Three-Dimensional Kardar--Parisi--Zhang Scaling in Polariton Condensates

Kardar--Parisi--Zhang (KPZ) universality provides an example of macroscopic scaling generated by microscopic violation of detailed balance. While one- and two-dimensional realizations have been explored in driven condensates and growing interfaces, demonstrating KPZ scaling in three spatial dimensions remains a major challenge. Here we propose a three-dimensional exciton-polariton crystal as a platform for observation of 3D KPZ universality. Starting from a stochastic driven-dissipative Gross-Pitaevskii equation for a condensate formed in a three-dimensional photonic-crystal lower-polariton band, we eliminate the massive density and reservoir modes and obtain an effective $3+1$-dimensional KPZ equation for the condensate phase. Numerical simulations of both the KPZ equation and the full driven-dissipative polariton model show an intermediate-asymptotic regime in which the first-order coherence obeys $-\ln |\gone(0,\Delta t)|\propto |\Delta t|^{2\beta}$ and $-\ln |\gone(\Delta r,0)|\propto |\Delta r|^{2\chi}$, with exponents consistent with the $3+1$ KPZ benchmarks $\beta= 0.1845$, $\chi= 0.3135$. Our results identify three-dimensional polariton crystals as a controllable quantum fluid route to higher-dimensional nonequilibrium universality.