using QDs [@ref:Rakher_NPhot_2010; @ref:Rakher_PRL_11; @ref:Ates_PRL12] such that the single photon produced by a QD could be frequency translated to another wavelength without destroying its quantum characteristics. Such techniques could bridge the frequency gap between In$_x$Ga$_{1-x}$As QD transitions and those of $^{87}$Rb. In addition, there are QDs composed of GaAs embedded in AlGaAs and InGaAs QDs embedded in AlGaAs which have been shown to emit in the 780 nm - 795 nm region [@ref:Finley_PRB02; @ref:Heyn_APL09; @ref:Akopian_APL_2010]. The optical properties of these QDs are much less well-known compared to standard InGaAs/GaAs QDs and detailed investigations are ongoing. Thus, either use of quantum frequency conversion techniques or proper material choices permit the study of broadband QD photon storage and retrieval in $^{87}$Rb. Notably, photons from a QD have been made to interact with $^{87}$Rb atoms in a recent experiment [@ref:Akopian_NPhot2011]. In this case the atoms acted as a passive medium whose dispersion near the $D_2$ transition was used to reduce the group velocity of the QD photons. In this present work, we are interested in actively manipulating the atoms to controllably store and retrieve the QD photon.

We proceed by determining which hyperfine states to assign to the four levels of the model. 