Thermodynamics and Excitations of Condensed Polaritons in Disordered Microcavities

We study the thermodynamic condensation of microcavity polaritons using a realistic model of disorder in semiconductor quantum wells. This approach correctly describes the polariton inhomogeneous broadening in the low density limit, and treats scattering by disorder to all orders in the condensed regime. While the weak disorder changes the thermodynamic properties of the transition little, the effects of disorder in the condensed state are prominent in the excitations and can be seen in resonant Rayleigh scattering.

Figure 1

(a) Plot of the squared coupling strength $|g_{\alpha ,\mathbf{p}}{|}^2$ vs energy (160 realizations of disorder) for $\mathbf{p}=0$; (b) contour plot of the squared averaged oscillator strength $g^2(\epsilon ,|\mathbf{p}|)$ vs energy and momentum. The resolution in momentum, $p_{\mathrm{step}}=2\pi /L=6.3\times {10}^4{\mathrm{cm}}^{-1}$ corresponds, for a cavity of ${\omega }_0=1.68\mathrm{eV}$, to an angle of ${\theta }_{\mathrm{step}}={tan}^{-1}(c{p}_{\mathrm{step}}/{\omega }_0)=36°$. The free particle dispersion $|\mathbf{p}{|}^2/2m_x$ (solid line), and the squared averaged oscillator strength for two values of momenta, $|\mathbf{p}|=0$ and $|\mathbf{p}|=10p_{\mathrm{step}}$ ($\theta =82°$) (plus symbols) are explicitly plotted.

Figure 2

Mean-field phase diagram for the dimensionless critical temperature $2T_c{k}_B/{\Omega }_{\mathrm{R}}$ vs the dimensionless density $\rho$ for effective zero detuning ${\omega }_0-E_x=-0.94\mathrm{meV}$ and ${\Omega }_{\mathrm{R}}=26\mathrm{meV}$. A detail of the low density region is shown in the inset (a), while the plot of the mean-field order parameter $\psi /\sqrt{N}$ vs the density for $T{k}_B=20\mathrm{K}$ ($2T{k}_B/{\Omega }_{\mathrm{R}}=0.13$) is shown in the inset (b).

Figure 3

Contour plot of the incoherent PL $P(\omega ,\mathbf{p})$ (a), (b), the spectral weight $W(\omega ,\mathbf{p})$ (c), (d), and the disorder averaged RRS intensity $⟨I_{\mathbf{p}\mathbf{q}}(\omega )⟩$ for $|\mathbf{p}|=|\mathbf{q}|$ (e), (f) vs the rescaled energy $2(\omega -\mu )/{\Omega }_{\mathrm{R}}$ and the dimensionless momentum $|\mathbf{p}|a_x$, for ${\omega }_0-E_x=-0.94\mathrm{meV}$, ${\Omega }_{\mathrm{R}}=26\mathrm{meV}$, $k_{B}T=20\mathrm{K}$: noncondensed (left column) ($\rho \simeq 0$, $\mu =-16.5\mathrm{meV}$) and condensed (right column) ($\rho =3.6\times {10}^{-3}$, $\mu =-11.3\mathrm{meV}$).