  A simplified model of an initially excited oscillator as a quantum system
interacting with a large number of oscillators acting as a reservoir has been
developed in this work. All these oscillators are in their ground state
uncoupled each other and at the limit of the weak coupling between the system
and the reservoir. This system could be an oscillator excited in a microcavity
that interacts with the vacuum's electromagnetic field at zero temperature.
This work's primary goal is to obtain the system's density matrix's exact
solution in these conditions. The general approach calculates all oscillators'
evolution as a single isolated entity using the evolution operator. Starting
from a total initial state that can be factored between the system and the
reservoir, the evolution is unitary, and the partial trace is taken in all the
degrees of freedom of the environment to obtain the density matrix of the
system at any instant of time; this procedure requires diagonalizing
Hamiltonian. The results are evaluated for a reservoir of N=1000 oscillators,
particular values of the coupling force, and ohmic order of the spectral
density, contrasted with the corresponding Markovian solution described in
section [2.3.1].
