  In this paper, we study a system of two-component Bose gas in an artificial
magnetic field trapped by concentric harmonic and annular potentials,
respectively. The system is realized by gases with two-internal states like the
hyperfine states of $^{87}$Rb. We are interested in effects of a Rabi
oscillation between them. Two-component Bose Hubbard model is introduced to
describe the system, and Gross-Pitaevskii equations are used to study the
system. We first study the Bose gas system in the annular trap by varying the
width of the annulus and strength of the magnetic field, in particular, we
focus on the phase slip and superflow. Then we consider the coupled Bose gas
system in a magnetic field. In a strong magnetic field, vortices form a
Abrikosov triangular lattice in both Bose-Einstein condensates (BECs), and
locations of vortices in the BECs correlate with each other by the Rabi
coupling. However, as the strength of the Rabi coupling is increased, vortices
start to vibrate around their equilibrium locations. As the strength is
increased further, vortices in the harmonic trap start to move along the
boundaries of the annulus. Finally for a large Rabi coupling, the BECs are
destroyed. Based on our findings about the BEC in the annular trap, we discuss
the origin of above mentioned phenomena.
