## 3 Results and Discussion
Figure 3: (color online) Measurements (hollow) and calculations (solid) of \(f_{AT}\) as a function of microwave Rabi frequency \(\Omega_{MW}\) for the probe Rabi frequency \(\Omega_{p}=2\pi\times 4.00\) MHz and two different coupling Rabi frequency \(\Omega_{c1}=2\pi\times 4.11\) MHz (red circles) and \(\Omega_{c2}=2\pi\times 2.06\) MHz (blue squares). \(f_{AT}\) shows linear increase as microwave field, and \(f_{AT}\approx\Omega_{MW}\) in a strong microwave coupling range, when \(\Omega_{MW}\gtrsim 2\pi\times 17\) MHz for the case \(\Omega_{c1}\) and \(\gtrsim 2\pi\times 10\) MHz for the case \(\Omega_{c2}\); whereas \(f_{AT}<\Omega_{MW}\), demonstrating the nonlinear dependence on the microwave field. Inset: a room-in of nonlinear region indicated by the gray rectangle. The black thin line indicates the reference \(f_{AT}=\Omega_{MW}\). Figure 2: (color online) Measurements of Rydberg EIT-AT spectra as a function of the coupling laser detuning \(\Delta_{c}\) with the probe/coupling Rabi frequency \(\Omega_{p}=2\pi\times 4.00\) MHz and \(\Omega_{c}=2\pi\times 4.11\) MHz and the indicated microwave power \(P=0.22\) mW (blue), \(0.63\) mW (green) and \(5.01\) mW (red). The frequency of microwave field is set to \(15.21\) GHz coupling Rydberg transition\(|66S_{1/2}\rangle\rightarrow|65P_{1/2}\rangle\). The microwave induced AT splitting, \(f_{AT}\), is extracted with the multipeak Lorentz fittings to the EIT-AT spectra (solid lines). The microwave-free EIT signal (black) is obtained with \(6S_{1/2}\)-\(6 PS_{3/2}\)-\(66S_{1/2}\) three-level scheme, corresponding EIT linewidth \(\gamma_{EIT}=2\pi\times(5.83\pm 0.2)\) MHz. To explain the experimental results in Figure 3, we numerically solve the master equation,  
\[\dot{\rho}=-\frac{i}{\hbar}[H,\rho]+\mathfrak{L} \tag{1}\]  
where \(H\) is the Hamiltonian of the four-level atomic system in Figure 1b, and \(\mathfrak{L}\) is the Lindblad operator that accounts for the decay processes of the atom, the details of the decay are described in Refs. [12; 24]. \[H=\frac{\hbar}{2}\left( \begin{array}{cccc}0&\Omega_{p}&0&0\\ \Omega_{p}&-2\Delta_{p}&\Omega_{c}&0\\ 0&\Omega_{c}&-2(\Delta_{p}+\Delta_{c})&\Omega_{MW}\\ 0&0&\Omega_{MW}&-2(\Delta_{p}+\Delta_{c}+\Delta_{MW})\\ \end{array} \right), \tag{2}\]  
where \(\Delta_{p}\), \(\Delta_{c}\), \(\Delta_{MW}\) are the detunings of the probe laser, couple laser, and the MW field, respectively. Lindblad operator, \(\mathfrak{L}\), is denoted as  
\[\mathfrak{L}=\left( \begin{array}{cccc}\Gamma_{2}\rho_{22}&-\gamma_{12}\rho_{ 12}&-\gamma_{13}\rho_{13}&-\gamma_{14}\rho_{14}\\ -\gamma_{21}\rho_{21}&\gamma_{3}\rho_{33}-\Gamma_{2}\rho_{22}&-\gamma_{23} \rho_{23}&-\gamma_{24}\rho_{24}\\ -\gamma_{31}\rho_{31}&-\gamma_{32}\rho_{32}&\Gamma_{4}\rho_{44}-\Gamma_{3} \rho_{33}&-\gamma_{34}\rho_{34}\\ -\gamma_{41}\rho_{41}&-\gamma_{42}\rho_{42}&-\gamma_{43}\rho_{43}&-\Gamma_{4} \rho_{44}\\ \end{array} \right), \tag{3}\]  
where \(\gamma_{ij}=\)(\(\Gamma_{i}+\Gamma_{j}\))/2 and \(\Gamma_{i(j)}\) is the spontaneous decay rate. We find the steady-state solution for various values of \(\Omega_{c}\), \(\Omega_{p}\) and \(\Omega_{MW}\) by setting \(\dot{\rho}=0\) and obtain the matrix element \(\rho_{12}\) that is proportional to the absorption coefficient of the probe laser [25]. The Beer's absorption coefficient for the probe is \(\alpha=2\pi\frac{Im(\chi)}{\lambda_{p}}\), with \(\chi\) the susceptibility of the medium seen by the probe laser. Considering atomic velocity distribution at temperature T, \(\chi\) is written as \(\chi\) (\(v\))d(\(v\)) \(\propto\)\(\rho_{12}\)\(N(v\))d(\(v\)), and \(N(v)\) is the Maxwellian velocity distribution function at T. In Figure 3, we also present calculated \(f_{AT}\) with filled circles (\(\Omega_{c1}\)) and squares (\(\Omega_{c2}\)).