Microwave electric field detection is widely used in radar, navigation, electromagnetic spectrum monitoring, and other areas^[1–5], and the microwave electric field detection technology based on atomic systems has significant advantages of repeatability, high accuracy, and wide-band response^[6–11]. Based on the Rydberg atomic quantum coherence effect, the microwave electric field measurement can be converted into the measurement of atomic transition spectral frequency, which has higher sensitivity and reliability than the traditional dipole antenna^[12]. Therefore, high-precision spectral frequency calibration is a prerequisite for ensuring detection accuracy in atomic optics experiments^[13]. However, most of the current research on microwave electric field detection based on Rydberg atoms focuses on resonant microwave electric field detection based on the Autler–Townes splitting effect, and non-resonant field detection based on the alternating current (AC) Stark shift effect is rarely studied due to the difficulty of spectral frequency shift detection. The AC Stark shift effect based on Rydberg atoms enables broadband, self-calibrating, and traceable microwave electric field detection^[2,29]. Researchers generally obtain spectral frequencies by means of atomic intermediate state energy level frequency spacing^[14] or wavelength meters^[15]. Yao et al. detected far-detuned fields by applying a near-resonant radio frequency field to magnify the AC Stark shift^[2]. Jiao et al. detected AC Stark shift under weak field action and completed the calibration of an electric field based on a series of exact Floquet-level intersections^[16]. Frequency shift detection methods based on wavelength meter-reading rely on the accuracy of the wavelength meter^[17], resulting in the accuracy of the measurement results needing to be further improved^[18,19].

Laser frequency shifting has important applications in laser control, cold atom preparation, and atom coherence detection^[20–25]. Acoustic-optic modulators (AOMs) use phonon scattering to control photon energy, which can typically shift the frequency of light in the range of KHz to a few GHz^[26]. Two laser beams with a fixed frequency interval can be generated by an AOM, thus completing the frequency shift of the laser^[27]. An efficient frequency shift can be used for spectral frequency detection^[28]. However, this technique is rarely used in Rydberg atomic microwave electric field detection. Therefore, it is necessary to carry out research on the combination of AOM spectral frequency shift detection technology and Rydberg atomic microwave electric field detection technology.

Based on the above research background, we designed and built a double Rydberg atomic excitation system that can support high-precision frequency shift detection and completed the microwave electric field detection. First, two beams of coupling light with a fixed frequency interval were generated by a frequency shifter system with an AOM, and the alkali metal atoms inside the atomic cell were excited, respectively, to complete the double Rydberg atomic excitation. Then, the double electromagnetically induced transparency (double EIT) spectrum was obtained. The double EIT has a fixed frequency interval, which can be used as a precise spectral frequency ruler. This eliminates the reading error from the wavelength meter compared to the traditional wavelength meter-reading scheme, ensuring a more accurate AC Stark shift detection. Combined with this double Rydberg atomic excitation system, we completed the microwave electric field detection based on the AC Stark shift effect of the Rydberg atoms.

The wavelength meter-reading scheme is shown in Fig. 1(a), and in the microwave electric field detection based on the AC Stark shift effect in Rydberg atoms (frequency shift $\mathrm{\Delta }{f}_{\text{Stark}}$), the wavelength meter (WS7-60, High Finesse, absolute accuracy of 60 MHz in 375–800 nm) is used to monitor the frequency changes of the coupling light to complete the frequency shift reading. The spectral frequency shift detection accuracy of this scheme is limited by the detection accuracy of the wavelength meter. In order to improve the accuracy of the AC Stark shift, a frequency shift measurement model based on double Rydberg atomic excitation is proposed, and the double Rydberg atomic excitation scheme is shown in Fig. 1(b). Two coupling lights (0-order beam and 1-order beam) with a fixed frequency interval were generated using an acousto-optic modulator (AOM, Z-532-350-L-D0.2-T-A-K, Qing-Jin, China), and the double Rydberg atomic excitation was completed. It will produce two EIT effects, which can complete double Rydberg atomic excitation, and will show two EIT transmission peaks, which are called double EIT spectra. When the two diffraction light beams have the same intensity, the double EIT spectra have nearly the same loss ratio and linewidth broadening. The frequency interval between the double EIT peaks is equal to the frequency offset of the AOM ($350\mathrm{MHz}\pm 1\mathrm{kHz}$), which provides an accurate optical frequency ruler. Therefore, we also call the double EIT spectra “frequency ruler, FR.” The conversion of the time and frequency can be completed by this “FR,” and the detection of frequency shift can be completed.

