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Abstract
We present an atom-based approach for determining microwave electric and magnetic fields by using a single rubidium vapor cell in a microwave waveguide. For a 87Rb cascade three-level system employed in our experiment, a weak probe laser driving the lower transition, 5S1/2→5P3/2, is first used to measure the microwave magnetic field based on the atomic Rabi resonance. When a counter-propagating strong coupling laser is subsequently turned on to drive the Rydberg transition, 5P3/2→67D5/2, the same probe laser is then used as a Rydberg electromagnetically induced transparency (EIT) probe to measure the microwave electric field by investigating the resonant microwave dressed Autler-Townes splitting (ATS). By tuning the hyperfine transition frequency of the ground state using an experimentally feasible static magnetic field, we first achieved a measurement of the microwave electric and magnetic field strength at the same microwave frequency of 6.916 GHz. Based on the ideal relationship between the electric and magnetic field components, we obtained the equivalent microwave magnetic fields by fitting the inversion to the measured microwave electric fields, which demonstrated that the results were in agreement with the experimental measurement of the microwave magnetic fields in the same microwave power range. This study provides new experimental evidence for quantum-based microwave measurements of electric and magnetic fields by a single sensor in the same system.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
With the development of atomic physics and quantum precision measurement technology, atom-based metrology standards for fundamental physical quantities, such as time, length, gravity, and magnetic fields play an important role in science, technology, and daily life. Atom-based detection techniques have also been successfully used to improve the measurements of microwave electric fields [1–3] and microwave magnetic fields [4–6]. Compared to the conventional detection techniques, these atom-based microwave detection techniques present unique advantages including a higher sensitivity, accuracy, resolution and reproducibility owing to the invariant atomic properties and self-calibrated capability. In particular, these detection techniques can translate the microwave field strength measurement into a higher accuracy Rabi frequency measurement via atomic constants and directly link microwave quantities with the International System of Units (SI) [7,8].
Most of atom-based microwave detection techniques demonstrated to date have mainly focused on the studies of single microwave parameter characteristics. For example, the resonance microwave electric field induced ATS of the Rydberg EIT spectrum in atomic vapor cells has been demonstrated as a practical approach for electric field measurement over a wide frequency ranging from megahertz (MHz) to terahertz (THz) [9–12]. Due to their large polarizability and transition dipole moments [13], Rydberg atom-based electric field sensing exhibits performance capabilities for the measurements of the electric field amplitude [14–17], polarization [10,18], phase [19,20], angle-of-arrival [21], power [22], as well as many applications for multifrequency microwave fields [23], spectrum analyzers [24,25], communication receivers [26–31], microwave and terahertz imaging [32–34], voltage measurement [35], antenna finite range gain [36], etc. The atom-based microwave magnetic field detection technique has also been demonstrated based on the Rabi resonances induced by a phase modulated resonant microwave field in atomic vapor cells [5,6], and was successfully explored for various applications including microwave magnetic field stabilization [37], microwave imaging [38], microwave power [39], and materials characterization [40].
Atomic vapor cells serve as key components of quantum systems and sensors have been studied extensively in electromagnetic field measurements and other applications. In the last decade, Several integration of atomic vapors with photonic structures have provided an effective path toward the mass fabrication and commercial deployment of atom-based quantum devices and sensors, several integrating and miniaturizing vapor cells with the micro- and nanoscale have been demonstrated such as the atomic cladded waveguide [41–43], hollow core waveguide [44,45], slot waveguide [46], hollow core photonic crystal fibers [47], tapered optical fibers [48], and anodic bonding cells [49,50]. In parallel to these efforts, laser spectroscopy techniques employing miniaturized vapor cells are attracting growing attention, and creating new opportunities in miniaturized and integrated optical frequency Refs. [51], magnetic and electric field sensors [52–54], and atomic clocks [55]. However, most of these practical quantum sensors and miniaturized quantum systems remain limited to single parameter measurements. Thus, the combination of the multifunctional characteristics of miniaturized atomic vapor cell sensing probes and multi-parameter quantum measurement techniques will be the future development trend for versatile quantum standards and future portable quantum sensing applications. The multi-parameter atom-based quantum measurement techniques and systems have become extremely important and need to be further studied; for example, to the best of our knowledge, the same frequency multi-parameter microwave measuring technique based on a single vapor cell has not been investigated yet.
