Spin-Textures of the Condensates with Two Kinds of Spin-1 Atoms Studied Beyond the Single Spatial Mode Approximation

This paper is based on the coupled Gross–Pitaevskii equations (CGP) for two-species BEC, where both the central and spin-dependent forces are included. Under the Thomas–Fermi approximation (TFA), an approach is proposed to obtain analytical solutions of the CGP. Recall that the spin-textures depend seriously on three factors \(I_A=\int \varphi _A^4\mathrm{d}\mathbf {r}\), \(I_B=\int \varphi _B^4\mathrm{d}\mathbf {r}\), and \(I_{AB}=\int \varphi _A^2\varphi _B^2\mathrm{d}\mathbf {r}\), where \(\varphi _A\) and \(\varphi _B\) are the spatial wave functions of the two species. In the single spatial mode approximation (SMA), these three factors remain unknown. However, in this paper, these factors have been obtained and analytically expressed by the input parameters (in which those for the central forces are essential). Besides, a number of formulae have been derived to describe the critical points of spin-transition. Thus, the present theory is a complement of the SMA so that the theoretical result can be compared with the experimental result quantitatively, and the effect of each parameter involved in experiments can be clarified. Finally, the results under the TFA are compared with those from the exact numerical solution of the CGP.