Phonon-Induced Rabi-Frequency Renormalization of Optically Driven Single $\mathrm{InGaAs}/\mathrm{GaAs}$ Quantum Dots

We study optically driven Rabi rotations of a quantum dot exciton transition between 5 and 50 K, and for pulse areas of up to $14\pi$. In a high driving field regime, the decay of the Rabi rotations is nonmonotonic, and the period decreases with pulse area and increases with temperature. By comparing the experiments to a weak-coupling model of the exciton-phonon interaction, we demonstrate that the observed renormalization of the Rabi frequency is induced by fluctuations in the bath of longitudinal acoustic phonons, an effect that is a phonon analogy of the Lamb shift.

Figure 1

($○$) A Rabi rotation measurement at 12.5 K. The full lines are calculated using Eqs. (1, 2). (red) [$K(\Omega )$ is real, and $\mathcal{P}=1$] In a low driving-field regime ($\Theta <6\pi$), the data are described by an ${\Omega }^2$ damping, but at higher fields this model overdamps the oscillation. (green) [$K(\Omega )$ is real, $\mathcal{P}(\Omega )=e^{-{\Omega }^2/{\omega }_c^2}$] The roll-off in the damping is reproduced by including the frequency dependence of the exciton-phonon form factor $\mathcal{P}(\Omega )$. With a constant rotation frequency, this model provides a “clock-signal” against which the nonlinearity of the rotation angle with $\Theta$ is clearly observed. (blue) [$K(\Omega )$ is complex] By including the imaginary part of $K(\Omega )$ the variation of rotation angle with $\Theta$ is reproduced.

Figure 2

(a) Temperature dependence of the Rabi rotations. Red lines are calculated using $A=11.2\mathrm{fs}{\mathrm{K}}^{-1}$, $\hslash {\omega }_c=1.44\mathrm{meV}$, and ${\Gamma }_2^{*}=0.025{\mathrm{ps}}^{-1}$, where $A$ and $\hslash {\omega }_c$ are extracted from a fit to all data. The bottom axis is scaled to the bare pulse-area $\Theta$, while the top axis indicates the peak Rabi energy of the laser pulse. (b) Square root of the incident power $\sqrt{P_{2\pi }}$ corresponding to the first minimum plotted vs temperature. $\sqrt{P_{2\pi }}$ is scaled to the pulse area $\Theta$ for the case of $T=5\mathrm{K}$. The trace is calculated assuming the scaling parameter $a=1$. (c) Calculation of the Lamb shift, $\mathrm{Im}[K]$ at $T=0\mathrm{K}$, as a function of Rabi energy.