Differing from the microwave electric field detection technique, the atom-based microwave magnetic field detection technique is performed only with hyperfine transitions of the atomic ground states, rather than the Rydberg states. In addition, both detection techniques depend on different optimal experimental conditions such as the resonant microwave frequency, microwave power, room temperature of vacuum vapor cells for the electric field, and temperature-controlled vapor cells coated with paraffin on the inner surface for the magnetic field. The Rydberg EIT-ATS method is only suitable for weak microwave electric field measurement, while the measurement of the microwave magnetic field based on Rabi resonances strength requires a higher microwave power. Therefore, the same frequency measurement of microwave electric and magnetic fields using the same vapor cell as a single sensing probe not only enables the extension of the application of the atom-based microwave sensing technique into a more demanding electromagnetic environment, but also may increase the strength dynamic range of the existing single parameter measurement technique by the fixed relationship between electric and magnetic components.
In this study, we first propose and demonstrate measurements of the microwave electric and magnetic fields in a single vapor cell via atom-based measurement techniques. Under a DC magnetic field, the microwave magnetic field is measured on the non-resonance ground state hyperfine transition based on the atomic Rabi resonance in the rubidium vapor cell. We then measure the same frequency of the microwave electric field by Rydberg EIT-ATS using the same probe laser. The equivalent microwave magnetic fields are obtained from the fitting inversion of the measured microwave electric fields, and compared to the experimental measurement results of the microwave magnetic fields in the same microwave power range.
2. Basic measuring method
The microwave field strength is determined from the measurement of the microwave Rabi frequency. For the microwave electric field, the strength is proportional to the microwave Rabi frequency Ω or frequency interval of ATS Δf, which is described by the following:
For the microwave magnetic field, when the Zeeman sublevels alkali atoms resonate with a phase modulated microwave field, the amplitude of the Rabi resonance signal P0 β can be expressed as a function of the phase modulation frequency ωm [56]:
3. Experimental setup
The experimental setup used for measurement of the microwave electric and magnetic fields and the relevant energy levels of 87Rb are schematically shown in Fig. 1(a) and Fig. 1(b). The experiment was performed in a C-band rectangular microwave waveguide with inner dimensions of 299.86 mm × 34.85 mm × 15.80 mm. A rectangular rubidium vapor cell with length of 50 mm was mounted at the internal center of the waveguide, and the cell temperature could be controlled with braided windings wrapped around the waveguide. The whole waveguide was centrally mounted inside three-dimensional solenoid coils that can produce a DC magnetic field up to 500 G, which can cause the continuously tunable frequency range to exceed 2 GHz.
Fig. 1. (a) Schematic of the experimental setup. P: polarizer; HWP: half waveplate; PD: photodetector. (b) Energy-level diagram of 87Rb atoms used for microwave electric and magnetic fields measurements.
A weak laser with wavelength of 780.24 nm was provided by an extended-cavity diode laser (DL100, Toptica), and the laser frequency was locked to the transition of 5S1/2 (F = 2)→5P3/2 (F'=3) using the saturated absorption spectroscopy technique. For the microwave magnetic field measurement, the laser as pump and probe beam passed through the rubidium vapor cell and a pair of 6 mm diameter holes perpendicular to the short side of the microwave waveguide. The probe laser beam had a diameter of 1 mm and power of 200 μW. The phase modulated resonance microwave field was generated by a commercial low noise microwave signal generator (E8257D, Keysight) and irradiated into the microwave waveguide by a C-band open-ended rectangular waveguide antenna. A polarizer and a half wave plate (HWP) were used to ensure the linear polarization of the output beams and consistency with the microwave field polarization. The output signal passing though the waveguide was detected using a low-noise Si photodetector, and then sent to a fast-Fourier-transform (FFT) spectrum analyzer for measurement of the Rabi resonance signal amplitude. The Rabi resonance lineshapes were obtained by varying the frequency of the microwave phase modulation for any achievable microwave magnetic resonance frequencies under an applied DC magnetic field. For the microwave electric field measurement, the probe and another coupling lasers with the same linear polarization were overlapped and counter-propagated through the same vapor cell. The weak probe laser as the probe beam was still locked to the transition of 5S1/2 (F = 2)→5P3/2 (F'=3), the laser parameters remained the same as the magnetic field. The strong coupling laser at a wavelength of 480 nm was produced from a commercial laser (SHG110, Toptica) with power of 60 mW and a diameter of 800 μm, and couples the transition of 5P3/2 (F'=3)→67D5/2. A ladder type EIT three-level configuration was formed by a ground state, 5S1/2, an excited state, 5P3/2 (F'=3), and a Rydberg state, 67D5/2, we scanned the frequency of the coupling laser across the corresponding transition, yielding the EIT spectroscopy. The microwave electric field without phase modulated from the same microwave signal generator couples the 67D5/2→68P3/2 Rydberg transition, and its polarization direction was parallel to the polarization of the both lasers. The ATS signal was detected with the same photodetector and recorded with an oscilloscope.
