We report on a fast, bandwidth-tunable single-photon source based on an epitaxial GaAs quantum dot. Exploiting spontaneous spin-flip Raman transitions, single photons at $780,$nm are generated on-demand with tailored temporal profiles of durations exceeding the intrinsic quantum dot lifetime by up to three orders of magnitude. Second-order correlation measurements show a low multi-photon emission probability ($g^{2}(0)sim,0.10-0.15$) at a generation rate up to $10,$MHz. We observe Raman photons with linewidths as low as $200,$MHz, narrow compared to the $1.1,$GHz linewidth measured in resonance fluorescence. The generation of such narrow-band single photons with controlled temporal shapes at the rubidium wavelength is a crucial step towards the development of an optimized hybrid semiconductor-atom interface.
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