  Deterministic generation of single photons is essential for many quantum
information technologies. A bulk optical nonlinearity emitting a photon pair,
where the measurement of one of the photons heralds the presence of the other,
is commonly used with the caveat that the single-photon emission rate is
constrained due to a trade-off between multiphoton events and pair emission
rate. Using an efficient and low noise photon-number-resolving superconducting
nanowire detector we herald, in real time, a single photon at telecommunication
wavelength. We perform a second-order photon correlation $g^{2}(0)$ measurement
of the signal mode conditioned on the measured photon number of the idler mode
for various pump powers and demonstrate an improvement of a heralded
single-photon source. We develop an analytical model using a phase-space
formalism that encompasses all multiphoton effects and relevant imperfections,
such as loss and multiple Schmidt modes. We perform a maximum-likelihood fit to
test the agreement of the model to the data and extract the best-fit mean
photon number $\mu$ of the pair source for each pump power. A maximum reduction
of $0.118 \pm 0.012$ in the photon $g^{2}(0)$ correlation function at $\mu =
0.327 \pm 0.007$ is obtained, indicating a strong suppression of multiphoton
emissions. For a fixed $g^{2}(0) = 7e-3$, we increase the single pair
generation probability by 25%. Our experiment, built using fiber-coupled and
off-the-shelf components, delineates a path to engineering ideal sources of
single photons.
