Reducing decoherence of the confined exciton state in a quantum dot by pulse-sequence control

We study the phonon-induced dephasing of the exciton state in a quantum dot excited by a sequence of ultrashort pulses. We show that the multiple-pulse control leads to a considerable improvement of the coherence of the optically excited state. For a fixed control time window, the optimized pulsed control often leads to a higher degree of coherence than the control by a smooth, single Gaussian pulse. The reduction of dephasing is considerable already for 2–3 pulses.

Figure 1

The evolution of the optical polarization excited with one or four infinitely short pulses. Solid: exact result; dashed: perturbative result.

Figure 2

Spectral characteristics of the phonon reservoir and of the driven dynamics. Solid line: The spectral density $R\left(\omega \right)∕{\omega }^2$ at $T=0\mathrm{K}$ for LA phonons. Dashed line: the nonlinear spectrum ${\mid K\left(\omega \right)\mid }^2$ for a sequence of identical short pulses for $\alpha =\pi ∕2$ and pulse numbers and total durations as shown. Dash-dotted line: the envelope calculated as the linear spectrum of a single pulse (up to an appropriate factor). In the upper figures the nonlinear spectrum for a sequence of $\delta$ pulses is also shown (dotted).

Figure 3

The relative polarization drop (with respect to the unperturbed case) for the $\alpha =\pi ∕2$ excitation performed by sequences of $N=2,3,4$ equidistant narrow pulses with ${\tau }_{\mathrm{p}}=100\mathrm{fs}$ within the time interval $t_{\mathrm{tot}}$ at $T=0\mathrm{K}$ for optimized pulse intensities (solid) and for identical pulses (dashed).

Figure 4

The optimized sequences of $N=2$ (upper) and $N=4$ (lower) narrow pulses $\left({\tau }_{\mathrm{p}}=100\mathrm{fs}\right)$ compared to the Gaussian pulses (dashed, arbitrary scale) yielding the same asymptotic polarization (at $T=0\mathrm{K}$).

Figure 5

The polarization drop as a function of the number of pulses $N$ for the total durations $t_{\mathrm{tot}}=1\mathrm{ps}$ (squares) and $t_{\mathrm{tot}}=5\mathrm{ps}$ (circles) at $T=0\mathrm{K}$ (left) and $T=10\mathrm{K}$ (right). Full symbols: optimized pulse intensities; empty symbols: equal pulses. Here, ${\tau }_p=100\mathrm{fs}$.

Figure 6

Left: the optimized pulse envelope formed by the sequence of $N=4$ overlapping narrow pulses (dashed: pulses with optimized amplitudes; dotted: equal pulses) for the sequence duration $t_{\mathrm{tot}}=0.3\mathrm{ps}$ at $T=0\mathrm{K}$. Right: the corresponding nonlinear spectra ${\mid K\left(\omega \right)\mid }^2$ along with the phonon spectral density (solid).

Figure 7

Left: the asymptotic polarization decay induced by LA and LO phonons for $N=4$ pulses and ${\tau }_{\mathrm{p}}=60\mathrm{fs}$; solid: optimized sequence, dashed: equal pulses. The inset shows part of the curves in magnification. Right: The nonlinear spectrum ${\mid K\left(\omega \right)\mid }^2$ corresponding to two values of the pulse sequence duration $t_{\mathrm{tot}}=1.74\mathrm{ps}$ (strong overlap with the LO line; dashed) and $t_{\mathrm{tot}}=1.92\mathrm{ps}$ (weak overlap; dotted). Also shown are the spectral densities for LA (solid) and LO (bar, arbitrary scale) phonons.

Figure 8

The pulse sequence (left) and the nonlinear spectra (right) corresponding to the optimal control in the model extended by adding LO-phonon-assisted excitation of a few lowest dark states $\left(X_1-X_3\right)$. In the right panel, the solid and dashed lines show the spectral functions ${\mid K\left(\omega \right)\mid }^2$ and ${\mid L\left(\omega \right)\mid }^2$, respectively, while the vertical bars represent the spectral features related to phonon-assisted transitions to optically inactive states.