Abstract
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.
Highlights
It is reasonable to expect that the critical phenomena found in 2-species Bose-Einstein condensates (2-BEC) might be affected and new critical phenomena might emerge
This paper is dedicated to a primary theoretical study on the 3-BEC based on the coupled Gross-Pitaevskii equations (CGP)
Under the Thomas-Fermi approximation (TFA), we provide an approach for obtaining analytical solutions
Summary
DarYel is given, a determinant obtained by changing the l column of the three Yl are known because they depend only on. If a wave function (say, ul/r) is nonzero in a domain but becomes zero when r = ro, a downward form-transition (say, from Form III to II) will occur at ro. Ro is named a form-transition-point, and it appears as the boundary separating the two connected domains In this way the formal solutions serve as the building blocks, and they will link up continuously to form an entire solution. They must be continuous at the form-transition-points because the wave functions satisfy exactly the same set of nonlinear equations at those points. Their derivatives are in general not continuous at the boundaries
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