Hydrodynamic nucleation of vortices and solitons in a resonantly excited polariton superfluid

We present a theoretical study of the hydrodynamic properties of exciton-polaritons in a semiconductor microcavity under a resonant laser excitation. The effect of a spatially extended defect on the superfluid flow is investigated as a function of the flow speed. The processes that are responsible for the nucleation of vortices and solitons in the wake of the defect are characterized, as well as the regimes where the superfluid flow remains unperturbed. Specific features due to the nonequilibrium nature of the polariton fluid are pointed out. For the present case of a resonant polariton excitation, an effective way to create, trap, and control arrays of vortices is proposed.

Figure 1

Normalized real-space photonic density for different values of the pump detuning ${\delta }_p=\hslash {\omega }_p-\hslash {\omega }_{\mathrm{LP}}({\mathbf{k}}_p)$ and reservoir pump amplitude $F_p^{\max }$ that correspond to decreasing polariton densities. More specifically, ${\delta }_p=0.5\hspace{0.5em}\mathrm{meV}$, $F_p^{\max }/{\gamma }_C=5.{10}^5\hspace{0.5em}\mathrm{\mu }{\mathrm{m}}^{-1}$ (a); ${\delta }_p=0.5\hspace{0.5em}\mathrm{meV}$, $F_p^{\max }/{\gamma }_C=250\hspace{0.5em}\mathrm{\mu }{\mathrm{m}}^{-1}$ (b); ${\delta }_p=0.4\hspace{0.5em}\mathrm{meV}$, $F_p^{\max }/{\gamma }_C=250\hspace{0.5em}\mathrm{\mu }{\mathrm{m}}^{-1}$ (c); ${\delta }_p=0.3\hspace{0.5em}\mathrm{meV}$, $F_p^{\max }/{\gamma }_C=250\hspace{0.5em}\mathrm{\mu }{\mathrm{m}}^{-1}$ (d) (same parameters as in Fig. 3); ${\delta }_p=0.1\hspace{0.5em}\mathrm{meV}$, $F_p^{\max }/{\gamma }_C=20\hspace{0.5em}\mathrm{\mu }{\mathrm{m}}^{-1}$ (e). The pump wave vector has a magnitude $k_p=1\hspace{0.5em}\mu {\text{m}}^{-1}$ and is directed along the negative $y$ axis. System parameters: $\hslash {\omega }_X(k=0)=1479\hspace{0.5em}\mathrm{meV}$, $\hslash {\omega }_C(k=0)=1483\hspace{0.5em}\mathrm{meV}$, $\hslash {\gamma }_{X,C}=0.02\hspace{0.5em}\mathrm{meV}$, and $\hslash {\Omega }_R=2.65\hspace{0.5em}\mathrm{meV}$. Cavity photon mass $m_C=40.{10}^{-6}\hspace{0.5em}{m}_e$. Exciton mass $m_X$ is taken as infinite. The defect potential is a purely photonic one of depth $V_C=20\hspace{0.5em}\mathrm{meV}$. Numerical simulations were performed on a $256\times 128$ grid. Note that the color scale is slightly saturated. The density patterns are stationary in time in all panels except (b).

Figure 2

Spatial profile of the pump intensity $|F_p(\mathbf{x})|{}^2$ in panel (a). Red dashed circles indicate the position and size of the defect. The smaller one corresponds to Fig. 1, while the bigger one to Fig. 3. In panel (b) a vertical section of the pump profile is shown. Red regions indicate defect positions.

Figure 3

Normalized real-space photonic density for a value of the pump detuning ${\delta }_p=\hslash {\omega }_p-\hslash {\omega }_{\mathrm{LP}}({\mathbf{k}}_p)=0.3\hspace{0.5em}\mathrm{meV}$ and pump amplitude of $F_p^{\max }/{\gamma }_C=250\hspace{0.5em}\mathrm{\mu }{\mathrm{m}}^{-1}$. Except for the defect size the parameters are the same as in Fig. 2.

Figure 4

(a) Real-space pump profile; the red dashed circle indicates the position and size of the defect. The external region is pumped at a value of $F_p^{\mathrm{out}}$ within the vortex nucleation regime of Fig. 1c, $v/c_{\mathit{sR}}=0.45>0.43$. The inner dark area is pumped at a lower intensity $F_p^{\mathrm{in}}$. (b), (c) Time average of the normalized photonic density for two values of the inside pump amplitude $F_p^{\mathrm{in}}/F_p^{\mathrm{out}}=0.1$ (b) and $0.32$ (c). Calculations were performed on a $128\times 128$ grid. Same system parameters as in Fig. 1. The intensity profile across the edge is Gaussian with a thickness $\sigma =1\hspace{0.5em}\mathrm{\mu }\mathrm{m}$.

Figure 5

Plot of the number of stationary vortex-antivortex pairs as a function of the pump amplitude inside the triangle. Same pump profile and system as in Fig. 4.