The microwave (MW) field can be measured by the Autler–Townes (AT) splitting of the electromagnetically induced transparency (EIT) spectrum in the Rydberg atomic system; however, the EIT-AT splitting method fails in weak MW fields. We used the amplitude modulation of the MW field to resolve the EIT-AT splitting in weak MW fields. The EIT-AT splitting interval can be directly obtained, and the minimum detectable MW strength is improved by six times compared with the traditional EIT-AT splitting method. The proposed method is more intuitive and convenient for measuring the strength of weak MW fields in practical applications.

Atom-based quantum sensors have recently been successfully developed.^{1,2} Rydberg atoms with at least one electron excited to a large principal quantum number orbit are quite sensitive to external electromagnetic fields.^{3} Electromagnetically induced transparency (EIT) is a quantum interference phenomenon caused by the coherent destructive effect between two possible excitation pathways, and it is extremely sensitive to atomic energy level shifts and splittings.^{4} Owing to the large transition dipole moments and the microwave (MW) band-matched transition frequencies between adjacent Rydberg states,^{3} the Rydberg EIT has been successfully used to accurately measure the MW electric field.^{5–10}

The MW field results in the splitting of the Rydberg EIT spectrum, a phenomenon called EIT-Autler–Townes (EIT-AT) splitting. When the strength of the MW field *E*_{MW} is large enough to create an obvious AT splitting in the Rydberg EIT spectrum, i.e., when *E*_{MW} falls in the AT regime, one can obtain the splitting interval directly in the EIT spectrum. The spectral interval of the AT splitting Δ*f*_{m} is proportional to the Rabi frequency of the corresponding MW transition. Therefore, *E*_{MW} can be determined using a resonant MW field,^{5}

where *λ*_{p} and *λ*_{c} are the wavelengths of the probe light and the coupling light, respectively; Δ*f*_{m} is the measured AT splitting interval; $D=\lambda p\lambda c$ accounts for Doppler mismatch of the probe and coupling lasers when the probe laser is scanned;^{11,12}*ℏ* is Planck’s constant; and *℘* is the electric dipole moment corresponding to the MW transition. The Rydberg-atom based MW electrometry achieves a direct, traceable, and calibration-free International System of Units (SI) measurement of the MW field strength because it is related to Planck’s constant and the invariant nature of atoms directly.^{5} In the AT regime, the lower limit of the strength of the MW field is ∼5 mV/cm in a room temperature vapor cell.^{13} When Δ*f*_{m} is less than the EIT linewidth Γ_{EIT}, one cannot obtain Δ*f*_{m} in the EIT spectrum directly. *E*_{MW} can be characterized by detecting the reduction in the transparency of the probe light near the Rydberg EIT resonance.^{5}

The method of characterizing the strength of the microwave field by detecting the change of the transparency near the Rydberg EIT resonance in weak microwave fields is feasible and has achieved good results, which can reach a minimum detectable microwave field strength of 8 *μ*V/cm.^{5} This method needs to be compared with the transmittance of the probe light near the Rydberg EIT resonance without the microwave field to obtain the microwave electric field strength. Through the EIT-AT splitting measurement method, the microwave electric field strength can be characterized by only one measured spectrum without comparing with the transmittance of the probe light near the Rydberg EIT resonance with no microwave field. Therefore, it is meaningful to expand the application of the EIT-AT splitting measurement method in weak fields, and many research groups have achieved good results. Simons *et al.* varied the MW frequency from resonance to detuning to extend Δ*f*_{m} under the same MW electric field strength.^{14} Jia *et al.* utilized an auxiliary microwave-dressed Rydberg EIT-AT system to extend the low bound of the direct SI-traceable microwave electric field strength measurement.^{15} Liao *et al.* dramatically reduced the residual Doppler effect by laser-cooling atoms to achieve the detectable microwave field strength as small as 100 *μ*V/cm.^{16} Shaffer *et al.* proposed a three-photon read-out scheme to reduce the residual Doppler shifts for narrowing the EIT linewidth to the order of the Rydberg natural linewidth.^{17,18} Meanwhile, the Rydberg atomic receiver for microwave communication by using the amplitude modulation of the microwave field is developing rapidly recently,^{19–22} and our work shows that the amplitude modulation of the microwave field can not only be used in Rydberg microwave communication but also can improve the lower limit of the detectable microwave field strength compared with the traditional EIT-AT splitting method.

