Towards spin turbulence of light: Spontaneous disorder and chaos in cavity-polariton systems

Recent advances in nanophotonics have brought about coherent light sources with chaotic circular polarization; a low-dimensional chaotic evolution of optical spin was evidenced in laser diodes. Here we propose a mechanism that gives rise to light with a spatiotemporal spin chaos resembling turbulent states in hydrodynamics. The spin-chaotic radiation is emitted by exciton polaritons under resonant optical pumping in arbitrarily sized planar microcavities, including, as a limiting case, pointlike systems with only three degrees of freedom. The underlying mechanism originates in the interplay between spin symmetry breakdown and scattering into pairs of Bogolyubov excitations. As a practical matter, it opens up the way for spin modulation of light on the scale of picoseconds and micrometers.

Figure 1

Dynamics of a pointlike polariton system. Numerical solutions of Eq. (1) for $2f^2=5\times {10}^{-4}$ [(a) and (c)] and $1.25\times {10}^{-3}$ [(b) and (d)]. [(a) and (b)] Phase trajectories over 2000 ps. The large circles indicate the unstable one-mode condensate states (${\mathrm{\Pi }}_x$ and ${\mathrm{\Sigma }}_{\pm }$). The small circles show the respective lower (L) and upper (U) scattered modes. [(c) and (d)] Explicit time dependences of $|{\psi }_{\pm }^2|$ over 500 ps.

Figure 2

Chaotic attractor typical of a pointlike system. (a) Phase trajectory over a 6000 ps interval in the space of the Stokes polarization parameters; the color represents linear polarization ${\rho }_l$. (b) Explicit time dependences of $|{\psi }_{\pm }^2|$ over 1000 ps.

Figure 3

Self-organized field in a circular cavity. (a) Full intensity $|{\psi }_{+}^2|+|{\psi }_{-}^2|$, in arbitrary units, as a function of in-plane coordinates; the color represents circular polarization ${\rho }_c$. (b) Explicit spatial dependencies of $|{\psi }_{\pm }^2|$ along the cross section $AB$.

Figure 4

Chaotic field evolution. Instantaneous emission snapshots drawn at 20 ps intervals. The color represents circular polariz-ation ${\rho }_c$.

Figure 5

(a) One-mode solutions of Eq. (B1), condensate intensity vs. pump intensity. The range of small pump intensities is shown in the inset. Unstable branches are indicated by dotted lines. The interval with no stable solutions is shadowed. (b) Corresponding degrees of circular polarization.

Figure 6

Bogolyubov excitations in the ${\mathrm{\Pi }}_x$ (a) and ${\mathrm{\Sigma }}_{+}$ (b) one-mode states at $2f^2=1.25\times {10}^{-3}$ [the pump intensity is marked by the vertical arrow at the top of Fig. 5]. Stars indicate the condensate mode at $k=0, E=E_p$. Lines show the reference (unsplit) dispersion law $E_{\mathrm{LP}}\left(k\right)$ (solid line) and its “idler” counterpart $2E_p-E_{\mathrm{LP}}(-k)$ (dashed line). Strips and ellipses represent polarization states. Full ellipses in (b) indicate unstable modes.

Figure 7

Periodic solution for $R=10\mu \mathrm{m}$. (a)–(d) Instantaneous spin and intensity patterns; (e) the pattern averaged over the period. (f) Time dependencies of ${\rho }_c$ in the $\mathbf{r}=0$ point of the real space (solid line) and $\mathbf{k}=0$ point of the momentum space (dashed line). The time moments represented in panels (a)–(d) are marked as A–D.