Vacuum-induced coherence in quantum dot systems

We present a theoretical study of vacuum-induced coherence in a pair of vertically stacked semiconductor quantum dots. The process consists in a coherent excitation transfer from a single-exciton state localized in one dot to a delocalized state in which the exciton occupation gets trapped. We study the influence of the factors characteristic of quantum dot systems (as opposed to natural atoms): energy mismatch, coupling between the single-exciton states localized in different dots, and different and nonparallel dipoles due to sub-band mixing, as well as coupling to phonons. We show that the destructive effect of the energy mismatch can be overcome by an appropriate interplay of the dipole moments and coupling between the dots which allows one to observe the trapping effect even in a structure with technologically realistic energy splitting of the order of milli–electron volts. We also analyze the impact of phonon dynamics on the occupation trapping and show that phonon effects are suppressed in a certain range of system parameters. This analysis shows that the vacuum-induced coherence effect and the associated long-living trapped excitonic population can be achieved in quantum dots.

Figure 1

Vacuum-induced coherence in a system of two identical ($\Delta =0$, ${\Gamma }_{11}={\Gamma }_{22}=1$ ns${}^{-1}$) two-level systems. (a) Exciton occupation of the system and occupations of states $|10\rangle$ ($n_{10}$) and $|01\rangle$ ($n_{01}$). (b) The off-diagonal density matrix element ${\rho }_{01,10}$.

Figure 2

(a) Exciton occupation for a pair of decoupled ($V=0$) QDs with identical transition energies ($\Delta =0$) but with nonidentical values of the spontaneous recombination rates (${\Gamma }_{11}\ne {\Gamma }_{22}$). (b–d) The total exciton occupation [solid (red) line] and the occupations of the individual dots [dashed (green) and dotted (blue) lines] for the three cases shown in (a). The value of ${\Gamma }_{11}=1$ ns${}^{-1}$ is the same for all the graphs.

Figure 3

Exciton occupation of a pair of uncoupled ($V=0$) QDs with identical transition energies ($\Delta =0$) and different spontaneous recombination rates (${\Gamma }_{11}\ne {\Gamma }_{22}$) for a few values of the angle $\theta$ between the dipole moments. The values of ${\Gamma }_{11}$ and ${\Gamma }_{22}$ shown in (a) are valid for (b) also.

Figure 4

(a) Impact of the energy mismatch on exciton occupation of a pair of uncoupled QDs. (b) Exciton occupation of a technologically realistic QDM for a few values of coupling between the emitters (${\Gamma }_{11}={\Gamma }_{22}=1$ ns${}^{-1}$).

Figure 5

Exciton occupation of a pair of technologically realistic QDs. (a) Total exciton occupation for $\Delta =1$ meV, $V=-5$ meV, ${\Gamma }_{11}=1$ ns${}^{-1}$ and for a few values of ${\Gamma }_{22}$. (b) The total occupation and the occupations of the individual dots for the last case shown in (a). (c) Exciton occupation for three sets of parameters stabilizing the occupation trapping. Solid (red) line: $\Delta =0.5$ meV, $V=-4.47$ meV, ${\Gamma }_{11}=0.8$ ns${}^{-1}$, ${\Gamma }_{22}=1$ ns${}^{-1}$. Dashed (green) line: $\Delta =0.5$ meV, $V=-9.99$ meV, ${\Gamma }_{11}=0.95$ ns${}^{-1}$, ${\Gamma }_{22}=1.05$ ns${}^{-1}$. Dotted (blue) line: $\Delta =1$ meV, $V=-5$ meV, ${\Gamma }_{11}=1$ ns${}^{-1}$, ${\Gamma }_{22}=1.49$ ns${}^{-1}$. (d) The total occupation and the occupations of the individual dots corresponding to the solid (red) line in (c).

Figure 6

Exciton occupation of a pair of nonidentical ($\Delta =1$ meV, ${\Gamma }_{11}\ne {\Gamma }_{22}$) and coupled ($V=-5$ meV) QDs for a few values of $\theta$ (a) and for two values of ${\Gamma }_{22}$ (b).

Figure 7

The phonon spectral density for $D=8$ nm.

Figure 8

Phonon impact on vacuum-induced coherence in a pair of QDs with a fixed $V/\Delta$ ratio (corresponding to $\varphi =-1.4$): comparison of the total exciton occupation in a system coupled to phonons and without phonons. (a) $\Omega =4.0$ meV; (b) short-time section of (a), with the solid (red) line representing the total exciton occupation, the dotted (magenta) line corresponding to the total occupation without phonons, and the dashed (green) and dash-dotted (blue) lines showing the occupations of the two dots; (c) $\Omega =8.07$ meV (corresponding to the third minimum of the spectral density); (d) $\Omega =13.36$ meV (fifth minimum of the spectral density).