Critical speed of a binary superfluid of light

We theoretically study the critical speed for superfluid flow of a two-dimensional miscible binary superfluid of light past a polarization-sensitive optical obstacle. This speed corresponds to the maximum mean flow velocity below which dissipation is absent. In the weak-obstacle regime, linear-response theory shows that the critical speed is set by Landau's criterion applied to the density and spin Bogoliubov modes, whose relative ordering can be inverted due to saturation of the optical nonlinearity. For obstacles of arbitrary strength and large spatial extent, we determine the critical speed from the conditions for strong ellipticity of the stationary hydrodynamic equations within the hydraulic and incompressible approximations. Numerical simulations in this regime reveal that the breakdown of superfluidity is initiated by the nucleation of vortex-antivortex pairs for an impenetrable obstacle, and of Jones-Roberts solitons for a penetrable obstacle. Beyond superfluids of light, our results provide a general framework for the critical speed of two-dimensional binary nonlinear Schr\"odinger superflows, including Bose-Bose quantum mixtures.