Decoherence in quantum dots due to real and virtual transitions: A nonperturbative calculation

We investigate theoretically acoustic phonon-induced decoherence in quantum dots. We calculate the dephasing of fundamental (interband or intraband) optical transitions due to real and virtual transitions with higher energy levels. Up to two acoustic phonon processes (absorption and/or emission) are taken into account simultaneously in a nonperturbative manner. An analytic expression of acoustic phonon-induced broadening is given as a function of the electron-phonon matrix elements and is physically interpreted. The theory is applied to the dephasing of intersublevel transitions in self-assembled quantum dots.

Figure 1

(Color online) Schematic of the electron-phonon levels involved in the dephasing processes. Levels are classified in three columns according to the number of phonon occupancies they differ from ${\varphi }_0$: the projector operator $P$, $P^{\prime }$, and $P^{″}$ corresponds, respectively, to zero-, one-, and two-phonon difference from the initial phonon state. The levels in solid lines illustrate coherent couplings involved in two-phonon virtual processes.

Figure 2

(Color online) Full width at half maximum (FWHM) $2{\Gamma }_0$ (full line) of the ZPL component of the $s\text{−}{p}_x$ intersublevel transition as a function of the energy detuning ${\Delta }_{pp}$ between the $p$ levels for $T=100\text{K}$. The Fermi golden rule for phonon absorption $2{\gamma }_{od}\left(-\Delta \right)$ is shown in dotted line. The contribution of real two-phonon transitions $2{\Gamma }_0^{\beta }$ appears in dashed line.

Figure 3

(Color online) Linewidth (FWHM) $2{\Gamma }_0$ of the ZPL of the $s\text{−}{p}_x$ intersublevel transition versus temperature for different values of the energy splitting ${\Delta }_{pp}$ between the $p$ states (the values of ${\Delta }_{pp}$ are indicated on the figure).

Figure 4

(Color online) Function $2h$ representing the contribution to the linewidth $2{\Gamma }_0^{\alpha }$ as a function of the acoustic phonon energy $\left(T=100\text{K}\right)$ for different $p$ splittings. (a) 1, (b) 3, (c) 5, and (d) 7 meV. The area under the curve (red) is equal to $2{\Gamma }_0$ (FWHM).