Polarization dynamics of trapped polariton condensates with $\mathrm{PT}$ symmetry

We propose a grated microcavity setup to form trapped polariton condensates with parity-time $\left(\mathrm{PT}\right)$ symmetry and study their polarization dynamics. The pseudoconservative dynamics of the Stokes vector in the proposed configuration is preserved in the presence of a polariton-polariton interaction. In the case of weak gain-dissipation imbalance, as compared to the linear polarization splitting, the polarization Stokes spheres are deformed into ellipsoids. When linear polarization splitting becomes weak, i.e., when $\mathrm{PT}$ symmetry is broken for a noninteracting system, the Stokes spheres are transformed into hyperboloids, but the dynamics is still described by closed trajectories, allowing manipulation of the polarization of the polariton condensate by changing the polarization splitting without losing its coherency.