Central vortex steady states and dynamics of Bose–Einstein condensates interacting with a microwave field

Highlights: • Vortex steady states of a microwave field-coupled Gross–Pitaevskii equation for BECs are studied. • For a given winding number S, the existence and nonexistence central vortex steady states are proved. • The results are established for small and large contact interaction parameters, respectively. • Dynamical equations of observables are derived and solved numerically to study transient dynamics. 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.