  We present a numerical path-integral iteration scheme for the low dimensional
reduced density matrix of a time-dependent quantum dissipative system. Our
approach simultaneously accounts for the combined action of a microscopically
modelled pure-dephasing type coupling to a continuum of harmonic oscillators
representing, e.g., phonons, and further environmental interactions inducing
non-Hamiltonian dynamics in the inner system represented, e.g., by Lindblad
type dissipation or relaxation. Our formulation of the path-integral method
allows for a numerically exact treatment of the coupling to the oscillator
modes and moreover is general enough to provide a natural way to include
Markovian processes that are sufficiently described by rate equations. We apply
this new formalism to a model of a single semiconductor quantum dot which
includes the coupling to longitudinal acoustic phonons for two cases: a)
external laser excitation taking into account a phenomenological radiative
decay of the excited dot state and b) a coupling of the quantum dot to a single
mode of an optical cavity taking into account cavity photon losses.
