  Single-photon sources are a fundamental resource in quantum optics. The
indistinguishability and purity of photons emitted from different sources are
crucial (necessary, essential) properties for many quantum applications to
ensure high-visibility interference between different sources. The state of a
single-photon source is described by the modes occupied by the single light
quanta. Thus the ability to determine the mode structure of a single-photon
source provides a means to assess its quality, compare different sources, and
provide feedback for source engineering. Here, we propose and demonstrate an
experimental scheme that allows for complete characterization of the
spectral-temporal state of a pulsed single-photon source. The density matrix
elements of the single-photon source are determined by spectral interferometry
with a known single-photon reference. Frequency-resolved coincidence
measurements are performed after the unknown single-photon source is interfered
with a single-photon reference pulse. Fourier analysis of the
frequency-resolved two-photon interference pattern reveals the
spectral-temporal density matrix of the broadband single-photon source. We
present an experimental realization of this method for pure and mixed state
pulsed, single-photon sources.
