$\mathcal{E}_{\textrm{in}}(\tau) = \Theta(\tau)\exp(-\tau/2 T_1)/\sqrt{T_1}$ where $\Theta(\tau)$ is the Heaviside step function and $T_1$ is the spontaneous emission lifetime (taken to be 1 ns). We take both fields to be exactly on resonance $\delta_g=\delta_s=0$. As a function of $d$, the storage and total efficiencies are determined by performing gradient ascent for each value of $d$. Details and limitations of the numerical implementation can be found in Appendix \[app:Num\]. The results of the numerical calculation are shown in Fig. \[fig:fig3\] where the storage (blue) and total (red) efficiencies are plotted as a function of $d$. Both curves reach an asymptotic value of $\approx$96.0$\%$ for $d\approx10^3$ limited by the finite computational domain and the instantaneous rise of the quantum field, which is non-physical but chosen for calculational simplicity (see App. \[app:Num\]).

![(Color online) Optimal control pulses for quantum dot photon storage with $d=\{10.0,26.4,78.0,230,678\}$ determined using gradient ascent. The Rabi frequencies are scaled by $\gamma = 2\pi\times3.035$ MHz.[]{data-label="fig:fig4"}](fig_4.eps){width="0.95\columnwidth"}

In addition to determining the efficiencies, it is instructive to see how the optimal control pulse $\Omega(\tau)$ changes as a function of $d$. Figure \[fig:fig4\] shows the optimal control pulse for a few selected values of $d$ and it can be seen that the required peak 