Parametric instabilities of microcavity polaritons in a periodic potential

We analyze parametric instabilities of microcavity polartions in the presence of one-dimensional periodic potentials. Instabilities appear in a certain window of pump momentum orientations with respect to the potential so that the momentum projection onto the direction of periodicity is large enough. Ring-shaped domains of parametrically amplified perturbations form a periodic chain in momentum space. Independently from the orientation of the pump momentum, the chain maintains its orientation along the direction of periodicity of the system. In the physical space the amplified structures emerge as stripes modulated along the direction of periodicity and of finite width in the orthogonal direction.

Figure 1

(Color online) Polariton lattices. (a) Average amplitude $\overline{E}=1/L\int dy{E}_0\left(y\right)$ of the lattice solutions as a function of $E_p$: $\theta =0$ and $\theta =\pi /2$. $U_0=W_0=0.75$, ${\delta }_p=-0.3$, $k_p=0.8$, and ${\gamma }_{ph}={\gamma }_e=0.1$. Solution for the no potential case $U_0=W_0=0$ is plotted with thin black lines. Solid/dashed lines correspond to stable/unstable branches. (b) Profiles of the lattice solutions for $E_p=0.3$, $\theta =0$ (blue solid line), $\theta =\pi /2$ (red dashed line), and no potential (thin black line) cases.

Figure 2

(Color online) Bogolyubov spectrum for the case of pump momentum oriented along the homogeneous direction $\left(\theta =0\right)$. (a) Spectrum of the homogeneous system, only two closest to zero branches are shown. Black/red curves correspond to branches with positive/negative Krein signatures. (b) Schematic folding of the spectrum to the first Brillouin zone (boundaries are indicated by dashed vertical lines) in the limit $U_0,W_0\to 0$, $L=3$. [(c) and (d)] Spectra of periodical system with $U_0=W_0=0.25$ and $U_0=W_0=0.75$, respectively. All figures correspond to the pump parameters $E_p=0.3$ and $k_p=0.8$.

Figure 3

(Color online) Bogolyubov spectrum for the case of pump momentum oriented along the direction of periodicity $\left(\theta =\pi /2\right)$. [(a)–(d)] The same as Fig. 2 but for $\theta =\pi /2$. [(e) and (f)] Gain of perturbations corresponding to figures (c) and (d), respectively. Shaded areas indicate range of unstable perturbations $\left(\lambda >0\right)$.

Figure 4

(Color online) Results of 2D stability analysis. Shaded areas indicate domains of unstable perturbations for $E_p=0.3$ and pump orientations (a) $\theta =\pi /2$ and (b) $\theta =3\pi /8$. Other parameters are the same as in Fig. 1.

Figure 5

[(a) and (b)] Maximal perturbation growth rate as functions of the pump tilt for $E_p=0.3$ and pump amplitude for $\theta =\pi /2$, respectively. Other parameters are the same as in Fig. 1.

Figure 6

(Color online) Numerical propagation results: full spectrum (left) and folded to the first Brillouin zone (right) at different times: [(a) and (b)] $t=200$ and [(c) and (d)] $t=500$. Dashed horizontal lines indicate boundaries of the first Brillouin zone. Spectrum of the polariton lattice is shown with solid circles. Parameters of the pump and lattice are the same as in Fig. 4a. Parameters of the seed pulse are: $T=2.5$, $t_0=200$, and $ϵ=0.01$.

Figure 7

(Color online) Field intensity patterns following evolution of unstable perturbations. (a) $\theta =\pi /2$ and $t=500$; (b) $\theta =3\pi /8$ and $t=850$. Other parameters as in Fig. 6. Polariton lattice solution $E_0$ is subtracted in the plots in order to isolate the perturbation field.