Pseudospin dynamics of exciton-polariton patterns in a coherently driven semiconductor microcavity

The influence of the exciton spin on the formation and stability properties of periodic cavity polariton patterns is studied in a semiconductor microcavity operating in the strong-coupling regime. A linearly polarized optical beam excites polaritons formed by excitons with different spin orientations and left- and right-circularly polarized photons. The perturbation analysis of homogeneous solutions reveals a competition between these two spin states. The outcome of this competition is determined by the sign of the cross-phase modulation parameter. In particular, it is shown that linearly polarized patterns are preferred, if this parameter is positive. Otherwise, a spontaneous symmetry-breaking instability leads to the formation of transverse patterns with a spatial polarization asymmetry. In the regime of bistable homogeneous solutions we observe the spontaneous formation of domains framed by one-dimensional dark half solitons.

Figure 1

(a)–(c) Analysis of homogeneous solutions (HSs) for $\alpha =-0.1$ with $\Delta =-0.45$: (a) HS branches $|{\Psi }^{\pm }|$ over the pump power $E_p$: symmetric HSs (bold black), asymmetric HSs (thin gray), and MI range (blue chain line). Continuous lines denote stable solutions, whereas dashed lines stand for unstable HSs; (b) growth rate $\mathrm{Re}\lambda (k,|{\Psi }_0\left|\right)$, boundary of instability range $\mathrm{Re}\lambda (k,|{\Psi }_0\left|\right)=0$ (blue line) and stable domain $\mathrm{Re}\lambda (k,|{\Psi }_0\left|\right)<0$ (white area); (c) maximal growth rates $\mathrm{Re}{\lambda }_{\mathrm{s}}(k,|{\Psi }_0\left|\right)$ and $\mathrm{Re}{\lambda }_{\mathrm{as}}(k,|{\Psi }_0\left|\right)$ for a given order parameter $|{\Psi }_0|$ of symmetric (thin red) and antisymmetric (bold blue) perturbations, respectively. The dashed white line in (b) shows the position of $\mathrm{Re}{\lambda }_{\max }(k,|{\Psi }_0\left|\right)$ in the $k\text{−}|{\Psi }_0|$ plane. Horizontal dashed lines denote the onset and the cessation of the modulational instability. (d)–(f) show the respective figures for $\alpha =-0.1$ with $\Delta =-0.1$.

Figure 2

Results of the perturbation analysis of homogeneous solutions: (a) critical intensity for symmetric (thin red) and antisymmetric perturbations (bold blue) as a function of $\alpha$ for ${\Delta }_0=-0.45$; (b) intensity $I_{0,\mathrm{crit},\mathrm{as}}^{-}$ at the lower bifurcation point and $I_{0,\mathrm{crit},\mathrm{as}}^{+}$ at the upper bifurcation point as a function of the detuning $\Delta$ for $\alpha =-0.1$ (bold blue lines); within the region encircled by the thin solid black curve also perturbations with $k=0$ are unstable; the dashed black lines frame the range between the turning points of the bistability loop for symmetric HSs.

Figure 3

(a) Two-dimensional profile $|E^{+}(x,y)|=|E^{-}(x,y)|$ of a hexagonal pattern for $\alpha >0$; (b) field intensity $I_E(x,y)=|E^{+}{(x,y)|}^2+{\left|E^{-}(x,y)\right|}^2$. Parameters are $\alpha =0.1,\Delta =-0.1$, and $E_p=0.35$.

Figure 4

Symmetry-breaking mechanism leading to the formation of vectorial hexagonal patterns for $\alpha <0$: (a) total intensity $I_E(x,y)=|E^{+}{(x,y)|}^2+{\left|E^{-}(x,y)\right|}^2$ of the photon field; (b) polarization degree ${\rho }_E(x,y)$. The white (black) hexagon indicates the maxima of $|E^{-}(x,y)| \left(|E^{+}(x,y)|\right)$. Parameters are $\alpha =-0.1,\Delta =-0.1$, and $E_p=0.35$.

Figure 5

Symmetry-breaking mechanism leading to the formation of modulated stripes for $\alpha <0$: (a) total intensity $I_E(x,y)=|E^{+}{(x,y)|}^2+{\left|E^{-}(x,y)\right|}^2$ of the photon field; (b) polarization degree ${\rho }_E(x,y)$. The white (black) hexagon indicates the maxima of $|E^{-}(x,y)| \left(|E^{+}(x,y)|\right)$. Parameters are $\alpha =-0.1,\Delta =-0.45$, and $E_p=0.2816$.

Figure 6

Formation of various vectorial patterns for $\alpha =-0.1$ and $\Delta =-0.1$: (a)–(c) labyrinthine stripe patterns for $E_p=1.2$: (a) $|E^{+}(x,y)|$; (b) total intensity $I_E(x,y)=|E^{+}{(x,y)|}^2+{\left|E^{-}(x,y)\right|}^2$ of the photon field; (c) polarization degree ${\rho }_E(x,y)$. (d)–(f) honeycomb patterns add to modulated stripes for $E_p=1.4$: (d) honeycomb pattern $|E^{+}(x,y)|$; (e) stripelike total intensity $I_E(x,y)=|E^{+}{(x,y)|}^2+{\left|E^{-}(x,y)\right|}^2$ of the photon field; (f) hexagonally patterned polarization degree ${\rho }_E(x,y)$.

Figure 7

Formation of a domain structure separated by multiple one-dimensional dark half solitons for $\Delta =-0.7,\alpha =-0.1$, and $E_p=0.19$: (a) three spatially distinct domains (enumerated by 1 to 3) are separated by slowly moving dark half solitons (direction of motion indicated by red arrows), (b) region 3 has vanished after the melting of the surrounding half solitons, (c) region 1 has vanished, resulting in two crossed stationary dark half solitons.

Figure 8

(a) Dark 1D half soliton: slice along dashed line in Fig. 7; (b) multistability diagram: symmetric HSs (black), asymmetric HSs (gray), and modulationally unstable HSs (red), stable (unstable) solutions are denoted with bold (dashed) lines, the blue curve denotes the branch $\min |{\Psi }^{-}|\left(E_p\right)$ of the half soliton in (a). Parameters are $\alpha =-0.1$ and $\Delta =-0.7$.