We present a variational study of pseudo-spin 1/2 Bose gases in a harmonic trap with weak 3D spin-orbit coupling of σ · p type. This spin-orbit coupling mixes states with different parities, which inspires us to approximate the single particle state with the eigenstates of the total angular momentum, i.e. superposition of harmonic s-wave and p-wave states. As the time reversal symmetry is protected by two-body interaction, we set the variational order parameter as the combination of two mutually time reversal symmetric eigenstates of the total angular momentum. The variational results essentially reproduce the 3D skyrmion-like ground state recently identified by Kawakami et al.. We show that these skyrmion-like ground states emerging in this model are primarily caused by p wave spatial mode involving in the variational order parameter that drives two spin components spatially separated. We find the ground state of this system falls into two phases with different density distribution symmetries depending on the relative magnitude of intraspecies and interspecies interaction: Phase I has parity symmetric and axisymmetric density distributions, while Phase II is featured with special joint symmetries of discrete rotational and time reversal symmetry. With the increasing interaction strength the transition occurs between two phases with distinct density distributions, while the topological 3D skyrmion-like spin texture is symmetry protected.
We present a variational study of the spin-1 Bose gases in a harmonic trap with three-dimensional spin-orbit (SO) coupling of Weyl type. For weak SO coupling, we treat the single-particle ground states as the form of perturbational harmonic oscillator states in the lowest total angular momentum manifold 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 degenerate single-particle ground-states and the weight coefficients are determined by minimizing the energy functional. Two ground state phases, namely the magnetic and the nematic phases, are identified depending on the spin-independent and the spindependent interactions. Unlike the non-SO-coupled spin-1 Bose-Einstein condensate for which the phase boundary between the magnetic and the nematic phase lies exactly at zero spin-dependent interaction, the boundary is modified by the SO-coupling. We find the magnetic phase is featured with phase-separated density distributions, 3D skyrmion-like spin textures and competing magnetic and biaxial nematic orders, while the nematic phase is featured with miscible density distributions, zero magnetization and spatially modulated uniaxial nematic order. The emergence of higher spin order creates new opportunities for exploring spin-tensor-related physics in SO coupled superfluid.
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