Role of Coulomb correlations for femtosecond pump-probe signals obtained from a single quantum dot

We present a theoretical analysis of femtosecond pump-probe experiments performed on a single negatively charged quantum dot. The influence of Coulomb-correlation effects as well as carrier relaxation on transient transmission change signals is investigated. Our model describes ultrafast disappearance of the fundamental trion absorption due to instantaneous Coulomb renormalizations and a delayed onset of gain at the same frequency, as found in the measurements. Going beyond previous experimental information, we predict that after spin-conserving carrier relaxation, new optical transitions exhibiting either gain or absorption should emerge that build up on a picosecond timescale. The time dependence of these new transitions provides insight into details of the carrier relaxation processes.

Figure 1

Schematic illustration of the analyzed pump-probe experiment. Initially the QD is charged with a single electron in the lowest conduction band state ($s$ shell). The pump-pulse in resonance to a higher shell ($p$ shell) creates an additional electron-hole pair resulting in a trion in an excited state. Subsequently, the generated electron and hole relax to the $s$ shell. The pump-induced dynamics is monitored by the absorption of a probe pulse resonant on the $s$-shell transition.

Figure 2

(Color online) Schematic illustration of the analyzed pump-probe experiment. Initially, the QD is charged by a single electron in the lowest conduction band state ($s$ shell). At time $t=0$, the pump-pulse excites an electron-hole pair in the $p$ shell, which subsequently relaxes toward the $s$ shell (left column). The various steps are monitored by a probe pulse on the $s$-shell transition. At negative delay times the probe pulse creates a trion in its ground state. Immediately after the pump pulse, it leads to optical transitions to an excited charged biexciton [states (1) and (2)]. After spin-conserving carrier relaxation, new optical transitions emerge resulting in stimulated emission to a single-electron state [states (3) and (4)] and absorption to a charged biexciton in the ground state [state (5)].

Figure 3

(a)–(c) Calculated trion absorption spectra of a singly charged quantum dot. Energies are given with respect to the lowest energy transition. The ratio $\beta$ between the confinement length of holes and electrons is taken as (a) $\beta =0.85$, (b) $\beta =1.00$, and (c) $\beta =1.15$. (d) Measured PL and PLE spectra of the singly charged QD used in the two color pump-probe experiment in Ref. 24. In (c), also the spectral envelopes of the pump pulse (dashed blue line) and probe pulse (thin red line) as used in the experiment in Ref. 24 are depicted.

Figure 4

Calculated spectroscopic shifts of the optical transitions starting from a trion state to the states shown schematically in Fig. 2 as a function of $\beta$. The energies are given with respect to the energy of the FTL.

Figure 5

Simulated temporal evolution of the trion state occupation probabilities generated by a $\pi$-pulse excitation with a 700 fs pulse on the $p$-shell transition at $t=0$ ps. Labels refer to the states shown in Fig. 2.

Figure 6

Probe-pulse absorption spectra calculated as a function of energy and delay time $\tau$ between pump and probe-pulse: (a) for the case of noninteracting carriers and (b) including Coulomb interactions. The pump pulse is in resonance to the lowest $p$-shell transition (see Fig. 2). The energy is measured with respect to the fundamental trion line. The labels (1)–(5) refer to the electron and charged biexciton states shown schematically in Fig. 2. Here, ${\left(2\right)}^t$ corresponds to the holes in a triplet state, while ${\left(2\right)}^s$ to the holes in a singlet state.

Figure 7

(Color online) Amplitude of the fundamental trion absorption line (FTL) as a function of delay time $\tau$ between pump and probe pulse with Coulomb interaction (red solid line) and without Coulomb interaction (green dashed line).

Figure 8

Upper panel: calculated differential transmission change $\Delta T/T$ as a function of the delay time $\tau$ between pump and probe pulse. The excitation conditions are the same as used in Fig. 6b. Lower panel: cross section through the three main transition lines [(3), (4), and (5)].