Coherent dynamics and parametric instabilities of microcavity polaritons in double-well systems

We investigate the physics of coherent polaritons in a double-well configuration under a resonant pumping. For a continuous-wave pump, bistability and self-pulsing regimes are identified as a function of the pump energy and intensity. The response to an additional probe pulse is characterized in the different cases and related to the Bogoliubov modes around the stationary state. Under a pulsed pump, a crossover from Josephson-like oscillations to self-trapping is predicted for increasing pump intensity. The accurateness of the effective two-mode model is assessed by comparing its predictions to a full solution of the nonequilibrium Gross–Pitaevskii equation.

Figure 1

(Color online) Energy-intensity $(\hslash \omega -n_2)$ diagram. The color scale corresponds to increasing values of $\mid F^s\mid$ in logarithmic scale and the thin isolines are geometrically spaced contour lines. The thick line separates the stability (S) regions from instability ones: depending on their character, these are marked by either 1M (one-mode instability) or P (parametric instability). The dash-dot line separates a one-mode instability region from a parametric instability one. The dots on the vertical $\hslash \omega =0.45\mathrm{meV}$ thin line correspond to the pump parameters used later on in Figs. 7, 8. The system parameters are inspired from the symmetric double-box polariton traps discussed in Sec. 6, namely, $J=0.5\mathrm{meV}$, ${\gamma }_1={\gamma }_2=\gamma =0.2\mathrm{meV}$, $g=1.1\times {10}^{-3}\mathrm{meV}$, and ${\omega }_1={\omega }_2$.

Figure 2

(Color online) Scheme of the effective energy levels $E_{\pm }$ of a two-mode model under a continuous-wave pump. The vertical dashed arrows indicate the nonlinear blueshift. The horizontal solid arrows indicates the pump energy $\hslash \omega$ in the different regimes discussed in the text: (a) optical limiter, (b) one-mode instability, (c) parametric instability, and (d) high-energy pumping.

Figure 3

(Color online) Intensities (a) $n_1$ and (b) $n_2$ as a function of the pump amplitude. Pump energy $\hslash \omega =0.45\mathrm{meV}$. The arrows highlight the hysteresis cycle due to optical bistability. Same system parameters as in Fig. 1.

Figure 4

(Color online) Intensity $n_1$ as a function of $n_2$ for a high pump energy $\hslash \omega =1.5\mathrm{meV}$ (dashed line). The solid line corresponds to $n_2$ and is a guide for the eyes in order to identify the regions where $n_1⪢n_2$, $n_1\approx n_2$, or $n_2⪢n_1$. The light and dark shaded regions indicate one-mode and parametric instabilities, respectively. Same system parameters as in Fig. 1.

Figure 5

Logarithmic gray-scale plot of the ratio $n_2∕n_1$ as a function of the pump energy and of the value $n_2$. Same parameters as in Fig. 1. The thick line separates the stability regions from the one-mode instability and the parametric instability ones as in Fig. 1.

Figure 6

(Color online) (a) Real and (b) imaginary parts of the linearized spectrum around the steady state solution as a function of $n_2$. Stable regions correspond to the imaginary part of the spectrum being negative. Pump frequency $\hslash \omega =0.45\mathrm{meV}$. The vertical lines indicate the $n_2$ values used in panels (a)–(d) of Figs. 7, 8. Same system parameters as in Fig. 1.

Figure 7

(Color online) Time evolution of the $n_1\left(t\right)$ (thin line) and $n_2\left(t\right)$ (thick line) intensities for growing pump amplitudes (a) $F_{\mathit{max}}=1\mathrm{meV}$, (b) $5\mathrm{meV}$, (c) $8\mathrm{meV}$, and (d) $20\mathrm{meV}$. Pump energy $\hslash \omega =0.45\mathrm{meV}$. Pump switch-on time $\tau =100\mathrm{ps}$. Probe amplitude $f_g^0={10}^{-2}{F}_{\mathit{max}}⪡F_{\mathit{max}}$, probe duration ${\tau }_g=0.3\mathrm{ps}⪡\pi \hslash ∕J$, and probe delay $t_0=400\mathrm{ps}⪢\tau$. Same system parameters as in Fig. 1. The pump parameters used for panels (a)–(d) correspond to the dots in Fig. 1.

Figure 8

(Color online) Fourier transform spectra of ${\psi }_1$ (thin line) and ${\psi }_2$ (thick line) for growing pump amplitudes $F_{\mathit{max}}=1,5,8,20\mathrm{meV}$. The central $\delta$ peak is at the pump frequency. The arrows indicate the frequencies of the Bogoliubov modes with the same line style code as in Fig. 6. Parameters of panels (a)–(d) correspond to the ones in Fig. 7.

Figure 9

(Color online) [(a), (c), (e), and (g)] Intensity dynamics and [(b), (d), (f), and (h)] corresponding Fourier spectra under a Gaussian pulsed pump [Eq. (16)] at time $t_p=3\mathrm{ps}$ of duration ${\tau }_p=0.2\mathrm{ps}$. Peak pump amplitudes [(a) and (b)] $F_{\mathit{max}}=1\mathrm{meV}$, [(c) and (d)] $50\mathrm{meV}$, [(e) and (f)] $75\mathrm{meV}$, and [(g) and (h)] $100\mathrm{meV}$. Thin (thick) lines corresponds to the 1 (2) modes. Spectra are obtained by Fourier transforming the signals in the whole time interval. Same system parameters as in Fig. 1.

Figure 10

(a) Trapping potential $U_{\mathit{ext}}$ [Eq. (18)] due to two adjacent rectangular wells and (b) the fundamental ${\varphi }_{gs}\left(\mathbf{r}\right)$ and (c) first excited ${\varphi }_{\mathit{exc}}\left(\mathbf{r}\right)$ modes of the corresponding equilibrium GPE [Eq. (31)]. The spatial profiles are displayed in gray tones.

Figure 11

(Color online) Dynamical evolution of $n_1$ and $n_2$ $(n_2>n_1)$, as obtained from the full numerical calculation of GPE (thin lines) and from the two-mode model (thick lines). (a) GPE simulations with pump energy $\hslash \omega =-2.8\mathrm{meV}$ and pump amplitude $f_0=5\mathrm{meV}$. Parameters of the corresponding effective two-mode model: $\hslash {\omega }_{1,2}=-3.15\mathrm{meV}$, $J=0.45\mathrm{meV}$, $g={10}^{-3}\mathrm{meV}$, and $F_1=5.8\mathrm{meV}$. (b) GPE simulations with pump energy $\hslash \omega =-2.8\mathrm{meV}$ and pump amplitude $f_0=7.5\mathrm{meV}$. Parameters of the corresponding effective two-mode model: $\hslash {\omega }_j=-3.1\mathrm{meV}$, $J=0.5\mathrm{meV}$, $g={10}^{-3}\mathrm{meV}$, and $F_1=8.5\mathrm{meV}$.