Spontaneous formation of vortices and gray solitons in a spinor polariton fluid under coherent driving

It is generally accepted that quantized vortices formed in coherent bosonic fluids are “excitations” and as such do not arise in a one-mode condensate at zero temperature. To excite them, one needs either inhomogeneities (impurities, rotation, etc.) or essentially finite fluctuations. Here, we predict a perfectly spontaneous formation of vortices even at zero temperature, which takes place in a homogeneous cavity-polariton system under one-mode optical excitation at normal incidence. In spite of the absence of equilibrium and U(1) invariance, this system shows a counterpart of the Berezinskii-Kosterlitz-Thouless crossover between single vortices and coupled vortex-antivortex states ranging from small dipoles to rectilinear filaments with long-range ordering.

Figure 1

One-mode response functions [solutions of Eq. (2)] at $\gamma =5\mu \mathrm{eV}$ and $g=D=1$ meV, so that $f_2^2/f_1^2=g/2\gamma =100$. Solutions shown by dotted lines are unstable even in the one-mode limit with $k=0$. (b) shows a magnified interval of (a) at small $f$ which will be mainly considered in this paper; arrows and ellipses schematically indicate polarization states.

Figure 2

Solution of Eq. (1) at $\gamma =1\mu \mathrm{eV}$, $g=D=1$ meV, and $f^2/f_1^2=11$ in the one-dimensional case; the initial state is homogeneous (zero). The boundary between the ${\mathrm{\Omega }}_{\pm }$ states behaves as a gray soliton; it runs with a constant speed and gets reflected from the 2-meV high potential walls at $x=\pm 200\mu \mathrm{m}$.

Figure 3

Intensity $I$ and polarization degrees (normalized Stokes parameters) $S_{1,2,3}$ for a steady solution of Eq. (1) formed in about 1 ns at $f^2/f_1^2\approx 6.8$. Parameters $\gamma$, $g$, and $D$ correspond to Fig. 1. The initial conditions were biased to the ${\mathrm{\Omega }}_{+}$ and ${\mathrm{\Omega }}_{-}$ states at $\left|\mathbf{r}\right|\le 20\mu \mathrm{m}$ and $\left|\mathbf{r}\right|>20\mu \mathrm{m}$, respectively, $r=0$ being the grid center. To ensure zero boundary conditions, $\gamma$ is increased up to 4 meV at $\left|\mathbf{r}\right|=125\mu \mathrm{m}$. The insets show a magnified fragment with two vortices of opposite topological charge.

Figure 4

Snapshots of unsteady solutions of Eq. (1). Parameters $\gamma$, $g$, and $D$ correspond to Fig. 1. The value of $f^2/f_1^2$ is 6.8 for (a) and 13 for (b). The initial conditions are homogeneous (zero) in both cases. In (a), a very high potential wall is set at $\left|\mathbf{r}\right|=75\mu \mathrm{m}$. In (b), $\gamma$ is increased from 5 to $200\mu \mathrm{eV}$ at the same $\left|\mathbf{r}\right|$, whereas the boundary conditions are periodic. The obtained pattern steadily rotates clockwise, making a complete turn in 2.6 ns. Supplemental video files explicitly show the evolution of (a) and (b).