show that the memory efficiency remains fairly robust. These results show that with existing technology, QD-generated photons can be reliably interfaced with ultracold atomic ensembles, paving the way for their future use in quantum information as well as opening interesting avenues in the study of hybrid quantum systems.

This paper is arranged as follows: In Sec. \[sec:3L\], we briefly review the physics of photon storage and retrieval in a $\Lambda$-system following the treatment of Gorshkov *et al* [@ref:Gorshkov_PRL07; @ref:Gorshkov_PRA2]. In Sec. \[sec:4L\], we extend this treatment to a four-level atom, which is relevant for broadband photon storage in atomic systems with non-negligible hyperfine structure in the excited state. Section \[sec:QD\] implements the model to study storage and retrieval of a broadband single photon emitted by a quantum dot using a $^{87}$Rb ensemble. In Sec. \[sec:dephase\], deleterious effects resulting from imperfect indistinguishability of the quantum dot photon are discussed. In sec. \[sec:highOD\] we investigate how the efficiencies increase for very high optical depth. The work is concluded and summarized in Sec. \[sec:conc\] and details of the calculations and numerical implementation are discussed in the Appendices A and B. Appendix C discusses the role of four-wave mixing (FWM) in the storage process.

