Near-field intensity correlations in parametric photoluminescence from a planar microcavity

We study the spatiotemporal pattern of the near-field intensity correlations generated by parametric scattering processes in a planar optical cavity. A generalized Bogolubov–de Gennes model is used to compute the second-order field correlation function. Analytic approximations are developed to understand the numerical results in the different regimes. The correlation pattern is found to be robust against a realistic disorder for state-of-the-art semiconductor systems.

Figure 1

(Color online) Real (a-g) and imaginary parts (b-h) of the Bogolubov dispersion as a function of the wave vector $k$, for the four regimes A-D marked in panel (i). The exciton interaction energy is $g{\left|{\Phi }_x\right|}^2={10}^{-4}\text{meV}$ (a, b), $g{\left|{\Phi }_x\right|}^2=0.04\text{meV}$ (c, d), $g{\left|{\Phi }_x\right|}^2=0.08\text{meV}$ (e, f) and $g{\left|{\Phi }_x\right|}^2=0.61\text{meV}$ (g, h). (i) Bistable loop in the in-cavity exciton field intensity as a function of the incident pump intensity. The red dashed line represents the unstable solutions. The points labeled with the letters $\mathbf{A}\text{−}\mathbf{D}$ mark the four regimes considered in panels (a-h).

Figure 2

(Color online) (a) Pattern of the reduced intensity correlations ${\overline{G}}^{\left(2\right)}\left(r,t\right)$ as a function of the spatial distance $r$ and of the temporal delay $t$, for the low-density regime A, corresponding to Figs. 1a, 1b. The two thin lines, $r=v_g^{s}t$ and $r=v_g^{i}t$, highlight the group velocities of the signal and idler modes. (b) Cut of the figure (a) along the zero delay $t=0$ line. (c) Cut along the $t=5\text{ps}$ line. In panels (b) and (c), the analytical prediction Eq. (45) is shown as a dashed line. In panel (c), the two vertical dash-dotted lines delimit the region where the analytical result (45) predicts vanishing correlations.

Figure 3

(Color online) (a) Pattern of the intensity correlations ${\overline{G}}^{\left(2\right)}\left(r,t\right)$, for the moderate density regime B, still well below the OPO threshold and corresponding to Fig. 1c, 1d. The two thin lines, $r=v_g^{s}t$, and $r=v_g^{i}t$, highlight the group velocities of the signal and idler modes.

Figure 4

(Color online) (a) Pattern of the intensity correlations ${\overline{G}}^{\left(2\right)}\left(r,t\right)$ for the density regime C just below the OPO threshold and corresponding to Figs. 1e, 1f. (b) Cut of the figure (a) along the zero delay $t=0$ line. (c) Cut along the $t=10\text{ps}$ line. In panels (b) and (c), the dashed line is the result of the analytical prediction Eqs. (47, 48).

Figure 5

(Color online) (a) Pattern of the intensity correlations ${\overline{G}}^{\left(2\right)}\left(r,t\right)$ for the high-density regime D, corresponding to Figs. 1g, 1h. In this regime, the large blueshift of the polariton modes prevents the parametric oscillation. The two thin lines $r=\pm v_g^{\text{max}}t$ mark the maximal group velocities. (b) Cut of the previous figures along the zero delay $t=0$ line. (c) Cut along the $t=5\text{ps}$ line. In panels (b) and (c), the analytical prediction Eq. (52) (red dashed line) and Eq. (53) (green dash-dotted line) are also shown. In panel (c), the two vertical thin dash-dotted lines delimit the region where both the numerical result and the analytical result (52) predict vanishing correlations.

Figure 6

(Color online) Geometric construction to determine the pairs of points in space-time that are correlated in the low-density parametric luminescence regime. The shaded area highlights the region of space-time which is not correlated with the point $\left(r=0,t=0\right)$. The point $\left(r=0,t=0\right)$ is parametrically correlated with the point $\alpha$, as a result of the parametric emission event occurring at the point marked with a star. The same holds for the point $\beta$. Solid lines mark the motion of signal (blue) and idler (red) particles. No parametric event can instead produce correlations between the point $\left(r=0,t=0\right)$ and the point $\delta$ located inside the shaded region.

Figure 7

(Color online) Intensity correlation pattern ${\overline{G}}^{\left(2\right)}\left(\mathbf{r},t\right)$ as a function of the spatial coordinate $x$ and of the temporal delay $t$ along a $y=0$ line for a 2D system. Panels (a)–(c) are for a uniform case, while panels (d)–(f) are for a disordered system. The pump intensity is varied from a low value (a, d), to a value close to the OPO threshold (b, e) and to a value well above the bistability loop (c, f) where parametric processes are forbidden. Panels (g)–(i) show the coherent photon field profiles ${\left|{\Phi }_c\left(r\right)\right|}^2$ corresponding to the disordered case of panels (d)–(f).

Figure 8

(Color online) Transmittivity (a) and transmission phase (b) across the $\lambda /4$ DBR mirror between cavity and air as a function of the frequency $\omega$, for a given wave vector $k=0$ (solid line), $k=1\mu {\text{m}}^{-1}$ (dashed), $k=1.5\mu {\text{m}}^{-1}$ (dot-dashed) and $k=2\mu {\text{m}}^{-1}$ (dotted). Parameters are $l_1=65\text{nm}$, $l_2=75\text{nm}$, $n_1=3$, $n_2=3.5$, $n_c=3.4$, $L_c=290\text{nm}$ and $N=22$. The cavity frequency ${\omega }_0=\pi c/n_c{L}_c$ is located at the center of the mirror stop band. (c) Transmission phase across the same DBR mirror as a function of the wave vector $k$, for a give frequency $\omega -{\omega }_0=0$ (thick solid line) and $\hslash \left(\omega -{\omega }_0\right)=\pm 2\text{meV}$ (thin lines). Dashed lines are obtained by fitting the numerical results with Eq. (54), which gives ${\Delta }_z\simeq 140\text{nm}$.