Polariton Condensation with Localized Excitons and Propagating Photons

We estimate the condensation temperature for microcavity polaritons, allowing for their internal structure. We consider polaritons formed from localized excitons in a planar microcavity, using a generalized Dicke model. At low densities, we find a condensation temperature $T_c\propto \rho$, as expected for a gas of structureless polaritons. However, as $T_c$ becomes of the order of the Rabi splitting, the structure of the polaritons becomes relevant, and the condensation temperature is that of a BCS-like mean-field theory. We also calculate the excitation spectrum, which is related to observable quantities such as the luminescence and absorption spectra.

Figure 1

Condensed and uncondensed spectra, superimposed on the mean-field phase diagram to show choice of mean-field density and temperature. (Parameters ${\Delta }^{*}=0$, $m^{*}=0.01$. $\mu /g\sqrt{n}=-1.4$ below transition; $\mu /g\sqrt{n}=-0.6$ above. Energies and densities plotted in units of $g\sqrt{n}$ and $n$.)

Figure 2

Absorption coefficient vs momentum ($x$ axis) and energy ($y$ axis) calculated above (top) and below (bottom) phase transition. Negative values indicate gain. (Parameters as for Fig. 1, inhomogeneous broadening width $0.3g\sqrt{n}$.) The color scale covers the range $-\infty$ to $\infty$.

Figure 3

Momentum distribution of photons, from which follows the angular distribution. Two uncondensed and two condensed distributions are shown for mean-field densities as shown in the inset. (Parameters ${\Delta }^{*}=0$, $m^{*}=0.01$.)

Figure 4

Phase boundaries calculated from mean-field theory, and including fluctuation corrections for four different values of photon mass. Temperature and density plotted in units of $g\sqrt{n}$ and $n$.