In this study, we developed a scheme that adopts the amplitude modulation of the MW field to resolve the AT splitting in weak MW fields. The modulated MW amplitude results in a periodic shift in the Rydberg atomic energy levels, which is equivalent to the frequency modulation of the EIT-AT splitting spectra to obtain the dispersion error signals. Then, the *E*_{MW} can be characterized by the spectral properties of the dispersion error signals.

Figure 1 shows the basic principle of the proposed MW amplitude modulation scheme. The relevant atomic energy levels of ^{87}Rb are shown in Fig. 1(a): 5*S*_{1/2}(*F* = 2), 5*P*_{3/2}(*F*′ = 3), 53*D*_{5/2}(*F* = 4), and 54*P*_{3/2}(*F* = 3). Figure 1(b) shows the calculations for the schematic diagram of MW sensing with the MW amplitude modulation scheme of Rydberg EIT spectroscopy. The MW Rabi frequency, Ω_{MW}, is so weak that we cannot obtain the AT splitting interval directly from the Rydberg EIT spectrum, as indicated by the red dotted line in Fig. 1(b). Then, the sinusoidal amplitude modulation signal is applied to the MW field to shift the Rydberg atomic levels periodically, which corresponds to the frequency modulation of the EIT-AT splitting spectra. During the modulation, Ω = Ω_{0} − ΔΩ cos(*ωt*), where Ω_{0} is the central Rabi frequency, ΔΩ is the amplitude of the modulation, and *ω* is the frequency of the modulation. The transmission signal of the probe light *I*(*v*, Ω) can be written as

where *v* is the frequency of the probe light and Ω_{MW} is the MW Rabi frequency. Since the influence of the higher-order term is much less than the first-order term, we neglect the effect of the higher-order term in Eq. (2). The dispersion error signal $\u2202Iv,\Omega MW\u2202\Omega MW$ can be obtained by demodulating the EIT spectra using a lock-in amplifier (LIA), as shown by the blue dashed line in Fig. 1(b). The interval between the two zero-crossing points of the dispersion error signal is defined as Ω_{eff} = *s* × Ω_{0}, where *s* is related to ΔΩ and *ω*. Figure 1(b) shows that Ω_{eff} corresponds to the interval of intersections of the black and red lines. This indicates that the interval between two zero-crossing points of the dispersion error signal can be used to characterize Ω_{MW} in weak MW fields. In practical applications, it is easier to apply amplitude modulation to the MW by using an MW switch with a Transistor-Transistor Logic (TTL) signal, as shown in Fig. 1(c).

The experimental setup is illustrated in Fig. 1(c). The experiments were performed in a room temperature cylinder rubidium vapor cell with a diameter of 25 mm and a length of 75 mm. A probe and coupling laser are overlapped and counter-propagated through the cell. The frequency of the weak probe beam (*λ*_{p} = 780 nm) was locked at the transition of ^{87}Rb [5*S*_{1/2}(*F* = 2) to 5*P*_{3/2}(*F*′ = 3)] by the Zeeman modulation saturation absorption spectrum.^{23} The strong coupling laser (*λ*_{c} = 480 nm) was locked at the resonance transition of ^{87}Rb [5*P*_{3/2}(*F* = 3) to 53*D*_{5/2}(*F* = 4)] by the Zeeman modulation of the Rydberg EIT.^{24,25} The linewidths of all lasers are estimated to be less than 500 kHz by the linewidth of the Rydberg EIT with 494 kHz in cold atom samples.^{24,25} The Rabi frequencies of the probe light and the coupling light were Ω_{p} = 2.1 × 2*π* MHz and Ω_{c} = 1.5 × 2*π* MHz, respectively. The power of the probe beam passing through the cell was detected using a photodiode (PD) and recorded with an oscilloscope.

