  Non-Markovian quantum state diffusion (NMQSD) is an exact method for
calculating the reduced density matrix of an arbitrary subsystem interacting
linearly with the radiation field. Applications of the theory have however been
few due to the intractable nature of the variational-differential NMQSD
evolution equation. Recently, we argued that the variational-differential
equation can be rewritten as an integrodifferential equation which can be
readily solved numerically. This manuscript provides an explicit derivation of
the modified equations. Applications to intermittent fluorescence in
$^{24}$Mg$^+$ are discussed in detail. Earlier speculations that quantum jumps
occur on all time scales are verified on a picosecond timescale. We show that a
plot of the probability density of the signal vs signal strength shows the two
characteristic peaks associated with the bright and dark manifolds, and that
the ratio of the areas under the peaks is 16 as observed experimentally. We
also show that the shape of this distribution is sensitive to bath memory, but
has a mathematical form common to both the Markovian and non-Markovian cases.
