Theory of two-component superfluidity of microcavity polaritons

We develop a microscopic mean-field theory describing the coexistence of Bose-Einstein condensates of upper and lower polaritons (UP/LP) in a semiconductor microcavity. Incorporating interbranch scattering within a modified polariton Hamiltonian, we introduce a phenomenological population-split parameter $\alpha$ that quantifies the relative LP/UP occupations. At zero detuning, the critical temperature becomes independent of $\alpha$, converging to a single value that marks the balanced, resonant regime. Away from resonance, variations in $\alpha$ lead to distinctive and experimentally resolvable changes in both the sound velocity $c_s$ and critical temperature $T_c$, relative to the single-component (LP-only) condensate limit. The system under study consists of excitons confined in a transition metal dichalcogenide (TMDC) monolayer, particularly WSe$_2$ embedded within a planar optical microcavity of GaAs where they strongly couple to cavity photons. Our analysis focuses on monolayer WSe$_2$ embdedded in a GaAs microcavity. We present results for GaAs/AlGaAs quantum wells embedded in a GaAs microcavity in the Appendix. While mean-field in scope, the framework provides analytic benchmarks and physical insight for future treatments that include dissipation and fluctuations in nonequilibrium polariton superfluids.