Theory of two-component superfluidity of microcavity polaritons

TL;DR

We develop a microscopic mean-field theory for two-component LP/UP polariton condensation in microcavities, introducing a population-split parameter $α$ to control LP/UP occupations and reveal detuning-dependent collective modes. Using a Hopfield-based LP/UP basis, a reduced contact interaction $U_0$, and a Bogoliubov treatment near $P=0$, we obtain analytic expressions for the Bogoliubov spectrum, the tunable sound velocity $c_s$, and the critical temperature $T_c$, all explicitly dependent on detuning $Δ_0$, Rabi splitting $Ω_R$, and $α$. The theory predicts enhanced $c_s$ and $T_c$ relative to single-branch LP condensation, with distinct $α$-dependence away from resonance; at $Δ_0=0$ the two components behave symmetrically and observables converge irrespective of $α$. The results provide measurable benchmarks for identifying genuine two-component polariton superfluidity and offer experimental protocols to extract $α$ from $T_c$ or $n_c$, applicable to GaAs quantum wells and TMDC monolayers in GaAs cavities. The framework lays groundwork for future nonequilibrium treatments and generalizations to other multi-component light–matter fluids.

Abstract

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 $α$ that quantifies the relative LP/UP occupations. At zero detuning, the critical temperature becomes independent of $α$, converging to a single value that marks the balanced, resonant regime. Away from resonance, variations in $α$ 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.

Figures (13)

• Figure 1: The Hopfield coefficients as a function of the detuning $\Delta_0$ defined above for WSe$_2$. We employed the definitions in Eq. (\ref{['HopfieldEqns']}). We kept the Rabi splitting constant at $\Omega_R =$ 8 meV.
• Figure 2: The spectrum of collective excitations in GaAs as a function of momentum. The parameters used in the calculation are $\Omega_R = 17$ meV, a bare photon of energy 1.6 eV, and a detuning $\Delta_0 = 0$ meV. The collective excitation spectrum is defined in Eq. (\ref{['eq:collective_spectrum_hopfield']}).
• Figure 3: Comparison of sound velocity $c_s$ in a two-component polariton condensate: (a) dependence on detuning $\Delta_0$ at fixed $\Omega_R = 8$ meV, and (b) dependence on Rabi splitting $\Omega_R$ at fixed $\Delta_0 = 5$ meV. In both cases the polariton density is $n = 1 \times 10^{11}~\text{cm}^{-2}$.
• Figure 4: Sound velocity $c_s$ in a two-component polariton condensate as (a) a function of the total polariton density $n_0$ for different values of the condensate population--split parameter $\alpha$, (b) and as a function of the condensate population--split parameter $\alpha$ for various fixed total polariton densities. The Rabi splitting is fixed at $\Omega_R = 8~\text{meV}$ and the detuning at $\Delta_0 = 5~\text{meV}$.
• Figure 5: Critical temperature $T_c$ in a two-component polariton condensate in monolayer WSe$_2$. (a) Dependence on detuning $\Delta_0$ at fixed Rabi splitting $\Omega_R = 8$ meV. (b) Dependence on Rabi splitting $\Omega_R$ at fixed detuning $\Delta_0 = 5$ meV. In both panels, the total polariton density is held constant at $n_0 = 1 \times 10^{11}~\text{cm}^{-2}$, and results are shown for multiple values of the population--split parameter $\alpha$. (c) Contour map of $T_c$ as a function of $\Delta_0$ and $\Omega_R$ at fixed density $n_0 = 1 \times 10^{11}~\text{cm}^{-2}$ and population split $\alpha=0.5$ and (d) $\alpha=0.75$. This panel highlights how cavity detuning and light--matter coupling together shape the critical temperature landscape.