Kardar-Parisi-Zhang universality in the phase distributions of one-dimensional exciton-polaritons

Exciton-polaritons under driven-dissipative conditions exhibit a condensation transition that belongs to a different universality class from that of equilibrium Bose-Einstein condensates. By numerically solving the generalized Gross-Pitaevskii equation with realistic experimental parameters, we show that one-dimensional exciton-polaritons display fine features of Kardar-Parisi-Zhang (KPZ) dynamics. Beyond the scaling exponents, we show that their phase distribution follows the Tracy-Widom form predicted for KPZ growing interfaces. We moreover evidence a crossover to the stationary Baik-Rains statistics. We finally show that these features are unaffected on a certain timescale by the presence of a smooth disorder often present in experimental setups.

Figure 1

(a) Distribution of $\tilde{\theta }$ for $L/\widehat{x}=2^{10}$, together with the theoretical centered GOE-TW distribution. (b) Skewness and (c) excess kurtosis (lower panel) of the phase field in the condensate for $L/\widehat{x}=2^9,2^{10}$, together with the theoretical values for TW-GOE and BR statistics (green and purple dashed line, respectively). TW-GOE values are reached on a plateau around times $t/\widehat{t}\simeq {10}^4$.

Figure 2

Distribution of $\mathrm{\Delta }q(x,t_0,\mathrm{\Delta }t)$ for $\mathrm{\Delta }t/t_0={10}^3$ (light-green symbols) and $\mathrm{\Delta }t/t_0=5\times {10}^{-3}$ (purple symbols) for $L/\widehat{x}=2^{10}$, together with the theoretical centered TW-GOE (green solid line) and BR (purple solid line) distributions [35]. Inset (same color code): numerical and theoretical values of $\mathrm{var}\left(\mathrm{\Delta }q\right)$ for different initial times $t_0/\widehat{t}={10}^2,2\times {10}^4$ (blue and brown color-scale, respectively) and sizes $L/\widehat{x}=2^8,2^9,2^{10}$ (increasing from lighter to darker).

Figure 3

Distribution of $\mathrm{\Delta }q(x,t_0,\mathrm{\Delta }t)$ for different disorder correlation lengths ${\ell }_d/L=0.02,0.07,0.15$ (from lighter to darker) for $\mathrm{\Delta }t/t_0\simeq {10}^2$ (blue symbols) and $\mathrm{\Delta }t/t_0\simeq {10}^{-1}$ (purple symbols), with $L/\widehat{x}=2^{10}$. The solid lines correspond to centered TW-GOE (cyan) and BR (violet) theoretical distributions [35]. Inset (same parameters and color code): numerical and theoretical values for $\mathrm{var}\left(\mathrm{\Delta }q\right)$.