Figure 2(a) shows the double Rydberg atomic excitation system we designed, using two cylindrical quartz vacuum cesium cells with a diameter of 25 mm and a length of 50 mm. The probe laser (DLC DL pro, TOPTICA) generates probe lights, one of which enters the saturation absorption frequency stabilization system for frequency locking. This light is divided into two beams by the beam splitter (BS); one is called the probe light (weak light), and the other is called the pump light (strong light). The probe light and the pump light inside the saturation absorption frequency stabilization system are transmitted in reverse coincidence inside the atomic cell 1, and the probe light is detected by an avalanche photodiode detector (PDA36A2, Thorlabs). The coupling light generated by the coupling laser (ECL801, UniQuanta, China) is divided into two coincident beams of coupling light (the green dotted arrow indicates the coincident second coupling light) by the frequency shifter system, and the beams are jointly transmitted with the probe beam in the reverse coincidence inside the atomic cell 2. The two probe beams inside atomic cell 2 are used as reference signals and detection signals, respectively, and are detected by the differential photodetector (PDB210A/M, Thorlabs) at the same time, and this differential optical path can eliminate the background noise and improve the spectral signal-to-noise ratio (SNR). The $1/e^2$ radius of the probe light with a power of 220 µW is 700 µm. The $1/e^2$ radius of the coupling light (before splitting the beam) with a power of 20 mW is 750 µm. Both the probe light and coupling light are linearly polarized light with the same direction of polarization.

The frequency shift system of coupling light mainly consists of two half-wave plates, a polarizing beam splitter (PBS), and an AOM. The coupling light completes the laser beam reduction through the lens group, and it passes through the AOM to produce the zero-order diffraction light (0-order) and the first-order diffraction light (1-order). The two diffracted light beams have a fixed frequency interval, and the laser intensity can be adjusted by the half-wave plate. The orientation of the AOM is adjusted to ensure that the intensity of the 0-order and 1-order beams is the main proportion of the input laser intensity. They are coincident in the same direction inside cell 2 after passing through the PBS. When the frequency shift system we designed is used to complete the transition frequency scanning of the coupling light (${6\mathrm{P}}_{3/2}\to {45\mathrm{D}}_{5/2}$), the frequencies of the 0-order diffraction light and 1-order diffraction light successively scan this atomic transition level.

Figure 2(b) shows the Rydberg atomic transition four-level system, where the probe light is locked at the transition resonance frequency of ${6\mathrm{S}}_{1/2}(\mathrm{F}=4)\to {6\mathrm{P}}_{3/2}({\mathrm{F}}^{\prime }=5)$, and the coupling light is scanned around the transition resonance frequency of ${6\mathrm{P}}_{3/2}({\mathrm{F}}^{\prime }=5)\to {45\mathrm{D}}_{5/2}$. The vector microwave source (N5183B, KEYSIGHT, 9 kHz - 40 GHz) generates a microwave signal that is radiated to the cell region (distance, $d=130\mathrm{mm}$) through a standard gain horn antenna (89901, 1–18 GHz, Ceyear, China).

We fix the frequency of the probe light and scan the frequency of the coupling light (${6\mathrm{P}}_{3/2}\to {45\mathrm{D}}_{5/2}$). By adjusting the half-wave plate, the intensity of the two coupling lights is reduced to 8.65 mW, respectively. The atoms in the cell 2 have an EIT effect, and the oscilloscope (MDO34, Tektronix) will produce an EIT transmission peak. By applying an electric microwave field (non-resonant microwave electric field) that is close to the frequency of the transition, the microwave electric field couples the Rydberg atom. The EIT transmission peak will produce a significant frequency shift $\mathrm{\Delta }{f}_{\text{Stark}}$, which is the AC Stark shift effect^[29]. The resonant transition frequency of ${45\mathrm{D}}_{5/2}\to {46\mathrm{P}}_{3/2}$ is 7.976 GHz, and the AC Stark shift spectrum can be obtained by applying the microwave electric field at a non-resonant frequency (9.976 GHz). $\mathrm{\Delta }{f}_{\text{Stark}}$ is proportional to the square of the microwave electric field $E_{\mathrm{MW}}$^[30]: $$\mathrm{\Delta }{f}_{\text{Stark}}=-\frac{1}{2}\alpha ({\omega }_{\mathrm{MW}})E_{\mathrm{MW}}^2,$$(1)Where $\alpha ({\omega }_{\mathrm{MW}})$ is the atomic polarizability, which can generally be calculated by the transition dipole moment and the detuned microwave electric field frequency that causes the Rydberg atomic transition. Taking ${45\mathrm{D}}_{5/2}$ as an example, the polarizability of the Rydberg state ${45\mathrm{D}}_{5/2}$ at 9.976 GHz is $-595.5\mathrm{MHz}/{(\mathrm{V}/\mathrm{cm})}^2$, which is calculated using the Alkali-Rydberg-Calculator (ARC)^[31]. The theoretical microwave electric field strength ($E_{\mathrm{MW}}$) is^[32]$$E_{\mathrm{MW}}=\sqrt{\frac{30\mathit{P}·g}{d^2}},$$(2)where $d$ is the distance from the horn antenna (gain $g$) to the probe beam, and $P$ is the power of the microwave source. The relation between the frequency shift $\mathrm{\Delta }{f}_{\text{Stark}}$ and the square root of microwave power can be obtained by connecting Eqs. (1) and (2): $$\mathrm{\Delta }{f}_{\text{Stark}}=-\alpha ({\omega }_{\mathrm{MW}})\frac{15g}{d^2}{\left(\sqrt{P}\right)}^2.$$(3)

