We investigate phase separation and hidden vortices in spin-orbit coupled ferromagnetic BoseEinstein condensates with rotation and Rabi coupling. The hidden vortices are invisible in density distribution but are visible in phase distribution, which can carry angular momentum like the ordinary quantized vortices. In the absence of the rotation, we observe the phase separation induced by the spin-orbit coupling and determine the entire phase diagram of the existence of phase separation. For the rotation case, in addition to the phase separation, we demonstrate particularly that the spin-orbit coupling can result in the hidden vortices and hidden vortex-antivortex pairs. The corresponding entire phase diagrams are determined, depending on the interplay of the spin-orbit coupling strength, the rotation frequency, and Rabi frequency, which reveals the critical condition of the occurrence of the hidden vortices and vortex-antivortex pairs. The hidden vortices here are proved to be long-lived in the time scale of experiment by the dynamic analysis. These findings not only provide a clear illustration of the phase separation in spin-orbit coupled spinor Bose-Einstein condensates, but also open a new direction for investigating the hidden vortices in high-spin quantum system.
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