excited state splitting.

![(Color online) The storage efficiency after gradient ascent optimization as a function of detuning $\delta=\delta_g=\delta_s$. The slight asymmetry about $\delta/\Delta_e=0.5$ is due to the finite accuracy of the numerics (see Appendix \[app:Num\]).[]{data-label="fig:fig7"}](fig_7.eps){width="0.95\columnwidth"}

Another question one could ask is if there is an optimal choice of the optical detuning $\delta=\delta_g=\delta_s$ in the four level system. To address this, we again apply the gradient ascent optimization to the 4L+ configuration described above, but we vary the detuning $\delta$ from $-2\Delta_e$ to $3\Delta_e$ in steps of $\Delta_e/20$. The storage efficiency is plotted in Fig. \[fig:fig7\] as a function of $\delta$. Surprisingly, the efficiencies are smallest near the mid-point between ${\ensuremath{|1\rangle}}$ and ${\ensuremath{|2\rangle}}$ at $\delta = \Delta_e/2$, while the maxima occur near $-\Delta_e$ and $2\Delta_e$. Nonetheless, the change in storage efficiency over the entire range is within $\approx5\%$, and there is not much to be gained or lost by changing the detuning at least for this set of $\tilde{\mu}_{\alpha \beta}$. We have also verified that varying the detuning has a small effect ($\lesssim 5\%$ variation) when using the dipole moments for the relevant $^{87}$Rb transitions. With these mathematical and numerical tools in hand, we can address storage of a quantum dot 