Figure 1

(a) Dispersion relation of the upper and lower polaritons. (b) Stationary state densities of Eq. (1) with uniform pumping. The parameters are $m_C=2\times {10}^{-5}{m}_{\mathrm{e}}$ with $m_{\mathrm{e}}$ being the electron mass, $\hslash {\Omega }_{\mathrm{R}}=5\mathrm{meV}$, $g=0.01\mathrm{meV}\mu {\mathrm{m}}^2$ [15, 33], and $\hslash \delta =1.53\mathrm{meV}$. We assume $E_0^X=E_0^C$ and $\gamma \equiv {\gamma }_X={\gamma }_C=(10\mathrm{ps}{)}^{-1}$ [23]. The energy $\hslash {\omega }_{\mathrm{p}}$ and peak intensity $F_0$ of the pump used in Figs. 2, 3 are marked in (a) and (b).Reuse & Permissions

Figure 2

(a)–(i) Snapshots of the density profile $|{\psi }_C{|}^2$ of photons. The pump parameters are $F_0=8.97\mathrm{meV}$ and $d=50\mu \mathrm{m}$ and the other parameters are the same as those in Fig. 1. The color scale in each image is normalized by the peak density, where the unit of density is $\mu {\mathrm{m}}^{-2}$. (j)–(l) Fourier-transformed profiles $|\int {\psi }_C{e}^{-i\mathit{k}·\mathit{r}}d\mathit{r}{|}^2$. (m) Time evolution of $N_C=\int |{\psi }_C{|}^{2}d\mathit{r}$ (red solid line) and $N_X=\int |{\psi }_X{|}^{2}d\mathit{r}$ (green dashed line). The vertical dotted lines show $t=400$, 550, and 610 ps. See Supplemental Material [30].Reuse & Permissions

Figure 3

(a)–(c) Snapshots of the density profile $|{\psi }_C{|}^2$ for $d=60\mu \mathrm{m}$. The insets show Fourier-transformed profiles, where the field of view is $4\times 4\mu {\mathrm{m}}^{-1}$. (d) Spatiotemporal image of the density on the dashed line in (a). (e) Time evolution of the norms $N_C$ (red solid line) and $N_X$ (green dashed line). In (a)–(e), all the parameters, except $d$, are the same as those in Fig. 2. See Supplemental Material [30].Reuse & Permissions

Figure 4

(a)–(c) Snapshots of the density $|{\psi }_C{|}^2$ and phase $\arg {\psi }_C$ profiles for Laguerre-Gaussian pumping with $F_0=21\mathrm{meV}$ and $d=50\mu \mathrm{m}$. (d) Time evolution of the normalized angular momenta $L_j=-i\int {\psi }_j^{*}{\partial }_{\theta }{\psi }_{j}d\mathit{r}/\int |{\psi }_j{|}^{2}d\mathit{r}$ of photons ($j=C$, red solid line) and excitons ($j=X$, green dashed line). The insets show Fourier-transformed profiles. Except for the pump function, the parameters are the same as those in Fig. 2. Supplemental Material [30].Reuse & Permissions