  This series of papers has two broader aims: 1) Construct a theory for
multi-partite open quantum systems comprising several layers of structure with
self-consistent back-actions. Develop the graded influence action formalism
\cite{BehHu10,BH11} to account for the influences of successive sub-layers on
the dynamics of the variables of interest. 2) Apply these methods to the study
of atom-field-medium interactions and highlight their merits over conventional
methods. We consider a system of $N$ harmonic oscillators, modeling the
internal degrees of freedom (idf) of $N$ neutral atoms (A), interacting with a
quantum field (F), scalar here, for simplicity, altered by the presence of a
dielectric medium (M). In this paper we use the coarse-grained and stochastic
effective actions in the influence functional formalism to derive the
stochastic equations for the reduced density matrices of the dynamical
variables in the successive layers of structure. The word `graded' refers to
the specific ordering of the coarse-graining procedures. Three layers of
coarse-graining are performed, firstly, integrating over the common bath of the
dielectric oscillators results not only in the appearance of necessary
dissipative properties of the dielectric but also essential nuanced features
such as nonMarkovian spatial correlations in the dielectric. Secondly,
integrating over the medium variables as a whole results in a
dielectric-modified quantum field, the influence of the medium on the quantum
field manifesting through a frequency-dependent permittivity function. Finally,
integrating over this dielectric-altered quantum field which interacts with the
idfs of the atoms yields an influence action. From it we obtain the stochastic
equation of motion which describes the nonequilibrium stochastic dynamics of
the idf of the atoms interacting with a dielectric medium-modified quantum
field.
