Phase ordering kinetics of a nonequilibrium exciton-polariton condensate

We investigate the process of coarsening via annihilation of vortex-antivortex pairs, following the quench to the condensate phase in a nonresonantly pumped polariton system. We find that the late-time dynamics is an example of universal phase-ordering kinetics, characterized by scaling of correlation functions in time. Depending on the parameters of the system, the evolution of the characteristic length scale $L\left(t\right)$ can be the same as for the two-dimensional $XY$ model, described by a power law with the dynamical exponent $z\approx 2$ and a logarithmic correction, or $z\approx 1$ which agrees with previous studies of conservative superfluids.

Figure 1

Phase ordering through annihilation of vortex-antivortex pairs. Density (left column) and phase (right column) of the condensate wave function at $t=50\mathrm{ps}$ (top) and $t=150\mathrm{ps}$ (bottom) after the quench. Parameters are $m^{*}=5\times {10}^{-5}{m}_{\mathrm{e}}, {\gamma }_C^{-1}=50\mathrm{ps}, {\gamma }_R^{-1}=8\mathrm{ps}, A=0, g_C=3.4\mu \mathrm{eV}\mu {\mathrm{m}}^2, g_R=7.2\mu \mathrm{eV}\mu {\mathrm{m}}^2, R=5.5\times {10}^{-3}\mu {\mathrm{m}}^2 {\mathrm{ps}}^{-1}, P=40\mu {\mathrm{m}}^{-2} {\mathrm{ps}}^{-1}, \xi =2.9\mu \mathrm{m}$.

Figure 2

Typical evolution of the mean condensate density and number of vortices. (a) The pair annihilation occurs in parallel with the saturation of the density. (b) The evolution after $t\gtrsim 20\mathrm{ps}$ corresponds to pure phase-ordering kinetics. In this case the density of defects decays according to the scaling law Eq. (4). Parameters in (a) are the same as described in the caption of Fig. 1, while in (b) we use ${\gamma }_C^{-1}=3.3\mathrm{ps}, {\gamma }_R^{-1}=3\mathrm{ps}, A=0.1, g_C=3\mu \mathrm{eV}\mu {\mathrm{m}}^2, g_R=6\mu \mathrm{eV}\mu {\mathrm{m}}^2, R=2.3\times {10}^{-4}\mu {\mathrm{m}}^2 {\mathrm{ps}}^{-1}, P=3\times {10}^3\mu {\mathrm{m}}^{-2} {\mathrm{ps}}^{-1}, \xi =1.6\mu \mathrm{m}$. Averaged over 16 realizations of the Wigner noise.

Figure 3

(a) First-order correlation function vs. distance, at several evolution times. The increase in spread of $g^{\left(1\right)}$ is a result of coarsening, and an increase of the characteristic length scale $L\left(t\right)$, which we define as the value of distance $d$ at which $g^{\left(1\right)}=0.25$. (b) Collapse of the correlation function after rescaling the $d$ axis by $L\left(t\right)$, confirming the scaling hypothesis. The parameters are the same as described in the caption of Fig. 2.

Figure 4

(a) Time dependence of the length scale $L\left(t\right)$ in the case of Fig. 2. The dynamical exponent attains the value $z\approx 1$, in agreement with previous studies of conservative superfluids [6, 12]. (b) The case of Fig. 2 with pure phase ordering. In this case the length scale follows the universal scaling law for vector systems in two dimensions with nonconserved order parameter, Eq. (4), with $z\approx 2$. Error bars correspond to the standard deviation of the estimation of $L\left(t\right)$, determined from the values of $L\left(t\right)$ that vary from one realization to another. The error bars in (b) are comparable to the size of a single point. The number of averaged realizations was 30 in (a) and 150 in (b).

Figure 5

Time dependence of the length scale $L\left(t\right)$ in the case of a high-quality sample with increased polariton lifetime, as in Fig. 4, with Dirichlet (hard-wall) boundary conditions.

Figure 6

Time dependence of the vortex number $N\left(t\right)$ presented in bilogarithmic scale, in the case of a low-quality sample with short polariton lifetime (as in Fig. 4 in the main text). The data corresponds to a single simulation.

Figure 7

Same as described in the caption of Fig. 4, but theoretical fits without the logarithmic correction.