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We present a variational study of the spin-1 Bose gases in a harmonic trapwith three-dimensional spin-orbit coupling of Weyl type. For weak spin-orbitcoupling, we treat the single-particle ground states as the form ofperturbational harmonic oscillator states in the lowest total angular momentummanifold with $j=1, m_j=1,0,-1$. When the two-body interaction is considered,we set the trail order parameter as the superposition of three degeneratesingle-particle ground-states and the weight coefficients are determined byminimizing the energy functional. Two ground state phases, namely the magneticand the nematic phases, are identified depending on the spin-independent andthe spin-dependent interactions. Unlike the non-spin-orbit-coupled spin-1Bose-Einstein condensate for which the phase boundary between the magnetic andthe nematic phase lies exactly at zero spin-dependent interaction, the boundaryis modified by the spin-orbit-coupling. We find the magnetic phase is featuredwith phase-separated density distributions, 3D skyrmion-like spin textures andcompeting magnetic and biaxial nematic orders, while the nematic phase isfeatured with miscible density distributions, zero magnetization and spatiallymodulated uniaxial nematic order. The emergence of higher spin order createsnew opportunities for exploring spin-tensor-related physics in spin-orbitcoupled superfluid.

Magnetic and nematic phases in a Weyl type spin-orbit-coupled spin-1 Bose gas