memory in order to be a viable source for long distance quantum information processing. Here, we investigate the storage and retrieval of a broadband, QD-generated single photon with a 1 ns lifetime in an ultracold, dense ensemble of $^{87}$Rb atoms. Taking previously measured experimental parameters, we find that the total efficiency ($\eta_{tot} = \eta_{s} \times \eta_{r}$, where $\eta_s$ and $\eta_r$ are the storage and retrieval efficiencies) can exceed 28$\%$ for storage and backwards retrieval of a photon with a 1 ns lifetime in a $^{87}$Rb ensemble with an on-resonance optical depth of 150. Because the bandwidth of the QD photon can approach the excited-state hyperfine splitting in $^{87}$Rb, this result was obtained by extending the $\Lambda$-system theory of Gorshkov *et al* [@ref:Gorshkov_PRL07; @ref:Gorshkov_PRA2] to a four-level system. Using the gradient ascent approach outlined in Ref. [@ref:Gorshkov_PRA4], we determine the maximum efficiency as well as the optimal control pulse for photon storage. We find that the control pulse can be easily generated using a 12 mW laser diode in contrast to the broadband, Raman-based scheme in Ref. [@ref:Reim_NatPhot10] where a mode-locked laser is required. Finally, we consider the effects of excess dephasing and spectral wandering of the QD optical transition and 