The signal generator (SDG 1062X, SIGLENT) can produce triangular wave signals with a fixed period (2.00 s, duty cycle 50%), which is connected to the scan frequency module of the coupling laser to complete the sweep of the coupled light. At the same time, a wavelength meter (WS7-60) is used to record the wavelength change of the coupling light. Figure 3(a) shows the EIT transmission peak. The right axis is the coupling light frequency change recorded by the wavelength meter (588.404732–588.405470 THz, before subtracting the base), and the horizontal axis is the time of the oscilloscope. According to the frequency changes of coupling light recorded by the wavelength meter, the frequency shift detection of EIT spectra can be completed, and the frequency shift detection accuracy of this scheme depends on the wavelength deviation accuracy of the wavelength meter. The AC Stark shift spectrum under the action of the microwave electric field with different powers is shown in Fig. 3(b), and the frequency shift $\mathrm{\Delta }{f}_{\text{Stark}}$ increases with the increase of microwave power.

We control the switching of the microwave source by a square wave with a fixed scan period (1.00 s, duty cycle 50%), and transmission peaks without and with microwave electric field coupling are obtained. The AC Stark shift spectrum of the ${45\mathrm{D}}_{5/2}$ state can be obtained by applying a microwave with a non-resonant frequency (9.976 GHz), as shown in Fig. 4(b), and the black curve is the double EIT spectrum as the “FR.” The left half of the other curves (red, green, and purple) is the microwave-off EIT spectrum, and the right half is the microwave-on AC Stark shift spectrum. The transmission peak of the AC Stark shift spectrum produces a frequency shift ($\mathrm{\Delta }{f}_{\text{Stark}}$), and its magnitude can be calibrated using the “FR.” We can find that the frequency shift ($\mathrm{\Delta }{f}_{\text{Stark}}$) increases gradually with the increase of the power of the microwave source.

The fitting curves (without “FR” and with “FR”) between the frequency shift ($\mathrm{\Delta }{f}_{\text{Stark}}$) and the square root of the microwave power are shown in Fig. 5(a). In these two frequency shift detection schemes, the two fitting curves basically reflect the quadratic relationship between the two, and the fitting curves are basically consistent with Eq. (3). The “FR” scheme could detect smaller frequency shifts compared to the wavelength meter-reading scheme. Due to the different detection accuracy of the two schemes, the detected frequency shifts will be slightly different.

The differences between the experimental values and the fitted values are shown in Fig. 5(b). The maximum uncertainty of the wavelength meter-reading scheme (without “FR”) is $\pm 8.09\mathrm{MHz}$ after multiple experimental tests. The frequency shift detection accuracy is limited by the detection accuracy of the wavelength meter, and the uncertainty of the frequency shift detection mainly comes from the wavelength meter’s deviation accuracy, laser frequency jitter, and fitting errors. The minimum detectable frequency shift of the microwave electric field (9.976 GHz) is 7.1 MHz, and the corresponding microwave electric field is 15.4 V/m. The maximum uncertainty of the “FR” scheme is $\pm 1.33\mathrm{MHz}$, and the uncertainty mainly comes from the double EIT spectral fitting error. This system could detect frequency shifts induced by smaller microwave powers. This precise spectral frequency ruler avoids errors from wavelength meters, and high-precision frequency shift detection can be accomplished simpler and faster than with wavelength meters. In this system, the minimum detectable frequency shift of the microwave electric field (9.976 GHz) is 2.27 MHz, and the corresponding microwave electric field is 8.7 V/m.

In this work, we designed a double Rydberg atomic excitation system with an accurate spectral frequency ruler and completed microwave electric field detection based on the Rydberg atomic AC Stark shift effect. The construction of a double Rydberg atomic excitation system was completed using an AOM with a fixed center frequency, and a double EIT spectrum with a fixed frequency interval ($350\mathrm{MHz}\pm 1\mathrm{kHz}$) was generated. Bimodal spectra with a fixed frequency interval can be used as an accurate spectral frequency ruler, allowing high-precision frequency shift detection without the use of a wavelength meter. The frequency shift detection accuracy of the “FR” scheme ($\pm 1.33\mathrm{MHz}$) is 6.08 times better than that of the wavelength meter-reading scheme ($\pm 8.09\mathrm{MHz}$). The minimum frequency shift of the “FR” scheme (2.27 MHz) is 3.13 times better than that of the wavelength meter-reading scheme (7.1 MHz), and the corresponding microwave electric field increases from 15.4 to 8.7 V/m. This study provides a simple and high-precision scheme for frequency shift detection, which has certain reference significance for the microwave electric field detection technology based on the Rydberg atom and is expected to be applied to broadband microwave electric field measurement and metrology^[2,30].