In this section, the interaction between QDs doped within the PBG reservoir and a probe field with slowly varying amplitude is investigated. The total semiclassical Hamiltonian of the system can be written as

Here, *H*_{Q}, *H*_{QF}, *H*_{R}, *H*_{QR} and *H*_{QQ} are the Hamiltonians of the four-level QD, the QD–field interaction, the PBG reservoir, the QD–PBG reservoir interaction and QDs–QDs interaction, respectively [36, 37].

Using Eq. (

2), the equation of motion of the density matrix elements can be written as follows [

38]:

In Eqs. (3-9),

*ρ*_{
ij
} (

*i* *j* = a, b, c or d) are density matrix elements (coherences),

*p* is the strength of quantum interference and is defined by

*p* =

*μ*_{ac}·

*μ*_{ab}/

*μ*_{ac}*μ*_{ab}. In this paper, the maximum quantum interface has been considered, which corresponds to a dipole transition moment

*μ*_{ac} that is parallel to

*μ*_{ab}. This gives

*p* = 1. Here,

*μ*_{ac} and

*μ*_{ab} are the electric dipole moments induced by the transitions |a〉 ↔ |b〉 and |a〉 ↔ |c〉, respectively. Since the two upper energy levels are very close (

*ε*_{bc} = 0.03 eV), it is reasonable to consider

*μ*_{ab} =

*μ*_{ab} =

*μ*.

*Ω* is the Rabi frequency of the probe field, defined as

*Ω* =

*μE*/2

*ħ*, where the dipolar transition moments and external field E are parallel. The parameters

*α* and

*β* are related to the interaction of QDs when the MPC is densely doped [

37]. This interaction is called dipole–dipole interaction (DDI), and its effect was calculated using mean-field theory. The dependency of all decay rates to energy and local density of states can be written as

Here, the function *Z*(*ε*) is called the form factor, which contains the information about the electron–photon interaction and is obtained in reference [39–41].

For simplicity, all parameters have been normalized with respect to (

*Γ*_{b}*Γ*_{c})

^{1/2}/2, which gives a constant value for the resonant energies

*ε*_{ab} and

*ε*_{ac}. Here, γ

_{0} is the decay rate (line-width) for an excited electron in a QD when it is located in a vacuum. The expression of the absorption coefficient is written in terms of density matrix coherence as [

37]:

Here, *N* is the concentration of quantum dots and ε is the energy of the incident laser beam.