The MW field was provided by a signal generator (8340 B, Keysight Technologies) and then radiated to the vapor cell using a horn antenna. The frequency of the MW field in this work was selected as ∼14 GHz so that the ^{87}Rb transition [53*D*_{5/2}(*F* = 4) to 54*P*_{3/2}(*F* = 3)] could be resonantly driven with a dipole moment *℘* = 1619e*a*_{0}. The MW field has polarization along the z-axis and is parallel to the polarization of the probe and coupling lights. We applied TTL amplitude modulation to the MW field. The EIT-AT spectrum can be obtained by linearly scanning the probe laser frequency using an acousto-optic modulator around the EIT resonance.^{24,25} The MW amplitude modulation frequency can be varied from 1 to 99 kHz and the MW amplitude modulation depth can be varied from 10% to 90%, which is defined as the ratio of the difference between the MW power corresponding to the high and low levels of the amplitude modulation signal to the MW power corresponding to the high levels.

Figure 2 shows the experimental Rydberg EIT spectra and the corresponding dispersion error signal obtained by MW amplitude modulation corresponding to the calculations for the schematic diagram shown in Fig. 1(b). It can be found that Ω_{MW} is too weak to create an obvious AT splitting in the Rydberg EIT spectrum with Ω_{MW} = 4.5 × 2*π* or Ω_{MW} = 2.2 × 2*π* MHz. In these cases, the intensity of the EIT signal at resonance will decrease and the width of the EIT signal will broaden with an increase in Ω_{MW}. It was revealed that the EIT-AT splitting can be obtained by spectrum analysis, which depends on the specific physical model and cannot be obtained *in situ*. When we apply the amplitude modulation of the MW field by the TTL signal, Ω_{MW} can be switched between 4.5 × 2*π* and 2.2 × 2*π* MHz periodically. We can obtain the corresponding dispersion error signal by LIA, and it can be found that there are two zero-crossing points near the EIT resonance in the dispersion error signal whose interval is defined as Δ*f*_{AM}. It is confirmed that Δ*f*_{AM} can be obtained directly from the dispersion error signal, though Ω_{MW} is too weak to create an obvious AT splitting in the Rydberg EIT spectrum. This result indicates that the MW amplitude modulation scheme is a novel method that can be used to directly characterize Ω_{MW} by using Δ*f*_{AM} where Δ*f*_{m} cannot be distinguished. Next, we studied the relationship between Δ*f*_{AM} and the MW amplitude modulation parameters, as well as *E*_{MW}, and explored whether it can be used to characterize *E*_{MW}.

First, we set the output strength of the MW field (*E*_{MW}) as 10.8 mV/cm and the amplitude modulation depth as 10%. Then, we changed the amplitude modulation frequency every 10 kHz from 10 to 99 kHz. We obtained the Δ*f*_{AM} from the dispersion error signal directly and recorded the effective MW output strength simultaneously. The results revealed that the effective strength of the MW field remains almost unchanged, and the fluctuation range of Δ*f*_{AM} is within 0.1% when the AM frequency varies from 10 to 99 kHz. The results indicate that the Δ*f*_{AM} is very robust against variations in the amplitude modulation frequency.

Then, we altered the amplitude modulation depth from 10% to 90% by changing the control voltage of the amplitude modulation. We fixed the set output strength of the MW field and amplitude modulation frequency with 10 kHz, and we recorded the effective output power of the MW field *P*_{eff} and the Δ*f*_{AM} at different modulation depths. The results revealed that Δ*f*_{AM} decreases as the MW amplitude modulation depth increases from 10% to 90%, and Δ*f*_{AM} shows a good linear relationship with $Peff$. When the amplitude modulation depth was less than 10%, the amplitude of the modulated signal of the EIT spectrum is so weak that a lock-in amplifier cannot demodulate the dispersion error signal. This method cannot work at this time. The results confirm that Δ*f*_{AM} can be obtained with a broad range of amplitude modulation depths, and the unmodulated original *E*_{MW} can be obtained with a known MW amplitude modulation frequency and MW amplitude modulation depth.

