracy of the g.s. caused by varying ip (which marks the features of the interaction) and M is notable. Since the root-mean-square radius Rms is an observable, efforts have been made to derive a set of formulas to relate Rms and N, to (frequency of the trap), and ip. These formulas provide a way to check the theories with experimental data.
An approach is proposed to solve the coupled Gross-Pitaevskii equations (CGP) of the 3-species BEC in an analytical way under the Thomas-Fermi approximation (TFA). It was found that, when the strength of a kind of interaction increases and crosses over a critical value, a specific type of state-transition will occur and will cause a jump in the total energy. Due to the jump, the energy of the lowest symmetric state becomes considerably higher. This leaves a particular opportunity for the lowest asymmetric state to replace the symmetric states as the ground state. It was further found that the critical values are related to the singularity of either the matrix or a sub-matrix of the CGP. These critical values are not arising from the TFA but inherent in the CGP, and they can be analytically expressed. Furthermore, a model (in which two kinds of atoms separated from each other asymmetrically) has been proposed for the evaluation of the energy of the lowest asymmetric state. With this model the emergence of the asymmetric ground state is numerically confirmed under the TFA. The theoretical formalism of this paper is quite general and can be generalized for BEC with more than three species.
A three-body bosonic Thomson atom, namely a system of three charged
bosons in a harmonic trap, with a very large Wigner parameter,
has been studied. The states with zero angular momentum and
even parity denoted as 0n+ have been calculated and
analysed; the basic modes of excitation of this system have been
revealed. It was found that each lower 0n+ state (n is
from 1 to 5) has its own means of internal motion. Although the
size of the system is very sensitive to the strength of the
trap, the other features are found to be insensitive to the
strength if the system remains in the Wigner regime. The
breathing mode was found to be the easiest to excite.
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