Bénard–von Kármán vortex street in an exciton-polariton superfluid

The dynamics of an exciton-polariton superfluid resonantly injected into a semiconductor microcavity are investigated numerically. The results reveal that a Bénard–von Kármán vortex street is generated in the wake behind an obstacle potential, in addition to the generation of quantized vortex dipoles and dark solitons. The vortex street can be observed in the presence of a disorder potential in a realistic sample, and it can be observed even in time-integrated measurements.

Figure 1

Density $|{\psi }_{\mathrm{C}}{|}^2$ and phase $\arg \left({\psi }_{\mathrm{C}}\right)$ profiles of the photon wave function at $t=400$ ps. (a) $k_{\mathrm{p}}=0.25$ $\mu {\mathrm{m}}^{-1}$ without an obstacle potential, (b) $k_{\mathrm{p}}=0.25$ $\mu {\mathrm{m}}^{-1}$ and $d=2$ $\mu \mathrm{m}$, (c) $k_{\mathrm{p}}=0.25$ $\mu {\mathrm{m}}^{-1}$ and $d=3$ $\mu \mathrm{m}$, (d) $k_{\mathrm{p}}=0.25$ $\mu {\mathrm{m}}^{-1}$ and $d=5$ $\mu \mathrm{m}$, and (e) $k_{\mathrm{p}}=0.5$ $\mu {\mathrm{m}}^{-1}$ and $d=3$ $\mu \mathrm{m}$. The solid circles in (b) and (c), respectively, indicate vortex dipoles and co-rotating twin vortices, which are released from the obstacle potential. The dashed circles indicate the locations of the obstacle potential and the filled blue circles and red squares indicate clockwise and counterclockwise vortices, respectively. The detuning is $\hslash \delta =0.7$ meV and the lifetime is ${\gamma }^{-1}=30$ ps. The field of view of each panel is $150\times 110$ $\mu \mathrm{m}$ and the origin is located at the center of the left edge of the panel. See supplemental material[21] for movies of the dynamics in (b)–(e).

Figure 2

(a) Density profile $|{\psi }_{\mathrm{C}}{|}^2$ at $t=400$ ps for the pumping function in Eq. (5) with $\alpha =0.2$. The position of the obstacle potential is $x_0=90$ $\mu \mathrm{m}$ and the lifetime is ${\gamma }^{-1}=20$ ps. (b) Density profile $|{\psi }_{\mathrm{C}}{|}^2$ at $t=500$ ps for rectangular pumping regions, where the lifetime is ${\gamma }^{-1}=30$ ps. The other conditions are the same as those in Fig. 1c. The circles indicate corotating twin vortices, as in Fig. 1c. The field of view of each panel is $250\times 110$ $\mu \mathrm{m}$ in (a) and $220\times 130$ $\mu \mathrm{m}$ in (b). See the supplemental material[21] for movies of the dynamics.

Figure 3

(a) Density profile $|{\psi }_{\mathrm{C}}{|}^2$ in the presence of the disorder potential shown in (b). The disorder potential is generated by setting random numbers on each site and cutting off short-wavelength Fourier components. The parameters are the same as those in Fig. 1c. The black and white circles indicate clockwise and counterclockwise vortices, respectively. See the supplemental material[21] for a movie of the dynamics.

Figure 4

(a) Time-integrated density of the photon wave function $I\left(\mathit{r}\right)=\int |{\psi }_{\mathrm{C}}{(\mathit{r},t)|}^{2}dt$ for the dynamics in Fig. 1b. (b) $I$ for the dynamics in Fig. 1c. (c) Density ${\left|G\right|}^2$ (left) and phase $\arg \left(G\right)$ (right) of the first-order coherence $G\left(\mathit{r}\right)=\int {\psi }_{\mathrm{C}}^{*}({\mathit{r}}_{\mathrm{ref}},t){\psi }_{\mathrm{C}}(\mathit{r},t)dt$ for the dynamics in Fig. 1b. (d) ${\left|G\right|}^2$ (left) and $\arg \left(G\right)$ (right) for the dynamics in Fig. 1c. The time integrations in $I$ and $G$ are taken between $t=400$ and 1400 ps (the patterns are independent of the upper integration limit if the time integration is sufficiently long). The position of the reference is $(x_{\mathrm{ref}},y_{\mathrm{ref}})=(46,4)$, which is marked by the crosses. The field of view of each panel is $150\times 110$ $\mu \mathrm{m},$ and the origin is located at the center of the left edge of the panel.