Amplitude-Mode Dynamics of Polariton Condensates

We study the stability of collective amplitude excitations in nonequilibrium polariton condensates. These excitations correspond to renormalized upper polaritons and to the collective amplitude modes of atomic gases and superconductors. They would be present following a quantum quench or could be created directly by resonant excitation. We show that uniform amplitude excitations are unstable to the production of excitations at finite wave vectors, leading to the formation of density-modulated phases. The physical processes causing the instabilities can be understood by analogy to optical parametric oscillators and the atomic Bose supernova.

Figure 1

Quasienergy spectrum of an oscillating polariton condensate. Lines (left axis) correspond to the real part of the quasienergy $\lambda$, and blue crosses to the nonzero imaginary parts (right axis). Modes with nonzero $\mathrm{Im}\lambda$ are unstable. The line colors indicate modes deriving from the phase mode or lower polariton (red solid) or the amplitude mode or upper polariton (black dashed). Inset: Spectrum of a weakly interacting Bose-Einstein condensate with attractive interactions, qualitatively reproducing the small-wave-vector behavior. The parameters are ${\omega }_0=-E=0.4{\Omega }_R$, $\Delta (0)\approx 0.71$, and $\varphi (0)\approx 0.44$, implying a renormalized oscillation frequency $\Omega \approx 0.48{\Omega }_R$.

Figure 2

Photon density from numerical simulations of Eq. (5), with resonant pumping of both upper and lower polaritons (left panel) and the upper polariton alone (right panel). The left panel shows the formation of a short-wavelength density modulation due to the wave-mixing instability, and the right the result of the attractive-gas instability. The pump field profile is a Gaussian at $x=0$ with $\sigma =10\mu \mathrm{m}$; ${\Omega }_R=20\mathrm{meV}$; $E-{\omega }_0=4\mathrm{meV}$.