In recent decades, atom-based metrology has had a tremendous impact on science, technology, and human life, such as optical atomic clocks^[1,2] and the global positioning system, and highly sensitive position-resolved magnetometers.^[3,4] Atom-based field measurement has clear advantages over other field measurement methods, due to their invariable level structure that can be used as a calibration-free criterion. Rydberg atom, a highly excited atom with principal quantum number , has a large electric polarizability, scaling , and big microwave-transition dipole moments, scaling .^[5] These properties make Rydberg atoms good candidates for measuring an external electric field.^[6–14] The electromagnetically induced transparency (EIT)^[15] involving Rydberg states that provides a non-destructive optical detection of Rydberg states^[16] has been widely investigated in recent years. Rydberg EIT is employed to measure the electric field of electromagnetic radiation with a large dynamic range,^[6] demonstrating a number of applications, such as measurements of microwave fields,^[6–10] millimeter waves,^[11] static electric fields,^[12–14] sub-wavelength imaging of microwave electric field distributions,^[17,18] field inhomogeneities,^[19] and back scattered electric field of identification tag.^[20]

The radio-frequency (RF) electric field with a frequency greater than 1 GHz has been measured based on an Autler–Townes (AT) effect^[21] where the RF field couples the nearby Rydberg levels.^[7] The AT splitting is known as an AC Stark effect. The AT splitting γ_AT, defined as the peak-to-peak distance of AT spectrum, is proportional to the amplitude of RF-field E_RF, i.e., , where Ω_RF is the RF-field Rabi frequency, μ_RF is the RF-field coupled dipole matrix element, and is the reduced Planck constant. Rydberg EIT is employed to measure the AT splitting, and further the RF electric field. The Rydberg EIT-AT spectrum yields a direct International System of Units (SI) traceable, self-calibrated, and broad band measurement of microwave electric field, which has the capability to realize measurements on a fine spatial resolution. The approach of EIT/AT-based electric-field measurement has recently been investigated by several groups around the world.^[6–9,11]

Rydberg EIT is employed to measure the RF-field induced AT splitting. The EIT linewidth would strongly affect the accuracy of field measurements, which was discussed theoretically in ^[9]. In this work, we provide experimental evidence that agrees with the theoretical calculation of ^[9]. We present Rydberg EIT-AT spectra in a cesium room-temperature vapor cell, where a 15.21-GHz RF field couples the Rydberg transition for producing the AT splitting γ_AT. We investigate the dependence of γ_AT on EIT linewidth by varying the coupling and probe laser Rabi frequency. The residuals are used to describe the deviations between calculations and measurements of the EIT-AT spectra. It is found that the smaller coupling and probe laser Rabi frequencies can decrease the uncertainty of the RF electric field measurements in a weak RF-field regime.

We perform Rydberg EIT-AT experiments in a vapor cell, and the experimental setup and a relevant four-level Rydberg-EIT-AT scheme are shown in Figs. 1(a) and 1(b). A weak probe laser at a wavelength of λ_p = 852 nm is produced by an external cavity diode laser (Toptica DLpro), and a strong coupling laser at a wavelength of λ_c = 510 nm is produced with a commercial laser (Toptica SHG110). The probe and coupling lasers overlap and counter-propagate through a room-temperature cylindrical cesium vapor cell with a length of 25 mm and a diameter of 25 mm. The probe frequency is stabilized to the transition using a polarization spectrum technique, while the coupling laser frequency is ramped over the Rydberg transition. The EIT spectrum is detected by measuring the transmission of the probe laser using a photodiode (PD) after a dichroic mirror (DM).

Fig. 1.

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Fig. 1. (a) Schematic diagram of the experiment. The coupling (λc = 510 nm) and probe (λp = 852 nm) beams counter-propagate through a cesium vapor cell along the y-axis. The horn placed 86 cm from the cell (not scaled) emits an RF electric field with a frequency ∼15.21 GHz for coupling the Rydberg transition and producing an EIT-AT spectrum. The horn is set such that the RF electric field is linearly polarized along the z-axis which is parallel to the polarizations of the probe and coupling laser beams. The probe beam passing through a cesium cell and a dichroic mirror is detected with a photodiode. The PBS indicates polarization beam splitter. (b) The energy level diagram of the cesium four-level system. The probe laser λp is resonant with the lower transition, and the coupling laser λc is scanned through the Rydberg transitions. The applied RF electric field couples the transition , yielding a Rydberg EIT-AT spectrum.

The probe and coupling beams have Gaussian waist and in the cell, respectively. The corresponding Rabi frequency can be calculated with the formula , where is the probe (coupling) laser power, is a beam waist, is the permittivity in vacuum, and c is the speed of light. The indicates the transition dipole moment of the probe (coupling) coupled transition. Both the coupling and probe laser Rabi frequencies affect the EIT linewidth γ_EIT. Here, we vary the probe/coupling laser power to change the Rabi frequency with a series of neutral attenuators.

