## 3 Results and Discussion  
In Figure 2b,c, we present microwave dressed AT spectra with the 3.99 GHz microwave source power fixed at 13 dBm. The horn is placed at \(d=+0.24\) m and \(-0.09\) m, respectively. The microwave dressed AT splitting, \(\gamma_{AT}\), is defined as the peak-to-peak distance of the Rydberg EIT-AT line pair, which is extracted using the multipeak Lorentz fittings, denoted as solid lines of Figure 2b,c. The fittings to the EIT-AT spectra yield \(\gamma_{AT}=2\pi\times 55.50\) MHz in Figure 2b and \(2\pi\times 103.52\) MHz in Figure 2c. As AT splitting is proportional to the the Rabi frequency [9], the microwave dressed AT splitting is  
Figure 1: (Color online) (**a**) Energy level diagram of our caesium four-level system. A weak probe laser \(\lambda_{p}\) is resonant with the lower transition \(|6\text{S}_{1/2},F=4\rangle\rightarrow|6\text{P}_{3/2},F^{\prime}=5\rangle\), and a strong coupling laser \(\lambda_{c}\) scans over the Rydberg transition \(|6\text{P}_{3/2},F^{\prime}=5\rangle\rightarrow|56\text{D}_{5/2}\rangle\). A 3.99-GHz microwave field couples the Rydberg transition \(|56\text{D}_{5/2}\rangle\rightarrow|57\text{P}_{3/2}\rangle\), yielding microwave dressed Rydberg electromagnetically inducted transparency (EIT)-Autler–Townes (AT) spectra. (**b**) Schematic of the experiment. The probe and coupling lasers are overlapped in a counter propagating geometry inside an atomic probe (caesium vapor cell). The probe beam is passed through a dichroic mirror (DM) and is detected with a photodiode (PD). The distance from the center of the atom probe to the guide rail is \(d_{0}=0.3\) m. The horn can be translated along the rail. \(d\) is the distance between the horn and the center of the rail. Not to scale. proportional to the microwave coupling, \(2\pi\gamma_{AT}=\Omega_{MW}\). We can then obtain the microwave electric field amplitude via the equation,  
\[E=\frac{\hbar}{\mu_{MW}}\Omega_{\mathrm{MW}}=2\pi\frac{\hbar}{\mu_{MW}}\gamma_{AT}. \tag{1}\]  
The AT splittings of Figure 2b,c yield microwave electric fields of 2.1 and 3.9 V/cm, respectively. It is noted that in addition to the peaks indicative of microwave dressed AT splitting, there is an additional peak at zero-detuning. This peak is attributed to the magnetic sub-levels of \(56D_{5/2}\) state that are not coupled to the \(57P_{3/2}\) state in the linear polarized microwave field or the component of the microwave field that is perpendicular to the laser polarization. Our experiment is the latter case that indicates that the microwave field has a slight elliptical polarization. The magnitude of this component increases at large \(d\), causing the central peak to reduce in size. For details see [33]. To investigate the distribution of the microwave field emitted by the horn, we perform a series of measurements at horn positions \(d\) between \(d_{max}=\pm 0.4\) m with a step size of 0.03 m. The microwave dressed AT spectrum is displayed with a linear color scale in Figure 2d. We find that (i) the microwave dressed AT splitting is largest at \(d=0\). Here, atoms experience the largest microwave field. (ii) Closer  
Figure 2: (Color online) **(a)**\(\Omega_{MW}=0\) Rydberg EIT spectrum. The main peak at zero detuning corresponds to the \(|6S_{1/2},F=4\rangle\rightarrow|6P_{3/2},F^{\prime}=5\rangle\rightarrow|56D_{5/2}\rangle\) cascade EIT spectrum. The small peak at \(-168\) MHz results from the intermediate-state hyperfine level \(|6P_{3/2},F^{\prime}=4\rangle\), and is used to calibrate the coupling laser frequency. **(b,c)** Microwave dressed Rydberg EIT-AT spectra as a function of the coupling laser detuning, \(\Delta_{c}\), for the horn position at \(d=+\) 0.24 m (**b**) and \(d=-0.09\) m (**c**), respective AT splittings are \(\gamma_{AT}/2\pi=55.50\) MHz and 103.52 MHz. AT splitting, \(\gamma_{AT}\), is defined as the peak-to-peak separation of the microwave dressed AT line pair, which are extracted using the multi-peak Lorentz fittings (solid lines).