  We introduce the "gapped coherent state" in the form of a single-photon
source (SPS) that consists of uncorrelated photons as a background, except that
we demand that no two photons can be closer in time than a time gap
$t_\mathrm{G}$. While no obvious quantum mechanism is yet identified to produce
exactly such a photon stream, a numerical simulation is easily achieved by
first generating an uncorrelated (Poissonian) signal and then for each photon
in the list, either adding such a time gap or removing all successive photons
that are closer in time from any photon that is kept than $t_\mathrm{G}$. We
study the statistical properties of such a hypothetical signal, which exhibits
counter-intuitive features. This provides a neat and natural connection between
continuous-wave (stationary) and pulsed single-photon sources, with also a
bearing on what it means for such sources to be perfect in terms of
single-photon emission.
