  We study the impact of mechanical vibrations on the performance of the
photonic "hourglass" structure, which is predicted to emit single photons
on-demand with near-unity efficiency and indistinguishability. Previous
investigations neglected the impact of vibrational modes inherent to this
quasi-1D geometry, relying instead on a three-dimensional bulk assumption for
the phonon modes. However, it has been shown that phonon decoherence has a much
stronger impact in 1D structures as compared with bulk media. Here, we
surprisingly demonstrate the robustness of the photonic hourglass design,
achieving close-to-unity indistinguishability even by incorporating a detailed
description of the vibrational modes. We explain this unexpected result in
terms of the large Purcell enhancement of the hourglass single-photon source,
which eliminates the negative effect of phonons. Our findings highlight the key
role of high-Q optical cavities in mitigating the detrimental effect of phonon
decoherence, even for structures of reduced dimensionality.
