Dicke states in multiple quantum dots

We present a theoretical study of the collective optical effects which can occur in groups of three and four quantum dots. We define conditions for stable subradiant (dark) states, rapidly decaying super-radiant states, and spontaneous trapping of excitation. Each quantum dot is treated like a two-level system. The quantum dots are, however, realistic, meaning that they may have different transition energies and dipole moments. The dots interact via a short-range coupling which allows excitation transfer across the dots, but conserves the total population of the system. We calculate the time evolution of single-exciton and biexciton states using the Lindblad equation. In the steady state the individual populations of each dot may have permanent oscillations with frequencies given by the energy separation between the subradiant eigenstates.

Figure 1

The planar arrangement of the three- (a) and four- (b) QD ensembles.

Figure 2

Exciton occupations [(a) and (c)] and coherences [(b) and (d)] of a system of three QDs prepared initially in a bright state $\left(5\right|1\rangle +|2\rangle )/\sqrt{26}$ for ${\Gamma }_{11}=2.44$ ns${}^{-1}$, ${\Gamma }_{22}=3.31$ ns${}^{-1}$, and ${\Gamma }_{33}=1$ ns${}^{-1}$. (a) and (b) show the results for uncoupled QDs ($V_{ij}=0$) with identical transition energies ($\Delta =0$), while (c) and (d) refer to energetically inhomogeneous ($\Delta \ne 0$) and coupled ($V_{ij}\ne 0$) systems arranged in a lateral array of an equidistant triangle shape with the side length $r=30$ nm. The inset to (a) shows the initial evolution of the exciton occupation of the QD no. 2.

Figure 3

The exciton occupation of the QD no. 1 ($n_1$) shown by the blue line in Fig. 2c. The time evolution is shown for three different time intervals at picosecond resolution scale.

Figure 4

Decay rate (in the collective regime) dependence on the spatial arrangement of a triple QD, i.e., on the angle $\alpha$ defined in Fig. 1 (a) and on the energy mismatch (b), and the corresponding steady-state occupations for the initial state $\left(5\right|1\rangle +|2\rangle )/\sqrt{26}$ (c) and (d), respectively. For all panels we assume $r_{12}=30$ nm and $r_{23}=20$ nm and a constant decay rate of the QD No. 2 (${\Gamma }_{22}=2$ ns${}^{-1}$).

Figure 5

Exciton occupation for a biexciton initial state spanned in a system of four QDs placed at the corners of a square of side length $r=30$ nm. (a) Super-radiant initial state. (b) and (c) Initial biexciton state which allows for a recombination of only one exciton. $d_{12}=\sqrt{{\Gamma }_{33}{\Gamma }_{44}}$, $d_{13}=-\sqrt{{\Gamma }_{22}{\Gamma }_{44}}$, $d_{14}={\Gamma }_{44}({\Gamma }_{33}-{\Gamma }_{22})\sqrt{{\Gamma }_{22}{\Gamma }_{33}}/({\Gamma }_{22}{\Gamma }_{33}-{\Gamma }_{11}{\Gamma }_{44})$, $d_{23}=-{\Gamma }_{44}({\Gamma }_{33}-{\Gamma }_{22})\sqrt{{\Gamma }_{11}{\Gamma }_{44}}/({\Gamma }_{22}{\Gamma }_{33}-{\Gamma }_{11}{\Gamma }_{44})$, $d_{24}={\Gamma }_{44}\sqrt{{\Gamma }_{11}{\Gamma }_{33}}/{\Gamma }_{11}$, and $d_{34}=-{\Gamma }_{44}\sqrt{{\Gamma }_{11}{\Gamma }_{22}}/{\Gamma }_{11}$. The values of the decay rates of individual dots are the same for all panels and take the values ${\Gamma }_{11}=2.26$ ns${}^{-1}$, ${\Gamma }_{22}=2.75$ ns${}^{-1}$, ${\Gamma }_{33}=0.88$ ns${}^{-1}$, and ${\Gamma }_{44}=1.5$ ns${}^{-1}$. The biexciton shifts are $B_{13}=B_{12}-0.702$ meV, $B_{14}=B_{12}+0.022$ meV, $B_{23}=B_{12}+0.218$ meV, $B_{24}=B_{12}-0.741$ meV, and $B_{34}=B_{12}+0.042$ meV with $B_{12}$ being an arbitrary parameter. The line types used in (b) are also valid for (a).

Figure 6

Exciton occupation for a system of four QDs placed at the vertices of a square of side length $r=30$ nm and prepared initially in a single-exciton subradiant state $\sim$$\left(\sqrt{{\Gamma }_{22}{\Gamma }_{33}{\Gamma }_{44}}\right|1\rangle +\sqrt{{\Gamma }_{11}{\Gamma }_{33}{\Gamma }_{44}}|2\rangle +\sqrt{{\Gamma }_{11}{\Gamma }_{22}{\Gamma }_{44}}|3\rangle -3\sqrt{{\Gamma }_{11}{\Gamma }_{22}{\Gamma }_{33}}|4\rangle )$. In regions I and III the decay rates are the same as in Fig. 5, i.e., selected such that the system interacts collectively with the radiative surrounding. The red solid line corresponds to changes of the decay rates of individual QDs, and the blue dotted line to the same changes plus changes of the relative angular orientation of the dipoles. In region II the decay rates and angles have been modified such that the ensemble does not interact collectively with its photon environment (see text).