Bessel vortices in spin-1 Bose-Einstein condensates with Zeeman splitting and spin-orbit coupling

We investigate the ground states of spin-orbit coupled spin-1 Bose-Einstein condensates in the presence of Zeeman splitting. By introducing the generalized momentum operator, the linear version of the system is solved exactly, yielding a set of Bessel vortices. Additionally, based on linear solution and using variational approximation, the solutions for the full nonlinear system and their ground state phase diagrams are derived, including the vortex states with quantum numbers m = 0, 1, as well as mixed states. In this work, mixed states in spin-1 SOC BEC are interpreted for the first time as weighted superpositions of three vortex states. Furthermore, the results also indicate that under strong Zeeman splitting, the system cannot form localized states. The variational solutions align well with numerical simulations, showing stable evolution and meeting the criteria for long-term observation in experiments.