Singularity in the matrix of the coupled Gross-Pitaevskii equations and the related state-transitions in three-species condensates

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.

Criticality and hidden criticality in multi-species Bose-Einstein condensates

“…Therefore, when more and more species take part in the coupling, the critical point (i.e., the singular point of the matrix) will shift. The shift has been confirmed in a previous paper on 3-species BEC [26].…”

“…This causes a great change in the composition of the state, i.e., a state-transition. Accompanying the transition, the total energy increases remarkably as shown in the references [25,26]. Since all the K species are involved in this transition, it is called a full-state-transition.…”

“…This paper is one along this line. it is a generalization of our two previous paper [26,35] from threespecies BEC to K-species, and from adopting a sphericalsymmetric trap to a more general trap (spherical-and axial-symmetric traps are considered as special cases). The aim is to find out the common features existing in various multi-species BEC.…”

“…During the progress a noticeable point is the relation between the BEC and other many-body systems. Recently, the similarity on the criticality and the hidden criticality between the multi-species BEC and the multi-band superconductivity was primary found [26][27][28][29][30]. It is believed that new and rich physical phenomena would emerge from multispecies BEC.…”

Spin-structures of the Bose-Einstein condensates with three kinds of spin-1 atoms

“…Since the kinetic energy increases linearly with particle number N, while the interaction energy increases with N 2 , the relative importance of the kinetic terms is very weak when N is very large. In this case, the TFA is a reasonable approximation [19][20][21] . By neglecting the kinetic terms, in a domain where all the ϕ J are nonzero, the CGP can be written in a matrix form as…”