A 15.21-GHz RF electric field, produced with a function generator (Agilent N5183B) and emitted by a horn antenna, is applied transversely to the laser beam through the cell, where the RF field couples the Rydberg transition and yields splitting of the EIT spectrum, e.g., Rydberg EIT-AT spectrum. The calculated radical dipole moment μ_RF is 3942 ea₀. The AT splitting γ_AT is defined as the peak-to-peak distance of EIT-AT spectrum, which is used to measure the RF electric field. The RF electric field is linearly polarized along the z-axis, parallel to the polarization of the probe and coupling lasers. This configuration yields the two-peak EIT-AT spectral profile. The RF electric field is expressed as ,^[22] where P_RF is the power of the microwave source, g is a gain of the antenna, and d is a distance from the antenna to the cell. The calculated far-field condition d₀ is about 65 cm in this work, and the horn antenna is set to 86-cm away from the cell, which is larger than the far-field condition d₀. The experimental region is surrounded by the microwave-absorbing material to avoid unwanted reflections.

Rydberg EIT is a quantum interference process in which the absorption of a weak probe laser, interacting resonantly with an atomic transition, is reduced in the presence of a coupling laser, which (near-) resonantly couples the upper probe level to a Rydberg state.^[16] Rydberg EIT is employed to perform an optical detection of Rydberg level and AT splitting induced by an RF electric field that is coupled to the nearby Rydberg states.

In Fig. 2(a), we present Rydberg EIT-AT spectra with an up level Rydberg state and a 15.21-GHz microwave electric field coupling transition. The probe and coupling beam Rabi frequencies are and , respectively. We obtain an EIT peak at two-photon detuning δ =0, as shown by the black symbols in Fig. 2(a). The EIT linewidth , extracted by a Lorentz fitting to the experimental data, is close to the natural linewidth of the intermediate state. The EIT spectrum is calibrated by the hyperfine level Rydberg EIT signal, which is not shown here; see ^[12] for more details. When we apply an RF electric field that drives the Rydberg transition, the EIT line peak decreases and separates into two peaks as the RF field increases, known as the AT splitting γ_AT. The AT splitting γ_AT is defined as the peak-to-peak distance of EIT-AT spectrum and extracted using the multipeak Lorentz fittings to the data. Figure 2(a) displays EIT-AT spectra with two indicated RF powers. The measured γ_AT values are and for microwave power of 0.32 mW and 1.99 mW, respectively. The γ_AT shows a linear increase with a square root of RF power, amplitude of the RF field, which is the chief gauge for measuring RF electric field based on the Rydberg atom.

Fig. 2.

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Fig. 2. (a) Measurements (symbols) and calculations (solid lines) of Rydberg EIT-AT spectra as a function of the coupling laser detuning Δc for fixed probe/coupling laser Rabi frequency and and an indicated microwave power PRF = 0.32 mW (blue) and 1.99 mW (red). The RF electric field with frequency 15.21 GHz couples the nearby Rydberg states and . The AT splitting γAT is obtained by multipeak Lorentz fittings to the Rydberg EIT-AT spectra. The RF field-free EIT signal (black) is obtained with a three-level scheme, and a Lorentz fitting to the data yields an EIT linewidth . (b) The residuals R between experiment and theory. The RMS values of the residuals are 0.052 and 0.079 for PRF = 0.32 mW and 1.99 mW, respectively, showing good agreement with the experimental data. The small fluctuations near the center of the line are attributed to the broadening that is mainly induced by the inhomogeneity of the RF field, see text.

For comparison with the theory, we simulate the EIT-AT spectrum by numerically solving the master equation 1 where H is the Hamiltonian of the four-level atomic system in Fig. 1(b), and is the Lindblad operator that accounts for the decay processes of the atom. The details of the decay term, , are described in our previous work^[23] and ^[9]. The simulations of EIT-AT spectra are displayed with solid lines in Fig. 2(a). In Fig. 2(b), we present the residuals between experiment and theory for further comparison, and the corresponding root mean square (RMS) values of residuals are respectively 0.052 and 0.079 for RF power P_RF = 0.32 mW and 1.99 mW, showing good agreement. There are some small but noticeable structures for the case of P_RF = 1.99 mW near the center of the line, which is attributed to the line broadening caused by the inhomogeneity of the RF field. Atoms interacting with the laser field experience different fields due to the inhomogeneity of the RF field in the cesium cell, leading to the broadened line and even multi-peak EIT-AT spectra.^[22]

For a weak RF electric field, it is noted that the measured AT splitting γ_AT strongly depends on the EIT linewidth γ_EIT that is relevant to the coupling and probe laser Rabi frequency. To investigate the dependence of EIT-AT splitting γ_AT on the EIT linewidth, we keep the RF Rabi frequency Ω_RF fixed and vary the Rabi frequency of probe/coupling laser to do a series of γ_AT measurements. As an example, in Fig. 3(a), we plot γ_AT as a function of Ω_p while keeping and (corresponding to ) fixed. The measured at small Ω_p and shows a decrease with Ω_p. The dependence of γ_AT on Ω_c is also plotted in Fig. 3(b), and γ_AT again displays a decrease with Ω_c. We attribute this dependence of γ_AT in Fig. 3 to the Rydberg EIT linewidth γ_EIT, which is used to measure the RF-induced AT splitting. In a weak field regime, when , RF induced AT splitting cannot be distinguished at all, whereas as , AT splitting is distinguishable but measured γ_AT would depend on the EIT linewidth. For the case here, , the EIT linewidth is expressed as 2 RF-free EIT γ_EIT depends on Ω_c and Ω_p. The other possible reason leading to is AT splitting being the nonlinear dependence regime, an intrinsic behavior of EIT or AT spectrum, which is beyond the scope of this work. See ^[9] for details.

