although an integrated source of heralded single photons has been recently realized [@kartik2012].

Generation of pure single-photon states requires a quantum nonlinear source, i.e. ideally a two-level emitter that is prepared in its excited state and produces exactly a single quantum of light per pulse when relaxing to its ground state. One of the measures of SPS efficiency is determined by the degree of antibunching in its second-order correlation function at zero-time delay, $g^{(2)}(0)<g^{(2)}(\tau)$, whereby an ideal SPS satisfies $g^{(2)}(0) \to 0$ [@loudon_book]. Tremendous progress in solid-state SPS has been made since the development of artificial quantum emitters, such as semiconductor quantum dots or atomic-like defects in solids (for a review, see Ref. [@focus_sps] and references therein). In practice, cavity quantum electrodynamics (CQED) is the most straightforward way of obtaining single-photon nonlinear behavior and hence a SPS on demand [@kuhn_review], allowing to efficiently access the underlying anharmonicity introduced by a single atomic-like emitter through a high-finesse resonator [@tian92; @imamoglu99]. In solid-state CQED, the most promising results on single-photon emission have been originally achieved with III-V materials, by using non-resonantly excited semiconductor quantum dots coupled to photonic microcavity modes, either in the Purcell regime [@Michler2000; @Pelton2002; @Santori2002], in strong light-matter coupling 