Decay dynamics of radiatively coupled quantum dots in photonic crystal slabs

We theoretically investigate the influence of radiative coupling on light emission in a photonic crystal slab structure. The calculation method is based on a formalism that combines the photon Green's tensor with a self-consistent Dyson equation approach and is applicable to a wide range of problems in nanophotonics. We apply the method to calculate how resonant interactions of neighboring quantum dots affect the spontaneous emission, and we observe a pronounced nonexponential decay in the intensity at the detector position. We analyze the decay based on detailed calculations of the Green’s tensor and show how interference between different light scattering pathways is responsible for this nontrivial detector response.

Figure 1

Sketch of photonic crystallite. Red and green spheres represent embedded QDs at two high-symmetry points of the central Wigner-Seitz cell and a detector is placed above the central air hole of the crystallite.

Figure 2

Schematic of a system with two QDs at the positions ${\mathbf{r}}_1$ and ${\mathbf{r}}_2$ and a detector at the position $\mathbf{R}$. Arrows indicate different scattering channels for the light.

Figure 3

Sketch of the crystallites. Of special importance are the high-symmetry points in the center and along the edge of the central Wigner-Seitz cell (hexagonal area). In this work we focus on the points $\Gamma =(0,0,0)$, $K_{\mathbf{r}}=(0,a/\sqrt{3},0)$, and $M_{\mathbf{r}}=(a/2,0,0)$.

Figure 4

Band diagram for even modes in the infinite PC slab calculated with both the plane wave method (red solid line) and using FDTD (circles). The light cone is indicated by the gray shading and hatching indicates the in-plane band gap. Inset shows the Brillouin zone along with high-symmetry points and the mode profile at the $K$ point.

Figure 5

Relative projected LDOS, $\text{Im}\{G_{\alpha \alpha }(\mathbf{r},\mathbf{r})\}/G_0$, for the two positions $K_{\mathbf{r}}$ and $M_{\mathbf{r}}$ in a crystallite with side length $L=6a$. Hatching indicates the in-plane band gap of the infinite crystal.

Figure 6

Absolute values, in units of $G_0$, of elements in the propagator between the points $K_{\mathbf{r}}$ and $M_{\mathbf{r}}$ in a crystallite with side length $L=6a$. Hatching indicates the in-plane band gap of the infinite crystal.

Figure 7

Left: Normalized mode profiles at constant frequency $\omega a/(2\pi c)=0.26$ as excited by an $x$-oriented source at the $K_{\mathbf{r}}$ point. Right: Zoom-in on the central region of the crystallite showing the mode profile (top) and scaled in-plane field vectors (bottom) with a double arrow indicating the source.

Figure 8

Intensity at the position $\mathbf{R}=(0,0,3.3a)$ of the light emitted from QD 1 at the position $K_{\mathbf{r}}$ when a second QD with relative detuning $\Delta \mathrm{\omega ̃}=\Delta \omega /{\omega }_1$ is positioned at $M_{\mathbf{r}}$. Both QDs have dipole moments pointing in the $y$ direction. For reference, we show also the case of a single QD only. The intensity is normalized to the intensity of the single-QD case at $t=0$.

Figure 9

Spectra (left) and decay curves (right) corresponding to the three different decay channels in the two-QD case with QD 1 initially excited. The calculations were performed as in Eqs. (25) and (26) with the electric field given by either the direct term ${\mathbf{E}}^{\text{D}}$ (red solid lines) or one of the two scattering terms ${\mathbf{E}}_1^{\text{S}}$ (green dashed line) or ${\mathbf{E}}_2^{\text{S}}$ (blue dashed-dotted line), as indicated. Gray solid lines show the results as calculated with the total electric fields (equal to the curves in Fig. 8). Results are shown for the three different detunings $\Delta \mathrm{\omega ̃}={10}^{-5}$ (top), $\Delta \mathrm{\omega ̃}=2\times {10}^{-6}$ (center), $\Delta \mathrm{\omega ̃}={10}^{-7}$ (bottom), where $\mathrm{\omega ̃}=\omega /{\omega }_1$. The curves corresponding to the scattering channels have been scaled as indicated to make them visible on the scale of the total field.

Figure 10

Intensity of $y$ polarized light at the position $\mathbf{R}=(0,0,3.3a)$ emitted from QD 1 at the position $K_{\mathbf{r}}$ with dipole moment in the $x$ direction. The light scatters off a second QD at the position $M_{\mathbf{r}}$ with relative detuning $\Delta \mathrm{\omega ̃}=\Delta \omega /{\omega }_1$ and dipole moment in the $y$ direction. The intensity is normalized as in Fig. 8.

Figure 11

Intensity at the position $\mathbf{R}=(0,0,3.3a)$ of light emitted from QD 2 at the position $M_{\mathbf{r}}$ with dipole moment in the $y$ direction. The intensity is normalized as in Fig. 8.

Figure 12

Intensity at the position $\mathbf{R}=(0,0,3.3a)$ of light emitted from QD 1 at the position $K_{\mathbf{r}}$ within a homogeneous slab of GaAs when a second QD with relative detuning $\Delta \mathrm{\omega ̃}=\Delta \omega /{\omega }_1$ is positioned at $M_{\mathbf{r}}$. Both QDs have dipole moments pointing in the $y$ direction, and the intensity is normalized as in Fig. 8.