Long-distance radiative excitation transfer between quantum dots in disordered photonic crystal waveguides

We theoretically investigate the magnitude and range of the photon-mediated interaction between two quantum dots embedded in a photonic crystal waveguide, including fabrication disorder both in the crystal and in the dot positioning. We find that disorder-induced light localization has a drastic effect on the excitation transfer rate—as compared to an ideal structure—and that this rate varies widely among different disorder configurations. Nevertheless, we also find that significant rates of $50\mu \mathrm{eV}$ at a range of $10\mu \mathrm{m}$ can be achieved in realistic systems.

Figure 1

(a) Band structure of the regular waveguide; the main guided band is shown as a blue line. The dashed line represents the light cone. (b) Zoom-in close to the edge of the guided band. (c) $y$ component of the electric field ($E_y$) of the mode at frequency $\omega =1.2980\mathrm{eV}$ [dashed line in (b)]; the white crosses indicate the elementary cell centers, where we assume a quantum dot can be placed. (d) $E_y$ of the mode at frequency $\omega =1.2993\mathrm{eV}$ [dashed-dotted line in (b)]. (e) Disorder-averaged zero-distance coupling $\langle \left|G^{11}\left({\omega }_0\right)\right|\rangle$, plotted as a function of the exciton frequency ${\omega }_0$. (f) Characteristic decay length of the disorder-averaged coupling $\langle \left|G^{12}\left({\omega }_0\right)\right|\rangle$ as a function of ${\omega }_0$. Full lines: Including disorder-induced localization. Dashed lines: No localization; the decay length is simply given by $2v_g/\gamma$. In both (e) and (f), three different magnitudes of disorder have been assumed (see legend). The vertical dot-dashed line denotes the band edge of the regular waveguide. For $\sigma =0.002a$, the $2v_g/\gamma$ curve in (f) cannot be distinguished from the band-edge line on the scale of the plot. The inset in (e) shows the disorder-averaged $\langle \left|G^{12}\left({\omega }_0\right)\right|\rangle$ as a function of distance, for three different values of the exciton frequency ${\omega }_0$, and $\sigma =0.004a$.

Figure 2

(a) Excitation transfer rate vs distance without light localization (dashed) or with for a single disorder realization (dashed-dotted) and the configuration average (solid), for $\sigma =0.004a$ and ${\omega }_0=1.2985\mathrm{eV}$ [indicated by an arrow in (b)]. (b) The two solid lines are the same as the $\sigma =0.004a$ lines in Figs. 1e and 1f; the dashed line shows the value of $|G^{11}|$ for which the CDF [panel (c)] is 0.95. (c): PDF and CDF of $|G^{11}|$, with $\langle \left|G^{11}\right|\rangle$ and the CDF $=$ 0.95 values explicitly indicated.

Figure 3

For two different positions of the first dot, the excitation transfer rate as a function of dot-dot distance, for the dots placed exactly at the center of the elementary cells (blue line) and with some positioning disorder corresponding to a finite value of ${\sigma }_D$ (black and red lines, averaged over dot positioning). The configuration average over PHC disorder is also shown (green line).