Theory of relaxation oscillations in exciton-polariton condensates

We provide an analytical and numerical description of relaxation oscillations in the nonresonantly pumped polariton condensate. Presented considerations are based on coupled rate equations that are derived from the open dissipative Gross-Pitaevskii model. The evolution of the condensate density can be explained qualitatively by studying the topology of the trajectory in phase space. We use a fixed points analysis for the classification of the different regimes of condensate dynamics, including fast stabilization, slow oscillations, and ultrashort pulse emission. We obtain an analytical condition for the occurrence of relaxation oscillations. Continuous and pulsed condensate excitations are considered and we demonstrate that, in the latter case, the existence of the second reservoir is necessary for the emergence of oscillations. We show that relaxation oscillations should be expected to occur in systems with relatively short polariton lifetimes.

Figure 1

Time evolution of polariton population $n_C\left(t\right)$ and incoherent reservoirs $n_I\left(t\right)$ and $n_R\left(t\right)$ generated by continuous wave excitation. Blue lines correspond to the analytical solution describing the evolution of reservoirs under the assumption of negligible polariton population. Red line presents condensate evolution obtained by nonlinear oscillator approximation, Eqs. (4). Grey points correspond to numerical solution obtained from the full open-dissipative Gross-Pitaevskii Eqs. (1, 2, 3). Condensate parameters are given in Table 1.

Figure 2

Evolution of polariton condensate density under ultrashort pulsed excitation. The above three panels show, respectively, (a) spiked pulse emission, (b) decaying harmoniclike oscillations, and (c) smooth stabilization. Simulation parameters are given in Table 1.

Figure 3

(a) Phase space of trajectory in the pulsed excitation case of Fig. 2 (red line). Purple lines correspond to the evolution projected, respectively, to $(n_I,n_C)$, $(n_I,n_R)$, and $(n_R,n_C)$ subspaces. The spiral evolution corresponds to the oscillatory behavior of the system, with a gradually decaying inactive reservoir density $n_I$. It follows the evolution of the stationary state at a particular value of $n_I$ (orange line). (b) Cross-section of the phase space corresponding to the blue surface in panel (a). The arrows indicate the gradient of the evolution in the ($n_C,n_R$) subspace at the time when the value of $n_I$ is equal to $1000\mu {\text{m}}^{-2}$. The orange dot is the center of the spiral, which is a stationary point within this subspace. (c) Same as (b) but in the case of decaying harmoniclike oscillations of Fig. 2.

Figure 4

Schematic presentation of the types of fixed points. The figure axes correspond to the trace $\tau$ and the determinant $\mathrm{\Delta }$ of the Jacobian A. The bold black line corresponds to the analytical condition for relaxation oscillations.

Figure 5

Bottom: Diagram showing region in parameter space which exhibits oscillations in the case of continuous wave excitation, according to the analytical formula, Eq. (25). (a)–(d) Examples of evolutions of condensate density corresponding to the points marked in the diagram.

Figure 6

(a) Diagram similar to Fig. 5, but in the case of pulsed excitation. Here, instead of power $P/P_{\text{th}}$, we plot the inactive reservoir density, which decays slowly during the evolution. The arrows indicate the paths in parameter space corresponding to the panels in Fig. 2. While all evolutions traverse the oscillatory region, in the case (c) there are no visible oscillations because the system “sticks” to the stationary state while passing through it, as shown in panel (b).