BCS Wave-Function Approach to the BEC-BCS Crossover of Exciton-Polariton Condensates

The crossover between low and high density regimes of exciton-polariton condensates is examined using a BCS wave-function approach. Our approach is an extension of the BEC-BCS crossover theory for excitons, but includes a cavity photon field. The approach can describe both the low density limit, where the system can be described as a Bose-Einstein condensate (BEC) of exciton-polaritons, and the high density limit, where the system enters a photon-dominated regime. In contrast to the exciton BEC-BCS crossover where the system approaches an electron-hole plasma, the polariton high density limit has strongly correlated electron-hole pairs. At intermediate densities, there is a regime with BCS-like properties, with a peak at nonzero momentum of the singlet pair function. We calculate the expected photoluminescence and give several experimental signatures of the crossover.

Figure 1

(a) The occupation probability $v_{\mathit{k}}^2$ of electron-hole pairs of momentum $k$ for a cavity photon energy of $\omega =-1$ and photon coupling $\Omega =0.1$. The dotted line shows the 1s exciton wave function for comparison. The (b) electron-hole pair, photon, and total particle density, (c) energy per electron-hole pair, (d) gap energy versus chemical potential.

Figure 2

The singlet pair function $\Psi (\mathit{k})=u_{\mathit{k}}{v}_{\mathit{k}}$ for various chemical potentials as shown for (a) zero detuning $\omega =-E_0$ (b) blue detuning $\omega =0$. (c) The singlet pair function peak momenta for detunings shown. (d) The particle-particle interaction energy $V$ and condensate interaction energy $Vn$ as a function of the density.

Figure 3

(a) The PL intensity of the BCS model in the high density limit for the photonic PL and excitonic PL. A reservoir coupling $\gamma$ is assumed for both cases, giving the linewidths as shown. (b) The photonic PL for the BCS model at all densities. Zero detuning is assumed in all cases.