## 2 The Model  
Imagine we have an initial atomic system, and by applying the strong coupling fields to the system, the state population is modified. We then add the weak probe field, and the state population will be slightly modified again. Thus we see, the difference made by the adding-on of the probe field is the optical response of the atomic system to the probe field. Correspondingly in calculation, we apply the strong coupling fields as perturbation to the Hamiltonian of the original atomic system in master equation, and do the first order perturbation calculation. Following that, we do the second order perturbation calculation, where we throw in the weak probe field, the effect of which will be used to deduce the expression of the atomic system's linear optical susceptibility. We point out that in the treatment of the perturbation, when including the strong coupling fields and weak probe field as a whole, the different orders of nonlinear susceptibility can all be looked into; however, the features of EIT we study only involve the one particular term corresponding to the linear probe susceptibility. As described above, treating all applied fields as a perturbation to the atomic system, we take the strong coupling fields as the first order perturbation and the weak probe field as the second order; the Hamiltonian of the system can be written as  
\[\hat{H}=\hat{H}_{0}+\lambda\hat{V}_{c}+\lambda^{2}\hat{V}_{p}, \tag{1}\]  
where \(\lambda\) is the parameter characterizing the strength of the perturbation, ranging between 0 and 1. \(\hat{H}_{0}\) is the Hamiltonian of the free atomic system, \(\hat{V}_{c}\) is the interaction energy corresponding to the strong coupling fields, and \(\hat{V}_{p}\) is the interaction energy corresponding to the weak probe field. We then seek a solution of the density matrix element in the power series of \(\lambda\) up to second order  
\[\rho_{nm}=\rho_{nm}^{(0)}+\lambda\rho_{nm}^{(1)}+\lambda^{2}\rho_{nm}^{(2)}. \tag{2}\]  
As discussed before, we expect the lowest order contribution of \(\hat{V}_{p}\) to be related to the atomic media's optical response to the probe field. Starting with the general form of master equation  
\[\dot{\rho}_{nm}=-\frac{i}{\hbar}[\hat{H},\rho]_{nm}-\gamma_{nm}\rho_{nm}, \tag{3}\]  
where \(\gamma_{nm}\) is the decay rate of \(\rho_{nm}\), we plug in the expression of \(\hat{H}\) from (1), noting that \([\hat{H}_{0},\rho]_{nm}=(E_{n}-E_{m})\rho_{nm}\), we then have  
\[\dot{\rho}_{nm}=-i\omega_{nm}\rho_{nm}-\frac{i}{\hbar}[\lambda\hat{V}_{c}+ \lambda^{2}\hat{V}_{p},\rho]_{nm}-\gamma_{nm}\rho_{nm}. \tag{4}\]  
In the above we define \(\omega_{nm}=\frac{E_{n}-E_{m}}{\hbar}=\omega_{n}-\omega_{m}\) as the resonant transition frequency between energy levels \(|n\rangle\) and \(|m\rangle\) of the atom. Then, when only adding in the strong coupling fields in interaction, from Equation (4) we have  
\[\rho_{nm}^{(1)}=\frac{-i}{\hbar}\int_{-\infty}^{t}[\hat{V}_{c}(t ^{\prime}),\rho^{(0)}]_{nm}e^{(iu\omega_{nm}+\gamma_{nm})(t^{\prime}-t)}dt^{\prime} \tag{5}\] \[\rho_{nm}^{(2)}=\frac{-i}{\hbar}\int_{-\infty}^{t}[\hat{V}_{c}(t ^{\prime}),\rho^{(1)}]_{nm}e^{(iu\omega_{nm}+\gamma_{nm})(t^{\prime}-t)}dt^{ \prime}. \tag{6}\]  
On the other hand, in the second round we turn on the weak probe field and only look at the modification it brings to \(\rho_{nm}^{(2)}\), because a modification of probe field to \(\rho_{nm}^{(1)}\), which is  
\[\frac{-i}{\hbar}\int_{-\infty}^{t}[\hat{V}_{p}(t^{\prime}),\rho^{(0)}]_{nm}e^ {(iu\omega_{nm}+\gamma_{nm})(t^{\prime}-t)}dt^{\prime}, \tag{7}\]means the effect of the weak probe field without presence of strong coupling fields, and this is obviously not the case of EIT we are considering.