Polariton Crystallization in Driven Arrays of Lossy Nonlinear Resonators

We investigate the steady states of a lossy array of nonlinear optical resonators that are driven by lasers and interact via mutual photon tunneling. For weak nonlinearities, we find two-mode squeezing of polaritons in modes whose quasimomenta match the relative phases of the laser drives. For strong nonlinearities the spatial polariton density-density correlations indicate that the polaritons crystallize and are predominantly found at a specific distance from each other despite being injected by a coherent light source and damped by the environment.

Figure 1

The steady state in the strong driving regime. Left: $n$ as given by Eq. (3). Right: $m/n$ as given by Eqs. (3, 5).

Figure 2

Density correlations of the steady state for 16 cavities in the weak driving regime with $\Delta =0$, $U/\gamma =10$, $\Omega /\gamma =2$, and $J/\gamma =2$. Top left: $g_r^{(2)}(j,l)$. Top right: $g_m^{(2)}(k,p)$. Bottom left: $g_r^{(2)}(8,l)$ and $g_{\mathrm{TG}}^{(2)}(8,l)$. Bottom right: $g_r^{(2)}(8,l)$ and $g_{\mathrm{TG}}^{(2)}(8,l)$, zoomed in on $8\le j\le 16$.

Figure 3

$g_r^{(2)}(8,j)$ and $\tilde{n}(8)$ as functions of $J/\gamma$. Left: $g_r^{(2)}(8,8)$ and $\tilde{n}(8)$ as functions of $J/\gamma$. Right $g_r^{(2)}(8,j)$ for $8\le j\le 16$ and $J/\gamma =0$, 0.1, 0.5, 1, and 2. $\Delta =0$, $U/\gamma =10$, and $\Omega /\gamma =2$.

Figure 4

Left: $g_r^{(2)}(8,j)$ for $\Delta =0$, $J/\gamma =1$, $\Omega /\gamma =2$, and $U/\gamma =5$. Right: $g_r^{(2)}(8,j)$ for $\Delta =0$, $U/\gamma =10$, $J/\gamma =1$, and $\Omega /\gamma =2$ for $\varphi =0$, $\pi /4$, and $\pi /2$.