Vortex Lattices in Coherently Pumped Polariton Microcavities

We propose a new class of vortex lattices supported by the parametric conversion of polaritons in wide aperture semiconductor microcavities operating in the strong coupling regime and pumped by a coherent beam. We present numerical and analytical results confirming the existence and robustness of the polaritonic vortex lattices in practically relevant settings and discuss their melting scenarios.

Figure 1

Homogeneous solution (HS, $|{\Psi }_0{|}^2$) and pump component of the lattice solution ($|{\Psi }_p{|}^2$) as functions of $E_p$: (a) $k_p=\pi /2$, ${\delta }_p=-0.05$ and (b) $k_p=1.5$, ${\delta }_p=-0.25$. Dashed lines correspond to the unstable parts of HS. ${\Psi }_p$ has been calculated from Eqs. (8, 9, 10) for $k_s=0.07$ (a) and $k_s=0.1$ (b). Open circles show $|{\Psi }_p{|}^2$ extracted from numerical modeling of Eqs. (1, 2). ${\gamma }_{\mathrm{ph}}={\gamma }_e=0.1$.

Figure 2

(a) Shaded circles indicate transverse momenta generating instability of the HS from Fig. 1a at $E_p=0.56$. The structure of the vortex lattice shown in the momentum space is indicated with dots. (b) Instability domain of HS and the momentum space structure typical for conventional hexagons [22].

Figure 3

Amplitudes of the signal (a) and idler (b) components of the vortex lattice solutions. Lines are analytical and circles are numerical results. Dashed and full lines correspond to the parameters as in Figs. 1a, 1b, respectively.

Figure 4

Intensities (a),(c) and phases (b),(d) of the signal component of $E$ corresponding to the vortex lattices generated by a seed pulse. (a),(b) are for the plane wave pump ($E_p=0.56$) and (c),(d) are for the Gaussian pump: $E_p=0.56\times e^{-|\overrightarrow{r}{|}^2/2w^2}$, $w=200$. In all the cases $p=\pi /80$, $t=100$, $T=0.6$, and $t_0=5$, see Eq. (11). The other parameters are as in Fig. 1a.

Figure 5

(a) Exciton densities in the signal $|{\Psi }_s{|}^2$ and idler $|{\Psi }_i{|}^2$ components of the vortex lattice. Parameters are the same as in Fig. 4a. (b) Solid lines show $k_s=k_{\mathrm{opt}}$ as a function of $E_p$ (from bottom to top: $p=\pi /80$, $\pi /40$, and $\pi /20$, respectively). Dashed lines indicate domain boundaries $p=0$ and $p=p_{\max }$, see text for details.

Figure 6

The same as Fig. 4a, but with the seed momentum $k_s$ shifted on either side from its optimal value $k_{\mathrm{opt}}=0.07$ and thus leading to the two different lattice melting scenarios: (a) $k_s=0$, (b) $k_s=0.14$.