Scaling properties of one-dimensional driven-dissipative condensates

We numerically investigate the scaling properties of a one-dimensional driven-dissipative condensate described by a stochastic complex Ginzburg-Landau equation (SCGLE). We directly extract the static and dynamical scaling exponents from the dynamics of the condensate's phase field, and find that both coincide with the ones of the one-dimensional Kardar-Parisi-Zhang (KPZ) equation. We furthermore calculate the spatial and the temporal two-point correlation functions of the condensate field itself. The decay of the temporal two-point correlator assumes a stretched-exponential form, providing further quantitative evidence for an effective KPZ description. Moreover, we confirm the observability of this nonequilibrium scaling for typical current experimental setups with exciton-polariton systems, if cavities with a reduced $Q$ factor are used.

Figure 1

Finite-size scaling collapse of $w(L,t)$ in the 1D KPZ universality class with $\alpha =1/2$ and $z=3/2$. Ensemble averages were performed over a number of $N_{\mathrm{Traj}}=1000$ stochastic trajectories. Values of the parameters used in the simulations are $r_c=-0.1,u_c=0.1,\sigma =0.1,K_c=3.0$.

Figure 2

Dependence of the crossover time $t_c$ on the nonequilibrium strength $\left|g\right|$. $t_c$ decreases pronouncedly as $\left|g\right|$ increases. In the numerical results presented here, $\left|g\right|$ is tuned by changing $K_c=2.4,2.2,2.0,1.8,1.6$ while keeping the other parameters unchanged. Their values are $L=2^{15},r_c=-0.1,u_c=0.1,\sigma =0.1$.

Figure 3

Behavior of the translation invariant two-point function ${\overline{C}}_x(x_1,x_2,t=2.9\times {10}^5>T_s)$ at linear system size $L=2^{12}$ on a semi-logarithmic scale. $N_{\mathrm{Traj}}=800$ stochastic trajectories are used to perform the ensemble average. Values of other parameters used here are $r_c=-0.1,u_c=0.1,\sigma =0.1,K_c=3.0$. The exponential decay of $\left|{\overline{C}}_x(x_1,x_2,t)\right|$ with respect to $|x_1-x_2|$ can be clearly identified from this plot.

Figure 4

The dependence of $-log(\left|{\overline{C}}_t(t_1,t_2)\right|/\left|{\overline{C}}_t(t_2,t_2)\right|)$ on $|t_1-t_2|$ for three different sets of parameters in the SCGLE and system sizes. The system size and the parameters for the uppermost (blue) curve are the same as those used in Fig. 3. For the lowermost (at large time differences) curve shown in red, the dimensionless linear system size is $2^9,$ and the parameters are chosen to match typical values in current experiments with exciton-polaritons (see Sec. 5 for details). Finally, assuming that a cavity with reduced $Q$ factor is used we obtain the parameters for the middle (yellow) curve, which corresponds to a system size of $2^7$. KPZ behavior is revealed by performing linear fits to the data points: with $|t_1-t_2|\in [{10}^2,{10}^3]$ we find $\beta =0.311$ and $\beta =0.317$ for the blue and yellow curves, respectively, while for the red curve a fit with $|t_1-t_2|$ lying in the last half decade in the above plot, gives rise to $\beta =0.307$. These values should be compared with the KPZ prediction $\beta =1/3$. For all curves $N_{\mathrm{Traj}}={10}^3$ stochastic trajectories are used.

Figure 5

Finite size scaling of $w_s\left(L\right)$. Points marked by “$\times$” denote the numerical value of $w_s\left(L\right)$ for system sizes $L=2^8,2^9,2^{10},2^{11},2^{12},2^{13}$. The blue line is a linear fit to the data on the log-log scale, from which we extract the roughness exponent $\alpha =0.499$. This is in good agreement with the roughness exponent ${\alpha }_{\mathrm{KPZ}}$ of the KPZ dynamics in 1D, ${\alpha }_{\mathrm{KPZ}}=1/2$. Values of parameters used in the simulations are $r_c=-0.1,u_c=0.1,\sigma =0.1,K_c=3.0$.

Figure 6

Upper panel: Time dependence of $w(L,t)$ for different system sizes $L=2^{16},2^{17}$ with $r_c=-0.1,u_c=0.1,\sigma =0.1,K_c=3.0$, on logarithmic scales. Lower panel: $t_e$ dependence of extracted $\beta$ at different system sizes $L=2^{16},2^{17}$. The growth exponent of EW dynamics ${\beta }_{\mathrm{EW}}$ and KPZ dynamics ${\beta }_{\mathrm{KPZ}}$ are indicated by two lines in the plot to facilitate direct comparison. From the results at system size $L=2^{17}$, we obtain the dynamical exponent $\beta =0.335$ from the maximum value of $\beta \left(t_e\right)$, which is in good agreement with ${\beta }_{\mathrm{KPZ}}=1/3$.