4. Experimental results and discussions
For 87Rb, the nine Zeeman transition frequencies can be observed via the double resonance method under the tuning DC magnetic field. In our experiment, we choose the end Zeeman level, σ+ transition 5S1/2 (F = 1, mF=+1)→5P3/2 (F'=2, mF'=+2) to measure Rabi resonance, because this transition can shift faster to 6.916 GHz at the lower DC field, and effectively minimize the influence of the larger DC magnetic field on the Rabi resonance lineshape. First, we only turned on the frequency-locked probe laser, and used a phase modulated microwave field to excite the Zeeman transitions and generate the Rabi resonance at about 6.916 GHz by applying a small DC magnetic field of 35 G, the temperature of the rubidium vapor cell was maintained at 50 °C. A typical Rabi resonance lineshape as a function of ωm at microwave frequency of 6.916 GHz and power of 26 dBm is shown in the inset of Fig. 2(a). The obtained Rabi frequency of the microwave magnetic field was about 55 kHz by fitting the experimental data using Eq. (2). With keeping all other experimental parameters unchanged, the measured Rabi frequencies as a function of the square root of the microwave power are shown in Fig. 2(a). By the Eq. (3), the obtained microwave magnetic field strength as a function of the square root of the microwave power at microwave resonance frequency of 6.916 GHz are shown in Fig. 2(b). The measured Rabi frequencies and field strengths of microwave magnetic field obviously increase with increasing microwave power, and exhibited a linear dependence on the square root of the incident microwave power. The measurement of the microwave magnetic field strength may consist of a number of possible systematic uncertainties [39,63].
Fig. 2. (a) Rabi frequency of the microwave magnetic field as a function of the square root of the microwave power at frequency of 6.916 GHz. The inset presents a typical Rabi resonance lineshape at power of 26 dBm, the red solid line is the fitting curve to the experimental data. (b) Measurements of the microwave magnetic field strength versus the square root of microwave power. The error bars indicate the corresponding measurement uncertainty.
Combined with the previous investigation on the characteristics of our system [6,8], the uncertainty sources of the microwave magnetic field measurement in our system mainly included the following: background stray magnetic field uncertainty ∼1.2%, peak fitting finding uncertainty ∼0.8%, accuracy of the FFT spectrum analyzer uncertainty ∼7%, frequency shifts uncertainty arising from atomic collisions between atoms and the cell walls, instability of laser power and microwave field-atom detuning ∼4%, nonlinearity uncertainty ∼5%, and standing wave disturbance uncertainty of the fixed vapor cell geometry ∼2%. The total measurement uncertainty was calculated as the root sum square of each component value, which was on the order of 10%.
Following the microwave magnetic field measurement, the counter-propagating coupling laser was turned on to further excite the atoms to the Rydberg states, and the probe laser and other experimental conditions were fixed, except for adjusting the polarization of the probe laser and turning off the DC field. We observed the Rydberg-EIT spectroscopy by scanning the coupling laser. In our experiment, both EIT peaks of 67D3/2 and 67D5/2 were accurately measured within the frequency sweep range of the coupling laser, the frequency interval between the two peaks was about 37.8 MHz, which could be used to calibrate the frequency scales of all the measured spectra. We then applied the microwave field at frequency of 6.916 GHz to couple states 67D5/2 and 68P3/2. The EIT resonance peak was split into two symmetric peaks, the measured ATS spectrum at power of -43.6 dBm is shown in the inset of Fig. 3(a). The peak-to-peak separation defined frequency interval of the ATS could be extracted using the multipeak Lorentz fittings to the ATS spectrum, as denoted with the solid red line in the inset of Fig. 3(a). By calculating the dipole moment of this microwave resonance transition and measuring the frequency interval of this ATS, the Rabi frequency and strength of the microwave electric field were then determined using Eq. (1), the dipole moment for this transition was accurately calculated to be ℘ = 2865.88 ea0 (where e is the elementary charge and a0 is the Bohr radius). The measured frequency interval of ATS increases with increasing microwave power. The results of the Rabi frequency and the microwave electric field strength versus the square root of the microwave power were plotted in Fig. 3(a) and Fig. 3(b). The measured microwave electric field strength also exhibited an excellent linear variation with the square root of the applied microwave power. The main uncertainty sources of microwave electric field measurement can be divided into the following two categories: atomic measurement related uncertainties and microwave related uncertainties [1,15,64]. The atomic measurement related uncertainties in our experiment mainly included the technical noise uncertainty from the laser intensity noise and detection ∼0.6%, background stray magnetic field uncertainty ∼0.1% under the compensation of three-dimensional solenoid coils, dipole moment calculation uncertainty ∼0.1%, deviation from linearity uncertainty ∼0.6%, frequency shifts uncertainty arising from atomic collisions, and instability of laser power and cell temperature ∼0.4%. The dominant sources of microwave related uncertainties included microwave signal generator frequency uncertainty ∼0.5%, and the standing wave disturbance uncertainty of fixed vapor cell geometry ∼1.7%. By optimizing the system parameters within the power range we measured in Fig. 3(b), the total uncertainties could be controlled to be less than 2.0%.
