Tight-binding model for exciton-polariton condensates in external potentials

We propose a mean-field model to describe exciton-polariton condensates in deep periodic external potentials. We derive a set of coupled discrete equations for both condensed and uncondensed components, with interaction and tunneling coefficients obtained within the tight-binding approximation. We use the model to explain the intriguing phenomenon of increasing density modulation in a one-dimensional valley with disorder, reported by F. Manni et al. [Phys. Rev. Lett. 106, 176401 (2011)].

Figure 1

(a) Stationary polariton density ${\left|\psi \right|}^2$ as predicted by the discrete model (dashed line) and the continuous model (solid line). The open circles show the amplitude of the discrete function $|c_j{|}^2$. (b) Phase difference between adjacent cells for discrete (triangles) and continuous (circles) models. For the continuous model the phase was evaluated in the cell center. Here $P=1.35P_{\mathrm{th}}$. All calculations were performed for $m^{*}={10}^{-4}{m}_e$, $g_{\mathrm{C}}=8.08\times {10}^{-4}$ $\mu$m ps${}^{-1}$, $g_{\mathrm{R}}=6.73\times {10}^{-5}$ $\mu$m ps${}^{-1}$, $R_{\mathrm{sc}}=6.73\times {10}^{-2}$ $\mu$m ps${}^{-1}$, ${\tau }_{\mathrm{R}}=6.58$ ps, ${\gamma }_{\mathrm{A}}=15.2$ ps${}^{-1}$, ${\gamma }_{\mathrm{I}}=1.97\times {10}^{-3}$ ps${}^{-1}$, ${\gamma }_c=0.99$ ps${}^{-1}$.

Figure 2

Same as in Fig. 1, but for $P=1.1P_{\mathrm{th}}$.

Figure 3

The contrast of the polariton density profile $\psi$ calculated from (12). The red line with points is the experimental result of [14], and the solid blue line is the result of the continuous model. The dotted green line corresponds to the discrete model, where we replaced the integral by a sum in (12).

Figure 4

Disordered potential used for calculating polariton densities in Fig. 5.

Figure 5

Normalized polariton density in the disordered potential at different pumping intensities.

Figure 6

Phase difference between adjacent cells vs the pumping intensity $p$, from Eq. (16).