We demonstrate a single-photon collection efficiency of $(44.3pm2.1)%$ from a quantum dot in a low-Q mode of a photonic-crystal cavity with a single-photon purity of $g^{(2)}(0)=(4pm5)%$ recorded above the saturation power. The high efficiency is directly confirmed by detecting up to $962pm46$ kilocounts per second on a single-photon detector on another quantum dot coupled to the cavity mode. The high collection efficiency is found to be broadband, as is explained by detailed numerical simulations. Cavity-enhanced efficient excitation of quantum dots is obtained through phonon-mediated excitation and under these conditions, single-photon indistinguishability measurements reveal long coherence times reaching $0.77pm0.19$ ns in a weak-excitation regime. Our work demonstrates that photonic crystals provide a very promising platform for highly integrated generation of coherent single photons including the efficient out-coupling of the photons from the photonic chip.
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