Effect of disorder potential on the dynamics of resonantly excited incoherent free exciton-polariton fluids in high-$Q$ GaAs microcavities

The temporal behavior of the lower polariton (LP) distribution in the reciprocal space, $n_{\mathrm{LP}}\left(k\right)$, and formation of long-range spatial coherence are investigated in a nonequilibrium incoherent LP fluid generated resonantly with picosecond optical pulses at 2 K in a high-$Q$ planar GaAs/AlAs microcavity with 12 InGaAs quantum wells. The dynamics of $n_{\mathrm{LP}}\left(k\right)$ is found to be independent of excitation density and well described within the framework of linear Schrödinger equations taking into account random potential disorder, $\delta E_{\mathrm{LP}}$, and finite lifetime of LPs, up to LP density $n_{\mathrm{LP}}(t=0)=7\times {10}^{10}{\mathrm{cm}}^{-2}$ (3.5 orders of magnitude greater than the threshold density of Bose-Einstein condensation for LPs). This is explained by the smallness of the ratios of LP interaction energy to both the mean kinetic energy and potential disorder. The contribution of interparticle interaction to the formation of spatial coherence in the LP fluid is insignificant at $E_{\mathrm{int}}\ll \delta E_{\mathrm{LP}}$, but becomes noticeable already at $E_{\mathrm{int}}\approx 0.2\delta E_{\mathrm{LP}}$, despite the fact that its effect on the $k$ distribution of LPs remains insignificant. Coherence length $L_c$ in LP fluid with $n_{\mathrm{LP}}(t=0)=2$ and $7\times {10}^{10}{\mathrm{cm}}^{-2}$ in the region with $\delta E_{\mathrm{LP}}=0.15$ meV at $t=160$ ps increases to 4.1 and $5.3\mu m$, respectively, whereas in an incoherent Bose gas with the same $n_{\mathrm{LP}}\left(k\right)$ it is equal to $3.6\mu m$.