Two Formulae for the Determination of the Interspecies Interaction Via a Binary Bose-Einstein Condensates

When the frequencies omega A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _A$$\end{document} and omega B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _B$$\end{document} for trapping A- and B-species, respectively, in a binary Bose-Einstein condensates are tuned so that the mixture is in a special status, then all the parameters would fulfill a special relation. It implies that in this case the unknown parameter can be determined by the known parameters. In this paper, omega A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _A$$\end{document} and omega B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _B$$\end{document}, which can be accurately controlled, are tuned so that the two clouds of this system either overlap completely with each other or one cloud just disappears from the center. In each case, based on the analytical solution of the coupled Gross-Pitaevskii equations which is obtained under the Thomas-Fermi approximation, two formulae relating to the parameters have been derived. When the two intraspecies interactions have been known, the interspecies interaction can be thereby determined via these formulae.