Using BBGKY hierarchies to study the effect of the local field on the rate of radiative relaxation of quantum systems in a dielectric medium

We consider a model of atomic radiators in a dielectric medium and its optical characteristics. In the framework of the microscopic approach and the Bogolyubov–Born–Green–Kirkwood–Yvon method, we obtain a chain of coupled equations for reduced density matrices and correlation operators of atomic particles and modes of a quantized field. We derive a closed system of equations describing the evolution of radiators in the model medium under the action of external radiation. We calculate the effective spontaneous relaxation rate in the presence of the local field produced by the medium. For the first time, we demonstrate the relation between micro and macro approaches to the solution of the problem involving a change in the spontaneous radiation rate in the medium.