Lindblad theory of dynamical decoherence of quantum-dot excitons

We use the Bloch-Redfield-Wangsness theory to calculate the effects of acoustic phonons in coherent control experiments where quantum-dot excitons are driven by shaped laser pulses. This theory yields a generalized Lindblad equation for the density operator of the dot, with time-dependent damping and decoherence due to phonon transitions between the instantaneous dressed states. It captures similar physics to the form recently applied to Rabi oscillation experiments [Ramsay et al., Phys. Rev. Lett. 104, 017402 (2010)] but guarantees positivity of the density operator. At sufficiently low temperatures, it gives results equivalent to those of fully non-Markovian approaches [Lüker et al., Phys. Rev. B 85, 121302 (2012)] but is significantly simpler to simulate. Several applications of this theory are discussed. We apply it to adiabatic rapid passage experiments and show how the pulses can be shaped to maximize the probability of creating a single exciton using a frequency-swept laser pulse. We also use this theory to propose and analyze methods to determine the phonon density of states experimentally, i.e., phonon spectroscopy, by exploring the dependence of the effective damping rates on the driving field.

Figure 1

Illustration of time scales required for validity of (time-dependent) Markov approximation, shown in the frequency domain. Solid red line shows the frequency-dependent decay rate, illustrated by the phonon spectral function $J\left(\omega \right)$. At low temperatures and slow driving (dashed line), the linewidth is sufficiently small that the decay rate does not vary significantly across the linewidth (bath correlation time is short compared to decay time). At higher temperatures or faster driving, the linewidth grows (dot-dashed line) so that the decay rate does vary (decay time or sweep time become comparable to bath correlation time). The lineshapes illustrated are the Lorentzians corresponding to transition rates $T_1^{-1}=0.2$ and 1.0 ${\mathrm{ps}}^{-1}$, which may be compared with Fig. 2.

Figure 2

Phonon absorption (top) and emission (bottom) rates, Eqs. (8) and (9), respectively, divided by the squared ratio of Rabi splitting to dressed-state energy splitting, ${\Omega }^2/{\Lambda }^2$, as a function of the dressed-state energy splitting $\Lambda =\sqrt{{\Omega }^2+{\Delta }^2}$; for resonant driving $\Delta =0$ and the scaling is one.

Figure 3

Phonon-induced change in the dressed-state splitting $\Delta E$ [see Eq. (10)] divided by the squared ratio of Rabi splitting to dressed-state energy splitting, i.e., $\Delta E/({\Omega }^2/{\Lambda }^2)$, as a function of the dressed-state energy splitting $\Lambda$. Curves are temperatures 1 (solid), 10 (dashed), and 50 K (dot-dashed).

Figure 4

Final exciton occupation probability following an ARP pulse of the form given in Eq. (12), calculated using the time-dependent Lindblad form given in Eq. (7). Calculated using ${\tau }_0=2$ ps, at temperatures 1 (a), 20 (b) and 50 K (c), for direct comparison with Fig. 2 of Ref. 9. For low temperatures and slow sweeps, the results are very similar; for larger temperatures and higher chirp rates, the limited Markov approximation used in deriving Eq. (7) becomes invalid.

Figure 5

Final exciton occupation probability for the sech pulse [see Eq. (13)] with ${\Omega }_0=\Theta /\tau \text{and}{\Delta }_0=\alpha \tau$ for $\tau =6$ ps at a temperature 4 K. The gray scale and contour labels show ${\log }_{10}(1-P)$, where $P$ is the final occupation probability. Insets show the case without phonon induced dephasing, as in Ref. 42, on the same gray scale (from $-$2 to 0, left inset) and on a gray scale with a larger range (from $-$10 to 0, right inset). Main panel shows the effects of dynamical dephasing on the same pulse shape. Dashed line indicates the condition ${\Omega }_0={\Delta }_0$, which gives the optimal transfer in the absence of dephasing.

Figure 6

Dependence of inversion on strength of phonon coupling in ARP. (a) indicates how for a specific value of $\alpha \text{and}{\Omega }_0$, there is a nonmonotonic dependence of inversion upon phonon coupling. (b) Time evolution of $s_z$ at two values of $q$, relative to its ideal value for adiabatically following the dressed states. (c) Maps where nonmonotonic dependence on $q$ arises. This is determined by two quantities: the gray scale indicates the value of ${\text{Max}}_{0<q<2}(d{s}_z/dq)$, and the solid red line is the boundary where this is zero. The blue dashed line indicates where $d{s}_z{/dq|}_{q=0}=0$. Between these lines, nonmonotonic dependence as seen in (a) occurs. The magenta cross indicates the conditions used for (a) and (b). $T=4$ K, $\tau =5.68$ ps.

Figure 7

Phonon spectroscopy using divided ARP pulse. Main figure: deviation from inversion following the protocol in Eq. (16) (solid), compared with the approximation of Eq. (15) (dashed line), plotted by varying ${\Omega }_w$. Inset: time dependence of $\Delta \left(t\right)$ (solid) and $\Omega \left(t\right)$ (dashed), with parameters ${\Delta }_w=2{\text{ps}}^{-1},{\Delta }_0=-10{\text{ps}}^{-1},t_{\Delta }=T_w/2=3\text{ps},{\tau }_{\Delta }=1.2\text{ps},t_{\Omega }=15\text{ps},{\tau }_{\Omega }=5\text{ps}$, and $T=4$K.

Figure 8

Steady-state value of $s_z$ with driving at $\Omega =0.1{\text{ps}}^{-1}$, with a spontaneous decay rate of $\kappa =2{\text{ns}}^{-1}$, as a function of the detuning, at $T=20$K. (Inset) Comparison of actual phonon density of states $J\left(\omega \right)$ with the values reconstructed by inverting Eq. (19) and from the modified ARP approach, inverting Eq. (15) (parameters for modified ARP as in Fig. 7).

Figure 9

Comparison of the dynamics obtained from the Lindblad [Eq. (7), dashed curves] and non-Lindblad [Eq. (4), solid curves] forms of time-dependent Markovian approximations, for the Gaussian ARP pulse, Eq. (12), with ${\tau }_0=2$ ps, $T=1$ K, $a=30{\mathrm{ps}}^2$, and ${\Theta }_0=5\pi$. For each approach, both the magnitude of the pseudospin (Bloch) vector $\left|s\right|$ (left axis) and inversion $s_z$ (right axis) are shown. At this low temperature, as discussed above, the Markov approximation holds reasonably well, yet the non-Lindblad form leads to unphysical results $\left|s\right|,|s_z|>0.5$.

Figure 10

Final exciton occupation probability following an ARP pulse of the form given in Eq. (12), calculated using the non-Lindblad form [see Eq. (4)], for comparison with the results obtained from the Lindblad form [see Eq. (7)] shown in Fig. 4. The parameters and scale are as in Fig. 4. The results are qualitatively similar for many parameters. However, there is a significant region in the top panel (white) where the density operator obtained from Eq. (4) does not remain positive, and the results become unphysical.