This paper and planned subsequent ones have two broad aims. First, we construct a theory for multipartite open quantum systems comprising several layers of structure with self-consistent backactions. We develop the graded influence action formalism [Behunin and Hu, Phys. Rev. A 82, 022507 (2010); Phys. Rev. A 84, 012902 (2011)] to account for the influences of successive sublayers on the dynamics of the variables of interest. Second, we 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 (IDFs) 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 term ``graded'' refers to the specific ordering of the coarse-graining procedures. Three layers of coarse graining are performed. First, integrating over the common bath of the dielectric oscillators results not only in the appearance of necessary dissipative properties of the dielectric but also in essential nuanced features such as non-Markovian spatial correlations in the dielectric. Second, 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 allows us to derive an equation of motion which describes the nonequilibrium stochastic dynamics of the IDFs of the atoms interacting with a dielectric medium-modified quantum field. In the future we plan to calculate the nonequilibrium covariant matrix elements of the correlation functions of the IDFS of
N atoms in a dielectric-altered quantum field, which are useful for probing many basic quantum information issues, such as the entanglement dynamics in AFM interactions.
This third paper in this series continues the investigation of atom-field interactions in the presence of a conductor or a dielectric medium, focusing on quantum information related basic issues such as decoherence and entanglement. Here we consider the entanglement between two atoms with internal degrees of freedom modeled by a harmonic oscillator, with varying separations between them and varying distances between them and a conducting surface. These are configurations familiar in the Casimir-Polder effect, but the behavior of atom-surface entanglement is quite different from the well-studied behavior of field-induced forces. For one, while the attractive force between an atom and a conducting surface increases as they come closer, the entanglement between the atom and the quantum field actually decreases as the atom gets closer to the conductor, as shown in \cite{Rong,AFD2}. We show how different factors play out, ranging from the coupling between the atoms and the field to the coupling between the atoms, going beyond the weak coupling restrictions often found necessary in the literature. Gathering our results for the entanglement dependence on each variable concerned, we can provide a spatial topography of quantum entanglement, thus enabling a visualized understanding of the behavior of quantum field-mediated entanglement. In particular we can quantify the definition of a three-dimensional \textit{entanglement domain} between the two atoms, how it varies with their coupling, their separation and their distances from the conducting surface, and for practical applications, how to exercise effective control of the entanglement between two atoms by changing these parameters. Our findings are expected to be useful for studies of atom-field-medium interactions in vacuum and surface physics.
In Part I [1], we presented a quantum theory of third-harmonic generation (THG) under two-photon absorption in a two-level atomic medium. We studied the time evolution of the third-harmonic field and its coherence properties by neglecting the depletion of the pump power. In this paper we extend the theory by taking the depletion of pump power into account. The density matrix for the third-harmonic field is derived. The study of the growth rate of the density matrix in time shows the quantum mechanical effects more clearly. The effects of detuning in the two-photon interactions and atomic motion are pointed out. The results are compared with the earlier results and differences are brought out.