In this chapter, we derive quantum master equations with explicit, microscopic coefficients,
for the systems of interest of a superradiant semiconductor structure: the active
electrons, the electromagnetic field, and the optical crystal vibrations. These vibrations determine
an important retardation in the field propagation (refractive index), and a spectrum
splitting (the Raman effect). For the active electrons we consider three environmental systems:
the quasi-free electrons/holes of the conduction regions, the crystal lattice vibrations
excited by electron transitions, and the free electromagnetic field. For the electromagnetic
field, we consider the absorption by coupling to the conduction electrons/holes, and to the
optical vibrations of the crystal, while these vibrations are damped by coupling with the
valence electron transitions to thermally released states. For the electron-field coupling we
consider the potential derived in chapter 2 from the Lorentz force, while the momentum
difference is supposed to be taken by the crystal lattce. For the coupling of the crystal vibrations
to the electromagnetic field and electron transitions we consider potentials obtained
from the momentum conservation. For the active electrons, we find a quantum master
equation with a Markovian term describing correlated transitions with the environmental
particles, and a non-Markovian term given by the self-consistent field of the environmental
particles. Since the dissipative environment of the electromagnetic field is contained inside
the quantization volume of this field, which is taken as a unit volume, its quantum master
equation includes a dissipative term of a space integral form. For the field mean values,
the dissipation integral, of propagation through the dissipative environment, can be divided
in two parts: an integral from the initial coordinate up to the boundary of the quantum
uncertainty region, taken for a coherent wave, which describes dephasing, and an integral
over the uncertainty region, which describes absorption. Similar equations are obtained for
the optical vibration field. When the vibrational field is eliminated from these equations,
we obtain a frequency splitting, corresponding to the Raman effect, and an absorption rate,
including the absorption of the electromagnetic waves by conduction electrons/holes, and
the absorption of the vibrational waves by the valence electrons, excited by the crystal
deformations in the thermally released states.
Keywords: Total dynamics, reduced dynamics, Gibbs statistics, Fermi-Dirac statistics,
Bose-Einstein statistics, Markovian dynamics, non-Markovian dynamics, dipole moment,
interaction picture, transition operator, two-body potential, self-consistent field, decay,
dephasing, creation-annihilation operators, Fermion operators, anticommutation relations,
commutation relations, optical phonon, field-dressed electron.