We study central vortex steady states and dynamics of a two-dimensional (2D), two-component, Gross–Pitaevskii equation (CGPE) system for two pseudo-spinor Bose Einstein condensates (BECs) interacting with an electromagnetic field (microwave) analytically and numerically. For the central vortex steady state at any given winding number $S$, we prove its existence in a reduced, single component detuning limit when contact-interaction strength $\beta <{\beta }_b$, and nonexistence when $\beta >{\beta }_b$, respectively, where ${\beta }_b$ is a threshold value, whose value is given in Theorem 3.1 in the paper. We extend the existence and nonexistence result to the general two pseudo-spinor case and prove that a central vortex steady state exists for any given $S$ if $\beta \le {\beta }_b$ while it does not exist when $\beta >2{\beta }_b$. We then derive dynamical equations for some observables (expectations of the matter wave function) such as the center-of-mass, position of dispersion, the linear and angular momentum of the two pseudo-spinor CGPEs. Finally, numerical computations are brought in to validate and extend the existence of vortex steady state results to ${\beta }_b\le \beta <2{\beta }_b$ for the two pseudo-spinor case and to explore transient dynamics of the observables.

我们研究了两个伪自旋玻色爱因斯坦凝聚物（BEC）与电磁场（微波）相互作用的二维（2D），两分量Gross-Pitaevskii方程（CGPE）系统的中心涡旋稳态和动力学，并且数值上。对于任何给定绕组数的中心涡流稳态$\mathrm{小号}$，我们证明了当接触相互作用强度降低时，单组分失谐极限的存在 $\beta <{\beta }_b$，以及何时不存在 $\beta >{\beta }_b$分别在哪里 ${\beta }_b$是一个阈值，其值在论文定理3.1中给出。我们将存在和不存在的结果扩展到一般的两个伪自旋情形，并证明对于任何给定都存在中心涡旋稳态$\mathrm{小号}$ 如果 $\beta \le {\beta }_b$ 当它不存在时 $\beta >2{\beta }_b$。然后，我们导出一些可观测值（物质波函数的期望值）的动力学方程，例如质心，色散位置，两个伪自旋CGPE的线性和角动量。最后，引入数值计算以验证并将涡旋稳态结果的存在扩展到${\beta }_b\le \beta <2{\beta }_b$ 对于两个伪旋转子情况，并探讨了可观测对象的瞬态动力学。