  High-quality sources of single photons are of paramount importance for
quantum communication, sensing and metrology. To these ends, resonantly excited
two-level systems based on self-assembled quantum dots have recently generated
widespread interest. Nevertheless, we have recently shown that for resonantly
excited two-level systems, emission of a photon during the presence of the
excitation laser pulse and subsequent re-excitation results in a degradation of
the obtainable single-photon purity. Here, we demonstrate that generating
single photons from self-assembled quantum dots with a scheme based on
two-photon excitation of the biexciton strongly suppresses the re-excitation.
Specifically, the pulse-length dependence of the multi-photon error rate
reveals a quadratic dependence in contrast to the linear dependence of
resonantly excited two-level systems, improving the obtainable multi-photon
error rate by several orders of magnitude for short pulses. We support our
experiments with a new theoretical framework and simulation methodology to
understand few-photon sources.
