Parametric instability in coupled nonlinear microcavities

We report the observation of a parametric instability in the out-of-equilibrium steady state of two coupled Kerr microresonators coherently driven by a laser. Using a resonant excitation, we drive the system into an unstable regime, where we observe the appearance of intense and well-resolved sideband modes in the emission spectrum. This feature is a characteristic signature of self-sustained oscillations of the intracavity field. We comprehensively model our findings using semiclassical Langevin equations for the cavity field dynamics combined with a linear stability analysis. The inherent scalability of our semiconductor platform, enriched with a strong Kerr nonlinearity, is promising for the realization of integrated optical parametric oscillator networks.

Figure 1

(a) Scanning electron microscope image of the coupled micropillar cavities composed by two distributed Bragg reflectors (DBR), a $\lambda$ spacer, and a quantum well (QW), shaded in blue, green, and purple, respectively. (b) Schematic representation of the hybridization of the two bare pillar modes into molecular bonding (B) and antibonding (AB) photonic modes. (c) Measured transmission as a function of the laser detuning relative to the uncoupled cavity resonance $\mathrm{\Delta }$. The dashed line is a fit with the Eq. (1) steady-state expectation value. The left and right insets show the transmission pattern measured at each resonance. The dashed lines indicate the edge of the microstructure and the half-maximum contour of the incident laser spot.

Figure 2

(a, b) Symbols, measured polariton occupation number of the (a) driven and (b) undriven cavity, as a function of pump power; dashed line, steady-state prediction deduced from Eq. (1) with $U$ being the only adjustable parameter. (c) Left panels: Intensity patterns measured at the five different pump powers indicated in panel (a). Right panels: Calculated intensity patterns obtained as time-averaged solutions of Eq. (1), multiplied with a Gaussian spatial profile. The spatial patterns in panels (c1) and (c5) correspond to near-resonant driving conditions of the linear antibonding and nonlinear bonding modes. (d, e) Imaginary and real parts of the stability matrix eigenvalues as a function of pump power, along the lower branch of the hysteresis cycle.

Figure 3

(a) Total interaction energy $U{n}_{\mathrm{tot}}$ deduced from the experiment as a function of the incident pump power along the lower bistability branch. Horizontal dashed lines correspond to $U{n}_{\mathrm{tot}}=(J-\gamma ,J,J+\gamma )$. Inset: Schematics of the parametric instability mechanism occurring as the bonding (B) and antibonding (AB) modes are blueshifted by interactions. (b, c) Integrated spectrally resolved emission of the microstructure measured while (b) increasing or (c) decreasing the pump power $P_{\mathrm{Fw}}$ ($P_{\mathrm{Bw}}$), expressed in units of $P_{\mathrm{thr}}$. (d–f) Position and energy resolved emission patterns (d) below, (e) within, and (f) well above the instability region. As schematized in panel (b), the spatial profile corresponds to a cut through the long symmetry axis of the structure, while the laser drives the right microcavity. (g) Measured (purple) and simulated (gray) emission spectrum of the coupled cavities at $1.1{\mathrm{P}}_{\mathrm{thr}}$. (h) Calculated trajectories of the cavity field amplitudes ${\alpha }_{1,2}$ showing a limit cycle behavior in the instability region ($2.5-\mathrm{ns}$ evolution).

Figure 4

Self-pulsing dynamics. (a–c) Simulations obtained using the generalized Gross-Pitaevskii equation for the coupled polariton modes, i.e., Eq. (1) in the absence of noise terms. (d–i) Simulations obtained via the coupled Langevin equations Eq. (1). (a, d, g) Dynamics of the driven (red line) and undriven (blue) cavity populations obtained with (a) the DDGPE for $U=0.1\mu \mathrm{eV}$, (d) Eq. (1) for $U=0.1\mu \mathrm{eV}$, and (g) Eq. (1) for $U^{\prime }=10U=1\mu \mathrm{eV}$. The first $1.5\mathrm{ns}$ of the dynamics are used to adiabatically ramp the pump power to $1.1P_{\mathrm{thr}}$, [see gray line in panel (a)]. In the insets of panels (a), (d), and (g), we show a $200-\mathrm{ps}$ snapshot of the dynamics, showing a self-pulsing behavior with period $T\approx 24\mathrm{ps}$. (b, e, h) Normalized power spectral density associated to the last $2.5-\mathrm{ns}$ evolution of the cavity field dynamics shown in panels (a), (d), and (g), respectively. In panels (e) and (h) we keep plotting the DDGPE results obtained in panel (b) for comparison (solid red line). We can observe that the energy spacing of the sidebands ($\delta E\approx 0.17\mathrm{meV}$) remains unaffected by fluctuations; however, the relative height of the peaks decreases for larger fluctuations of the intracavity field (i.e., increasing $U$). (c, f, i) Phase-space limit cycle dynamics of the intracavity fields associated to the last $2.5-\mathrm{ns}$ evolution shown in panels (a), (d), and (g), respectively. In panels (c), (f), and (i) the red and blue lines correspond to the driven and undriven cavity field dynamics computed via the DDGPE. Gray dots in panels (f) and (i) correspond to the stochastic dynamics obtained via Eq. (1) sampled every $\mathrm{ps}$ for $U=0.1$ and $1\mu \mathrm{eV}$, respectively. As $U$ increases the number of excitations in the cavity at the threshold decreases; thus the relative amplitude of the fluctuation increases, as shown by the scatterplot of the dynamics. The random phase shifts induced by fluctuations tend to dephase the parametric oscillations, thus reducing the temporal coherence of the process as the reduced sideband contrast in panels (e) and (h) indicates.