We adopted the linear region of EIT-AT splitting, that is, when the interval of AT splitting is linear with the microwave Rabi frequency Ω_{MW}, to calibrate the microwave electric field strength. Then, we used a calibrated linear attenuator to obtain the required microwave electric field strength. Holloway *et al.* made the comparisons between the *E*_{MW} obtained by the Rydberg EIT-AT method and those obtained with a conventional power meter, and their results indicated that the EIT-AT splitting method can be used to calibrate the microwave field strength.^{26} A series of measured EIT-AT spectra and the corresponding dispersion error signals when *E*_{MW} varied from 17.1 to 0.43 mV/cm are shown in Figs. 3(a) and 3(b), respectively. The results reveal that the EIT-AT splitting interval Δ*f*_{m} gradually becomes indistinguishable when *E*_{MW} < 2.7 mV/cm in the traditional EIT-AT splitting method. Interestingly, Δ*f*_{AM} is still very clear with *E*_{MW} as small as 0.43 mV/cm, as shown in Fig. 3(b).

The interval Δ*f*_{m} and Δ*f*_{AM} corresponding to *E*_{MW} obtained from Fig. 3 are shown in Fig. 4, and the calculated results are obtained by a theoretical four-level model using the AtomicDensityMatrix package,^{27,28} and the weak probe condition was adopted for a quick calculation. The interval Δ*f*_{m} exhibits linear behavior when *E*_{MW} > 4.0 mV/cm. Furthermore, it decreases dramatically with a decrease in *E*_{MW} when *E*_{MW} < 4.0 mV/cm and becomes indistinguishable until *E*_{MW} < 2.7 mV/cm. The linear relationship fails because Δ*f*_{m} is comparable to or smaller than Γ_{EIT}. The calculated results of Δ*f*_{m} reveal the same relationship with *E*_{MW} and are consistent with the experimental results, as indicated by the black solid line in Fig. 4. The interval Δ*f*_{AM} can be obtained directly from the MW amplitude-modulated Rydberg EIT spectrum. More interestingly, the Δ*f*_{AM} shows a monotonic relationship with *E*_{MW}, and it can still be measured when *E*_{MW} is as small as 430 ± 37 *μ*V/cm. The lower limit of *E*_{MW} is increased by approximately six times compared with Δ*f*_{m} in a room temperature atomic vapor. The calculated results of Δ*f*_{AM} are qualitatively in agreement with the experimental results, as indicated by the solid red line in Fig. 4. More complete numerical calculations are needed to explain the small difference between the experimental and theoretical Δ*f*_{AM} in the weak *E*_{MW} in the future.

In conclusion, we developed a scheme that adopts the amplitude modulation of the MW field to resolve the EIT-AT splitting in weak MW electric fields. The spectral interval between the two zero-crossing points near the EIT resonance of the dispersion error signal was used to measure the strength of the MW electric field. The minimum detectable MW strength can be as small as 430 ± 37 *μ*V/cm in a room temperature vapor cell, which is improved by six times compared with the traditional EIT-AT splitting method. This method has an easier and more practical data reading and analysis process because the zero point of the signal in the experiment can be read directly, therefore eliminating the need to fit the spectrum to obtain the peak position or width. This MW amplitude modulation technique provides an important approach for enhancing the sensitivity of Rydberg-atom based microwave electrometry that can be easily introduced in practical applications.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

This work was supported by the Funding National Key Research and Development Program of China (Grant Nos. 2017YFA0304900 and 2017YFA0402300), the Beijing Natural Science Foundation (Grant No. 1212014), the National Natural Science Foundation of China (Grant Nos. 11604334 and 11604177), the Key Research Program of the Chinese Academy of Sciences (Grant No. XDPB08-3), the Open Research Fund Program of the State Key Laboratory of Low-Dimensional Quantum Physics (Grant No. KF201807), and the Fundamental Research Funds for the Central Universities.