  We develop methods to find the limits to finite-time single photon extraction
from emitter-cavity systems. We first establish analytic upper and lower bounds
on the maximum extraction probability from a canonical $\Lambda$-system before
developing a numeric method to optimise generic output probabilities from
$\Lambda$-systems generalised to multiple ground states. We use these methods
to study the limits to finite-time photon extraction and the wavepackets that
satisfy them, finding that using an optimised wavepacket ranging between a
sinusoidal and exponentially decaying profile can considerably reduce photon
duration for a given extraction efficiency. We further optimise the rates of
quantum protocols requiring emitter-photon correlation to obtain
driving-independent conclusions about the effect of system parameters on
success probability. We believe that these results and methods will provide
valuable tools and insights for the development of cavity-based single photon
sources combining high efficiency and high rate.
