Structure formation, superfluid velocity patterns, and induced vortex-antivortex oscillations and rotations in microcavity polaritons

We use the nonequilibrium Gross-Pitaevskii equation in the form recently derived [H. Haug et al., Phys. Rev. B 89, 155302 (2014)] to study spontaneous pattern formation and the connected superfluid current patterns. Using a nonresonant excitation beam with ring structure we get depending on the details of the structure and of the pump power, different spatial patterns of the condensate density. The corresponding phase profiles allow to identify, e.g., vortex-antivortex pairs, but beyond that, yield an image of the superfluid flow patterns linked with the structured condensate density. The fast superfluid flow driven by the spatially changing phase with velocities of the order of several $m\mu /ps$ is found to be often supersonic. In order to test dynamically the stability of the spontaneously formed flow patterns under external perturbations, we apply an additional resonant Laguerre-Gauss beam with angular momentum. This beam causes complex response of the phase patterns. This response is shown to be basically an oscillation or rotation of the vortex-antivortex pair depending on the strength of the extra beam. The rotation is induced via a ring of vortices induced by the Laguerre-Gaus beam. The main result of these studies is the extraordinary stability of the vortex-antivortex pair even under strongly perturbing external fields.

Figure 1

Superfluid flow patterns (colored arrows), phase patterns (black and white shaded), and condensate density contours. The phase changes from 0 (black) to $2\pi$ (white). At the endpoints of the sickle-shaped black area on the right-hand side a vortex V and on the left-hand side an antivortex AV are visible.

Figure 2

Superfluid flow patterns (colored arrows), phase patterns (black and white shaded), and condensate density contours. The phase changes from 0 (black) to $2\pi$ (white). At the endpoints of the black phase singularity lines, again vortices and antivortices, respectively, occur.

Figure 3

Phase (surface), amplitude (contour plot), and velocity (quiver plot) of the condensate for the same incoherent excitation as in Fig. 1 but now with an additional resonant Laguerre-Gauss beam with $p_{\mathrm{LG}}/p_0=0.026$. The position of the vortex (antivortex) is marked by a red (green) circle, an arrow marks the sense of rotation. The time between the plots increases by 1 $\mathrm{ps}$ from left to right and top to bottom, respectively.

Figure 4

Periodic motion of the vortex (V, red) and antivortex (AV, green) in Fig. 3

Figure 5

Time dependence of the angle of the oscillating vortex in Fig. 4. The green line is a fit with a sine function.

Figure 6

Phase (surface), amplitude (contour plot), and velocity (quiver plot) of the condensate for the same incoherent excitation as in Fig. 1 but now with a 5 times larger additional resonant Laguerre-Gauss beam which yields the intensity ratio $p_{\mathrm{LG}}/p_0=0.65$. The position of the vortex (antivortex) is marked by a red (green) circle. The time between the plots increases by 1 $\mathrm{ps}$ from left to right and top to bottom, respectively.

Figure 7

Rotational motion of the vortex (V, red) and antivortex (AV, green) pair in Fig. 6.

Figure 8

Linear increase of the rotational angle of the V-AV pair in Fig. 7 until it reaches the boundaries of the angle at $\pm \pi$.