  Semiconductor quantum dots embedded in optical cavities are promising
on-demand sources of single photons. Here, we theoretically study single photon
emission from an optically driven two-photon Raman transition between the
biexciton and the ground state of a quantum dot. The advantage of this process
is that it allows all-optical control of the properties of the emitted single
photon with a laser pulse. However, with the presence of other decay channels
and excitation-induced quantum interference, on-demand emission of the single
Raman photon is generally difficult to achieve. Here we show that laser pulses
with non-trivial shapes can be used to maintain excitation conditions for which
with increasing pulse intensities the on-demand regime is reached. To provide a
realistic picture of the achievable system performance, we include
phonon-mediated processes in the theoretical caluclations. While preserving
both high photon purity and indistinguishability, we find that although based
on a higher-order emission process, for realistic system parameters on-demand
Raman photon emission is indeed achievable with suitably tailored laser pulses.
