  Non-Markovian quantum state diffusion (NMQSD) is a non-relativistic but
otherwise exact theory which expresses the reduced density matrix of an
arbitrary subsystem, interacting linearly with an uncoupled harmonic oscillator
bath, as an average of diadics formed from state vectors which obey stochastic
variational-differential equations. The vacuum radiation field can be
represented as such an oscillator bath, and so this model is in widespread use
in quantum optics. Prior to the development of NMQSD, exact subsystem solutions
could only be obtained in a few special cases (e.g. spin-1/2, harmonic
oscillator). Unfortunately, it has not yet been possible to obtain exact
solutions to new problems using NMQSD due to the difficulty of solving the
variational-differential equations. Here we show that these equations can be
transformed into a pair of coupled nonlinear integrodifferential equations. We
develop exact numerical methods for the integrodifferential equations and show
that solutions can be readily obtained to good accuracy for quite general
subsystems. We exactly solve various examples including tunneling in a double
well representing molecular isomerization or racemization, suppression of
fluorescence from a two-level atom in a band gap, and intermittent fluorescence
from a driven three level system representing electronic states of singly
ionized magnesium.
