## 3 Examples: Comparison of Our Model with a Set of Multilevel Structures  
To demonstrate our results from the previous section, we use the example of multilevel atomic structure given by [6], the schematic diagram of which is drawn in Figure 1, to compare the EIT curves of the most typical special cases of this multilevel structure with the result of Equation (20). The structure of the multilevel system is as follows: There are 1 upper energy level state and \(N\) lower energy level states, coupled by \(N\) coherent laser fields between each upper-lower level pair. For convenience of comparison we label the upper energy state as \(|k\rangle\), the first lower energy state as \(|l\rangle\), and the rest of the lower energy states as \(|2\rangle\) to \(|N\rangle\) accordingly. The Rabi frequencies of each coupling field between \(|l\rangle\leftrightarrow|k\rangle\), \(|2\rangle\leftrightarrow|k\rangle\), \(\ldots|N\rangle\leftrightarrow|k\rangle\) are \(\Omega_{p}\), \(\Omega_{2}\), \(\ldots\Omega_{N}\), and the corresponding laser field detunings are \(\Delta_{p}\), \(\Delta_{2}\), \(\ldots\Delta_{N}\). As shown in [6], the linear probe susceptibility \(\chi_{kl}\) of the above multilevel structure can be expressed as  
\[\chi_{kl}(\Delta_{p})=4\pi\mathcal{N}|\mathbf{d}_{lk}|^{2}\frac{1}{\Delta_{p}+ i\gamma_{kl}-\sum\limits_{n=2}^{N}\frac{|\Omega_{n}|^{2}}{\Delta_{p}-\Delta_{n}+i \gamma_{ln}}}, \tag{21}\]which is obtained under the weak probe field limit, and where we have assumed \(\rho_{ll}(t)\approx 1\). In this multilevel structure all the \(\gamma_{ln}\) (\(n\geq 2\)) are set to be zero. For the case study, we now look at the structures of \(n=2\) and \(n=3\). When \(n=2\), the multilevel system is the \(\Lambda\) type configuration, and when \(n=3\), the system turns into the tripod type configuration. As for the parameter setting, we take the value of \(\gamma_{kl}\) as unit and choose \(\Omega_{2}=1\), \(\Omega_{3}=1\) as the typical value setting, together with the assumption \(\Delta_{2}=0\), \(\Delta_{3}=0\) (on-resonance strong field coupling). Then dropping the constant part in both Equation (20) and Equation (21) and adding a minus sign to Equation (20), the imaginary and real part as a function of \(\Delta_{p}\), in cases of \(\Lambda\) type, tripod type, and result of Equation (20), are plotted in Figure 1a,b. As can be seen from Figure 2a,b, when \(|\Delta_{p}|\) is small enough, Equation (20) gives the very close description of EIT curve to that given by \(\Lambda\) type and tripod type from Equation (21). However, when \(|\Delta_{p}|\) gets larger, there is obvious deviation of our estimation from the curves of \(\Lambda\) and tripod configuration. This can be easily understood. Under large detuning the weak probe field is not  
Figure 1: Schematic diagram of the \(N+1\) level atomic system. The upper level state is labeled \(|k\rangle\), and the lower level states are labeled \(|l\rangle\), \(|2\rangle\), \(\ldots|N\rangle\). The energy levels are coupled by \(N\) coherent laser fields on each upper-lower pair. The corresponding Rabi frequencies and field detunings of the coupling fields are \(\Omega_{p}\), \(\Omega_{2}\), \(\ldots\Omega_{N}\) and \(\Delta_{p}\), \(\Delta_{2}\), \(\ldots\Delta_{N}\). Figure 2: The comparison of our estimation from Equation (20) and two special cases from Equation (21) for linear probe susceptibility \(\chi_{kl}\) as function of probe field detuning \(\Delta_{p}\). The \(\Lambda\) type configuration corresponds to the \(n=2\) case in Equation (21), and the tripod type corresponds to the \(n=3\) case in Equation (21). The parameter settings are \(\Delta_{2}=0\), \(\Delta_{3}=0\), \(\Omega_{2}=1\), \(\Omega_{3}=1\) in unit of \(\gamma_{kl}\). The results are separately plotted as: (**a**) The absorption spectra (imaginary part of \(\chi_{kl}\)) as function of \(\Delta_{p}\). Here besides the minus sign, there is a offset of unit 1 in y direction in the plot of our model. (**b**) The dispersion spectra (real part of \(\chi_{kl}\)) as function of \(\Delta_{p}\).