Formation of vortex rings and hopfions in trapped Bose–Einstein condensates

The topological transition of vortex lines to vortex rings and hopfions is numerically investigated by the Gross–Pitaevskii equation in three-dimensional trapped Bose–Einstein condensates. The shape of the vortex rings formed by the two vortex lines of the vortex dipole depends strongly on the initial separation of the lines. An approximately perfect vortex ring can be obtained by choosing some suitable values of the separation. The deformation of the formed rings depends on the shape of the rings in turn. Furthermore, we show a feasible approach to generate vortex hopfions by imprinting a vortex line in the center of the generated vortex rings. Specifically, the movement of the vortex rings can excite helical waves along the central vortex line of the hopfion structure if the vortex ring is not perfect.