  An optimal single-photon source should deterministically deliver one and only
one photon at a time, with no trade-off between the source's efficiency and the
photon indistinguishability. However, all reported solid-state sources of
indistinguishable single photons had to rely on polarization filtering which
reduced the efficiency by 50%, which fundamentally limited the scaling of
photonic quantum technologies. Here, we overcome this final long-standing
challenge by coherently driving quantum dots deterministically coupled to
polarization-selective Purcell microcavities--two examples are narrowband,
elliptical micropillars and broadband, elliptical Bragg gratings. A
polarization-orthogonal excitation-collection scheme is designed to minimize
the polarization-filtering loss under resonant excitation. We demonstrate a
polarized single-photon efficiency of 0.60+/-0.02 (0.56+/-0.02), a
single-photon purity of 0.975+/-0.005 (0.991+/-0.003), and an
indistinguishability of 0.975+/-0.006 (0.951+/-0.005) for the micropillar
(Bragg grating) device. Our work provides promising solutions for truly optimal
single-photon sources combining near-unity indistinguishability and near-unity
system efficiency simultaneously.
