## 3 Results and Discussion  
In Figure 2, we present the measurements of four-time averaged Rydberg EIT-AT spectra with an up level \(66S_{1/2}\) Rydberg state, and a 15.21-GHz microwave electric field coupling \(|66S_{1/2}\rangle\to|65P_{1/2}\rangle\) transition. The Rabi frequencies for the probe and coupling beams are \(\Omega_{p}=2\pi\times 4.00\) MHz and \(\Omega_{c}=2\pi\times 4.11\) MHz, respectively. The black curve in Figure 2 shows a field-free Rydberg EIT spectrum. EIT linewidth \(2\pi\times(5.83\pm 0.2)\) MHz (close to the natural linewidth \(2\pi\times 5.2\) MHz of intermediate state) is extracted by Lorentz fitting to the EIT spectrum. EIT spectrum is calibrated with the hyperfine level \(|6P_{3/2}F^{\prime}=4\rangle\) Rydberg EIT signal, not shown in here, see ref [14]. It is noted that the observed hyperfine EIT interval of \(6P_{3/2}(F^{\prime}=5)\) and \((F^{\prime}=4)\) is \(\eta\times 251\) MHz = 168 MHz, here \(\eta=\lambda_{p}/\lambda_{c}-1\) is the Doppler factor. The EIT line splits into two-peak spectra, known as AT effect, when we apply a microwave electric field that couples \(|66S_{1/2}\rangle\to|65P_{1/2}\rangle\) transition. AT splitting, \(f_{AT}\), defined as the peak-to-peak separation of the line pair, is extracted using the multipeak Lorentz fittings to the spectrum, as denoted with the solid lines in Figure 2. The measured \(f_{AT}\) are \(2\pi\times 12.90\) MHz and \(2\pi\times 39.26\) MHz for microwave power of 0.63 mW and 5.01 mW, respectively. The measured AT splitting, \(f_{AT}\), shows increasing with the microwave electric field. For further investigating the dependence of EIT-AT splitting \(f_{AT}\) on the microwave field, we do a series of measurements by varying the microwave power. Figure 3 demonstrates \(f_{AT}\) dependence on the microwave Rabi frequency \(\Omega_{MW}\) with probe \(\Omega_{p}=2\pi\times 4.00\) MHz and two coupling Rabi frequency \(\Omega_{c1}=2\pi\times 4.11\) MHz (red circles) and \(\Omega_{c2}=2\pi\times 2.06\) MHz (blue squares). It is seen that \(f_{AT}\approx\Omega_{MW}\) and linearly increases with microwave at the region \(\Omega_{MW}\gtrsim 2\pi\times 17\) MHz, in a strong field region, for both cases. According to \(f_{AT}\approx\Omega_{MW}\) and \(\Omega_{MW}=\frac{\mu_{MW}E}{\hbar}\), the microwave electric field \(E\) can be measured [10]. However, in a weak region, measured \(f_{AT}\) is less than \(\Omega_{MW}\) and denotes a nonlinear dependence, see the inset of Figure 3. The nonlinear behavior is attributed to the EIT effect when probe and coupling fields interact with atoms at a weak-coupling region [24; 25]. It is found, from Figure 3's inset, that the nonlinearity of \(f_{AT}\) strongly depends on the microwave-free Rydberg EIT condition, i.e., the EIT linewidth. \(\gamma_{EIT}\) is expressed as \((\Omega_{c}^{2}+\Omega_{p}^{2})/\Gamma_{eg}\) with \(\Gamma_{eg}\) the spontaneous decay rate of intermediate state. EIT linewidth depends on the probe and coupling laser Rabi frequency. For the case \(\Omega_{c1}\) = \(2\pi\times 4.11\) MHz in the inset of Figure 3, the \(\gamma_{EIT}=2\pi\times 5.83\) MHz, the \(f_{AT}\) begins to show the nonlinear effect at \(\Omega_{MW}\simeq 2\pi\times 17\) MHz, that is \(\sim\)\(3\gamma_{EIT}\), and demonstrate the highly nonlinear behavior at \(\Omega_{MW}\lesssim\gamma_{EIT}\). For the smaller coupling Rabi frequency, \(\Omega_{c2}\) = \(2\pi\times 2.06\) MHz and \(\gamma_{EIT}=2\pi\times 4.35\) MHz, we see the nonlinear effect appear at \(\Omega_{c1}\simeq 2\pi\times 12.0\) MHz (\(\sim\)\(3\gamma_{EIT}\)). At a weak-field region, the AT peaks are artificially pulled closer together for simultaneously that leads to \(f_{AT}<\Omega_{MW}\). The phenomenon about nonlinear dependence are similar with the theoretical simulation in Ref. [12]. At a nonlinear region, \(f_{AT}\lesssim\Omega_{MW}\), yielding large uncertainty and prevents the precise measurement of the microwave field.