Fig. 3.

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	Fig. 3. (a) Measurements of γAT as a function of the probe Rabi frequency Ωp for fixed and . (b) Similar measurements analogous to (a) as a function of the coupling Rabi frequency Ωc for fixed Ωp = 2π ×3.99 MHz.

In order to investigate how the probe/coupling laser Rabi frequency affects the AT splitting γ_AT and further the RF electric field measurement, we demonstrate EIT-AT spectra (symbols) and simulations (solid line) with a fixed RF field and two different probe laser Rabi frequencies in Fig. 4 to analyze the uncertainty in the field-measurements. The RF Rabi frequency is set to be , corresponding to the electric field amplitude of 0.47 V/m. For the smaller in Fig. 4(a), the EIT-AT spectrum has two peaks that separate well, and the multipeak Lorentz fitting yields AT splitting , which is closer to the Ω_RF. We define the deviation to describe the relative error of RF field measurement, and D_Err = 1.68% in Fig. 4(a). For the larger in Fig. 4(b), two peaks in EIT-AT spectrum are just distinguishable, measured is −18.89% that is less than the expectation. We attribute this to an increased probe laser Ω_p that results in the broadened EIT linewidth. When the EIT linewidth approaches the AT splitting, the two peaks of EIT-AT spectrum become indistinguishable, see Fig. 4(b). The EIT-AT spectrum even turns into one peak when γ_EIT is larger than γ_AT. In the weak field regime, the EIT-AT splitting γ_AT decreases with Ω_p, which causes big uncertainty D_Err.

Fig. 4.

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Fig. 4. Measurements (symbols) and calculations (solid lines) of Rydberg EIT-AT spectra as a function of the coupling laser detuning Δc for fixed and at two different probe Rabi frequencies (a) and (b) 4.51 MHz. The deviation DErr is obtained with the ratio ( . The bottom panel shows the residuals R between the measured and calculated EIT-AT spectra, and the RMS values of residuals are (a) 0.063 and (b) 0.050.

Closely inspecting Fig. 4, we find that the signal-to-noise ratio of the EIT-AT spectrum for small Ω_p in Fig. 4(a) is less than that for large Ω_p in Fig. 4(b). This is because the small probe laser power results in a large background noise. At the bottom of Fig. 4, we plot the residuals between the measurements and calculations of the EIT-AT spectra. The RMS values of the residuals are 0.063 and 0.050 for (Fig. 4(a)) and 4.51 MHz (Fig. 4(b)), respectively. Figure 4 demonstrates that the narrow EIT linewidth using the small probe Rabi frequency could increase the accuracy of a weak RF-field measurement. However, further decrease of the probe Ω_p will lead to large background noise, see Fig. 4(a), which may also yield large measurement uncertainty. Furthermore, we also do the test of the coupling Rabi frequency as Fig. 4 and obtain similar results, as the coupling Rabi frequency broadens the EIT linewidth as demonstrated in Eq. (2). In experiments, we need to balance the EIT linewidth and spectrum sensitivity in the RF-field measurements.

In conclusion, we have investigated Rydberg EIT-AT spectra employing a cesium ladder four-level system involving Rydberg , in which a 15.21-GHz RF field couples the nearby Rydberg states transition . An RF field results in the AT splitting of the EIT resonance γ_AT which is proportional to the RF-coupled Rabi frequency , providing a calibration-free and broad band measurement of RF electric field.^[7] However, in a weak RF field regime, the AT splitting strongly depends on the RF-free EIT linewidth that is related to the coupling/probe laser Rabi frequency. It is found that the measured AT splitting γ_AT decreases with probe and coupling Rabi frequency . We have attributed this behavior to two causes. The first one is broadened Rydberg EIT linewidth due to the laser Rabi frequency. When the EIT linewidth is , two peaks of EIT-AT spectrum become indistinguishable that leads to smaller γ_AT. The experimental results are analyzed by the four-level model of ^[9]. The second one is the nonlinear dependence of the EIT-AT splitting on the RF field in the weak RF field regime. For RF electric field measurements, using narrow linewidth EIT would compress the nonlinear zoom of EIT-AT splitting and increase the field measurements accuracy, but it is accompanied by the disadvantage of low signal-noise ratio of EIT spectrum. The possible solution of increasing signal-to-noise ratio is using the Mach–Zehnder interferometer detection^[24] or a frequency modulation technique.^[25] In future work, the squeezed probe beam will be used to decrease the shot noise and to improve the Rydberg-atom-based field measurements.