Excitations in a Nonequilibrium Bose-Einstein Condensate of Exciton Polaritons

We develop a mean-field theory of the dynamics of a nonequilibrium Bose-Einstein condensate of exciton polaritons in a semiconductor microcavity. The spectrum of elementary excitations around the stationary state is analytically studied by means of a generalized Gross-Pitaevskii equation. A diffusive behavior of the Goldstone mode is found in the spatially homogeneous case and new features are predicted for the Josephson effect in a two-well geometry.

Figure 1

Top (a) panel: sketch of the pumping and condensate formation scheme. (b)–(e) panels: real and imaginary part of the excitation spectrum of a homogeneous polariton condensate as a function of the wave vector $k$ (in units of the healing length $\xi =\sqrt{\hslash /m_{\mathrm{LP}}\mu }$). Parameters: ${\gamma }_R/\gamma =5$ and $\alpha =\beta =\gamma /\mu =1$ (b),(c); ${\gamma }_R/\gamma =1$ and $\alpha =0.5$, $\beta =\gamma /\mu ={\gamma }_R/\mu =1$ (d),(e). Rescaled diffusion constant $D{m}_{\mathrm{LP}}/\hslash \approx 5\times {10}^{-4}$; the spectra are indistinguishable from the $D=0$ case. Dashed lines in (b),(d): standard equilibrium Bogoliubov dispersion.

Figure 2

Time evolution of the population $N_1$ after an excitation sequence of duration $\gamma T_{\mathrm{exc}}=1$ (dashed vertical lines). Upper (a) panel: Josephson oscillations ($J=\gamma /2$). Lower (b) panel: overdamped relaxation ($J=\gamma /20$). Other parameters: $U=J/10$, ${\gamma }_R=10\gamma$, $R_0=\gamma$, $P_0/P_{\mathrm{thr}}=2$, $\Delta P/P_0=0.5$.