Dynamics of elastic and inelastic energy transfer between quantum dots in a microcavity

We develop a theory of exciton energy transfer dynamics between two quantum dots in a planar microcavity. We study the dynamics of quantum dot excitons in three different cases: (i) a single dot coupled only to cavity modes, (ii) two dots coupled to cavity modes, and (iii) two dots coupled to cavity modes and to acoustic phonons. In the latter case, we focus on the process of phonon-assisted inelastic exciton energy transfer between the quantum dots. We take into account phonon effects by introducing a light-matter Hamiltonian with operators describing the exciton-photon-phonon coupling and we truncate the Hilbert space by assuming that at most one excitation is present in the system. Using this approach we simulate the exciton dynamics with realistic parameters in the zero temperature limit. From the dynamics we extract the dependence of the characteristic energy transfer rate as a function of the interdot separation. This theoretical approach can be used to optimize exciton energy transfer by designing structures with engineered photon and phonon density of states.

Figure 1

Schematic view of the two dots in a planar cavity. $L$ is the separation between the two planar mirrors in the cavity. We assume the dots to be disk-like. $L_{\text{dot}}$ is the size of the dot in the growth direction ($z$), and $\beta$ is the in-plane radius of the dot. $R_{ab}$ is the separation between the two dots.

Figure 2

Photon dispersion relation in a planar cavity, the photon energy ($\hslash \omega$) is on the horizontal axis and the in-plane photon momentum ${\mathbf{k}}_{\gamma }$ is on the vertical axis. Note that photon frequency has a minimum, which is determined by the photon wave vector quantization in the $z$ direction. Arrows indicate the exciton energy of QD${}_a$ (red) and QD${}_b$ (green).

Figure 3

Real part of $F(\tau ,\omega ,\mathbf{R})$ in the case of quantum dots (a) and (b) resonant and (c) and (d) off-resonant with respect to the lowest cavity mode. (a) and (c) On-site memory effects (single dot case). (b) and (d) Memory effects in the light propagation at finite distance.

Figure 4

Single dot dynamics with two different energy for the dot. (a) Quantum dot exciton energy resonant with the lowest cavity mode. (b) Quantum dot exciton energy far detuned at half the energy of the lowest cavity mode. The decay in both cases is the result of the coupling of the dot to the continuum of cavity modes.

Figure 5

Exciton dynamics in two dots system. (a) and (b) show the dynamics of the exciton population in QD${}_a$ and QD${}_b$. (c) and (d) Energy transfer time $T_{\text{tran}}$ for different detuning $\Delta$ and inter-dot separation $R_{ab}$. Here ${\omega }_a$ and ${\omega }_b$ are the excitation energy for QD${}_a$ and QD${}_b$, resepectively. ${\omega }_{{\mathbf{k}}_{\gamma }=0}$ is the frequency of the lowest cavity mode.

Figure 6

Phonon assisted energy transfer. (a) No phonon included. (b) The phonon-assisted exciton energy transfer with phonon energy equal to the detuning between the two dots (phonon-assisted resonance).