Spin-orbit coupling and nonlinear modes of the polariton condensate in a harmonic trap

We consider a model of the exciton-polariton condensate based on a system of two Gross-Pitaevskii equations coupled by the second-order differential operator, which represents the spin-orbit coupling in the system. Also included are the linear gain, effective diffusion, nonlinear loss, and the standard harmonic-oscillator trapping potential, as well as the Zeeman splitting. By means of combined analytical and numerical methods, we identify stable two-dimensional modes supported by the nonlinear system. In the absence of the Zeeman splitting, these are mixed modes, which combine zero and nonzero vorticities in each of the two spinor components, and vortex-antivortex complexes. We have also found a range of parameters where the mixed-mode and vortex-antivortex states coexist and are stable. Sufficiently strong Zeeman splitting creates stable semi-vortex states, with vorticities 0 in one component and 2 in the other.