## 4 Conclusions  
In conclusion, we have investigated Rydberg EIT-AT spectra employing cesium ladder four-level system involving Rydberg \(|665_{1/2}\rangle\), where a 15.21-GHz microwave field coupling the transition of \(|665_{1/2}\rangle\rightarrow|65P_{1/2}\rangle\). Microwave induced AT splitting \(f_{AT}\) is proportional to coupled Rabi frequency \(\Omega_{MW}=\frac{\mu_{MW}E}{h}\), providing a self-calibrated and broadband measurement of microwave electric field [10]. The \(f_{AT}\) dependence on \(\Omega_{MW}\) at a weak microwave field region shows a nonlinear effect, \(f_{AT}<\Omega_{MW}\). We find that the nonlinear region depends on the microwave-free EIT linewidth. When \(\Omega_{MW}\gtrsim 3\gamma_{EIT}\), the measured \(f_{AT}\) displays a good linear behavior, the uncertainty of the measured microwave field is less than 1%, while \(\Omega_{MW}\lesssim 3\gamma_{EIT}\), \(f_{AT}\) enter the nonlinear region, yielding the large measurement uncertainty using this technology. When \(\Omega_{MW}\lesssim\gamma_{EIT}\), it shows a highly nonlinear effect, corresponding uncertainty even up to 40%. Figures 4 and 5 demonstrate that using the weak probe and coupling laser beams, forming the narrow linewidth EIT spectrum, can enlarge the linear region and decrease the measurement uncertainty. The narrow linewidth EIT spectrum also can be obtained with a three-photon EIT scheme [26]. To observe a narrow EIT linewidth, one must decrease the probe and coupling laser power. This will increase the background noise as in Figure 5c. We need to balance the linewidth and spectrum sensitivity in experiments. It is noted that this kind of nonlinear  
Figure 5: (color online) Measurements (symbols) and calculations (solid lines) of Rydberg EIT-AT spectra as a function of the coupling laser detuning \(\Delta_{c}\) and indicated coupling Rabi frequencies for fixed probe \(\Omega_{p}=2\pi\times 4.00\) MHz and two microwave field \(\Omega_{MW}=2\pi\times 30.71\) MHz **(a,b)** and \(2\pi\times 7.73\) MHz **(c–e)**. The black dashed lines display multipeak Lorentz fittings to the EIT-AT spectra for extracting \(f_{AT}\). The deviation \(D_{Err}\) is defined as the ratio \((f_{AT}-\Omega_{MW})/\Omega_{MW}\). In (**a,b**), \(f_{AT}\) locates in the linear region where the measured microwave electric field is independent on the coupling laser, corresponding \(D_{Err}\) is smaller than 1%. In (**c–e)**, \(f_{AT}\) is in the nonlinear region where measured microwave electric field strongly dependent on the coupling laser \(\Omega_{c}\).  
presented in this work and Ref. [12] is inevitable result of the quantum effect, when the \(\Omega_{MW}\) decrease to the critical value, the system become the quantum interference region, as discussed in our previous work [24]. In future work, the modulation and demodulation technique will be used to improve the Rydberg-atom-based field measurements.  
Conceptualization, [PERSON], [PERSON] and L.H.; investigation, [PERSON], [PERSON] and L.H.; software, [PERSON] and Y.X.; validation, [PERSON], Y.X. and J.F.; formal analysis, [PERSON], Y.X. and J.F.; Writing--original draft, L.H.; writing--review and editing, [PERSON]; project administration, [PERSON], [PERSON] and Y.J.; funding acquisition, S.J., [PERSON] and Y.J.  
The work was supported by the National Key R&D Program of China (Grant No. 2017 YFA0304203), the National Natural Science Foundation of China (Grants No. 61675123, No. 61775124 and 11804202), Changjiang Scholars and Innovative Research Team in University of Ministry of Education of China (Grant No. IRT13076), and the State Key Program of National Natural Science of China (Grant No. 11434007 and 61835007).  
The authors declare no conflict of interest.