  Spin-orbit coupling (SOC) is responsible for a range of spintronic and
topological processes in condensed matter. Here we show photonic analogs of
SOCs in exciton-polaritons and their condensates in microcavities composed of
birefringent lead halide perovskite single crystals. The presence of
crystalline anisotropy coupled with splitting in the optical cavity of the
transverse electric (TE) and transverse magnetic (TM) modes gives rise to a
non-Abelian gauge field, which can be described by the Rashba-Dresselhaus
Hamiltonian near the degenerate points of the two polarization modes. With
increasing density, the exciton polaritons with pseudospin textures undergo
phase transitions to competing condensates with orthogonal polarizations.
Unlike their pure photonic counterparts, these exciton polaritons and
condensates inherit nonlinearity from their excitonic components and may serve
as quantum simulators of many-body SOC processes.
