Self-rotation and synchronization in exciton-polariton condensates

Self-rotation occurs in an exciton-polariton condensate in a two-dimensional semiconductor microcavity pumped by a nonresonant Gaussian laser beam. A wave packet of the condensate spontaneously rotates around the center of the pumped region at a constant frequency breaking the rotation symmetry of the system. When two self-rotating condensates are created with an appropriate distance, synchronization occurs between the dynamics of the self-rotating condensates.

Figure 1

Modulational instability in the uniform system with ${\gamma }_c^{-1}=0.15$ ps, ${\gamma }_R^{-1}=3$ ps, $g_c=6\times {10}^{-3}\mathrm{meV}\mu {\mathrm{m}}^2, g_R=2g_c, R=0.04{\mathrm{ps}}^{-1}\mu {\mathrm{m}}^2, P=160{\mathrm{ps}}^{-1}\mu {\mathrm{m}}^{-2}\simeq 2.9P_{\mathrm{th}}$, and $m=5\times {10}^{-5}$ electron mass. (a) Time evolution of the condensate and reservoir density profiles, ${\left|\psi \right|}^2$ and $n_R$, where the initial state is the homogeneous stationary state. (b) Positive imaginary part of the Bogoliubov spectrum.

Figure 2

Time evolution of the system pumped by a Gaussian laser beam with $P_0=160{\mathrm{ps}}^{-1}\mu {\mathrm{m}}^{-2}$ and $\rho =5\mu \mathrm{m}$. The other parameters are ${\gamma }_c^{-1}=0.15$ ps, ${\gamma }_R^{-1}=3$ ps, $g_c=6\times {10}^{-3}\mathrm{meV}\mu {\mathrm{m}}^2, g_R=2g_c$, and $R=0.04{\mathrm{ps}}^{-1}\mu {\mathrm{m}}^2$. (a) Time evolution of the center-of-mass (c.m.) position $(\langle x\rangle ,\langle y\rangle )$ and the angular momentum $\langle L\rangle$ of the condensate. (b) Magnification of (a). (c) Trajectory of $(\langle x\rangle ,\langle y\rangle )$, where the red and blue points are $t<200$ ps and $t>200$ ps, respectively. The points are plotted every 0.17 ps. (d) Snapshots of the density and phase profiles, ${\left|\psi \right|}^2, \arg \psi$, and $n_R$. The white arrow indicates the direction of rotation for the c.m. position. See the Supplemental Material for a movie of the dynamics [30].

Figure 3

Parameter dependence of the period $T$ of the self-rotation. The parameters except for the abscissas are ${\overline{P}}_0=160{\mathrm{ps}}^{-1}\mu {\mathrm{m}}^{-2}, \overline{R}=0.04{\mathrm{ps}}^{-1}\mu {\mathrm{m}}^2, {\overline{\gamma }}_c=0.15$ ps, ${\overline{\gamma }}_R=3$ ps, ${\overline{g}}_c=6\times {10}^{-3}\mathrm{meV}\mu {\mathrm{m}}^2$, and ${\overline{g}}_R=2{\overline{g}}_c$.

Figure 4

Dynamics of two self-rotating condensates pumped according to Eq. (6) with $P_{\mathrm{left}}=160{\mathrm{ps}}^{-1}\mu {\mathrm{m}}^{-2}, P_{\mathrm{right}}=0.99P_{\mathrm{left}}$, and $\rho =5\mu \mathrm{m}$. The other parameters are ${\gamma }_c^{-1}=0.15$ ps, ${\gamma }_R^{-1}=3$ ps, $g_c=6\times {10}^{-3}\mathrm{meV}\mu {\mathrm{m}}^2, g_R=2g_c$, and $R=0.04{\mathrm{ps}}^{-1}\mu {\mathrm{m}}^2$. (a) Time evolution of ${\langle x\rangle }_{\mathrm{left}}$ and ${\langle x\rangle }_{\mathrm{right}}$ with $X_{\mathrm{right}}-X_{\mathrm{left}}=20\mu \mathrm{m}$. The two self-rotations are not synchronized. (b) Time evolution of ${\langle x\rangle }_{\mathrm{left}}$ and ${\langle x\rangle }_{\mathrm{right}}$ (upper panel) and snapshots of the density profile ${\left|\psi \right|}^2$ (lower panels), where $X_{\mathrm{right}}-X_{\mathrm{left}}=16\mu \mathrm{m}$. Synchronization occurs between the two self-rotating condensates. (c) The same conditions as those in (b), except a random number seed used to produce the initial noise. See the Supplemental Material for movies of the dynamics in (b) and (c) [30].

Figure 5

Time evolution of $I$ defined in Eq. (8), where (a)–(c) correspond to the dynamics shown in Figs. 4–4.

Figure 6

Time evolution of (a) $N_c=\int {\left|\psi \right|}^{2}d\mathit{r}$ (solid curve) and (b) the c.m. position $(\langle x\rangle ,\langle y\rangle )\mathrm{for}{\gamma }_c^{-1}=0.19$ ps, ${\gamma }_R^{-1}=0.86$ ps, $g_c=3.6\times {10}^{-3}\mathrm{meV}\mu {\mathrm{m}}^2, g_R=17.3g_c$, and $R=0.04{\mathrm{ps}}^{-1}\mu {\mathrm{m}}^2$, and $P_0=272{\mathrm{ps}}^{-1}\mu {\mathrm{m}}^{-2}$, where $\rho =4.5\mu \mathrm{m}$ in (a) and $\rho =5.6\mu \mathrm{m}$ in (b). The dashed curve in (a) shows $N_c$ for the same parameters as those in Fig. 2.