Fig. 3. (a) Rabi frequency of the microwave electric field as a function of the square root of the microwave power at frequency of 6.916 GHz. The inset indicate the EIT-ATS spectrum at power of -43.6 dBm, the red solid line indicate the multipeak Lorentz fitting to the experimental data. (b) Measurements of the microwave electric field strength versus the square root of microwave power, the solid line displays the linear fitting result of the experimental data. The error bars indicate the corresponding measurement uncertainty.
Through the above measurements, we experimentally demonstrated the measurement of microwave electric and magnetic fields in a single rubidium vapor cell at the same frequency of 6.916 GHz. The high sensitivity measurements of the Rydberg atom-based microwave electric field mainly benefit from the large microwave-transition dipole moments between the energetically adjacent Rydberg states. The dipole moment in Rydberg atoms increase scaling as the square of the principal quantum number n, and can be several thousand times larger than that of the lower states [65], so the Rydberg atoms exhibit a strong response to the microwave electric fields. While the coupling strength between the hyperfine levels of ground state is much smaller than that of between Rydberg states, which will result in the microwave magnetic field measurements require much higher microwave power than electric field, it is difficult to accurately measure the microwave electric and magnetic fields in the same microwave power range using the existing methods. Considering the good linear relationship between their strength and square root of power, we will investigate and compare the microwave electric and magnetic field components within the same power range by their inherent relationship.
We linearly fitted to the experimental data of the microwave electric field strengths in Fig. 3(b). The strengths of the microwave electric field for each microwave power could therefore be directly obtained from the linear fitting equation, the linear fitting values for the gradient with an uncertainty of 0.6% were obtained from many measurements. According to the electromagnetic field theory, the ideal relationship between E and B can be expressed as E/B = c, where c is the speed of light in a vacuum. By this relationship and the microwave electric field measurement results in Fig. 3(b), the equivalent microwave magnetic field strengths could be obtained at the power points of the electric field measurements, which were shown in the inset of Fig. 4, the error bars represent the measurement uncertainty of about 2%. This linear fitting equation was used to accurately calculate the strengths of the microwave electric field at the power point of the microwave magnetic field measurement in the same manner. The corresponding equivalent strengths of the microwave magnetic field can be obtained by the relationship between them. The validity of the above method can be verified by comparison with the experimental results of the magnetic field measurement.
Fig. 4. Measurements (blue squares) and the equivalent calculated results of the microwave magnetic field as a function of the square root of the microwave power with m = 2 rad. The inset indicate the equivalent microwave magnetic fields (black squares) in microwave electric field measured power range. The shaded area signifies the 6% systematic uncertainty from the microwave electric field.
As the microwave power increased, we experimentally observed that the EIT-ATS peaks gradually broadened due to the microwave electric field inhomogeneity inside the vapor cell, and the similar behaviors were also observed in a previous study [66]. When the broadened ATS linewidth was larger than the linewidth of the EIT above a certain power, the two pairs of the symmetric EIT-ATS peaks appeared and determined the two electric field strengths by the corresponding frequency separation. The obtained average electric field strengths remained to present the expected linear dependence of the square root of the microwave power. With the further increase of the power, the separation of the two pairs of the EIT-ATS peaks becomes more significant, and the state mixing and high-order couplings caused the nonlinear response gradually occurs, which leads to decreased of spectral measurement accuracy. This makes experimental measurement difficult and increases the microwave electric field uncertainty, and the Rydberg EIT-ATS method is no longer valid in higher power range [67]. The maximum uncertainty we measured with EIT-ATS method in its applicable power range was about 6%, the main reason for the increase of uncertainty was the linearity decreased with increasing power. According to the linear fitting equation for the microwave electric field measurement results and the inherent relationship between E and B, the equivalent strengths of the microwave magnetic field within entire power range could be obtained with 6% as the average systematic uncertainty. The uncertainty may be larger in higher power range, which will be further investigated in a future study. The equivalent results (gray area) along with the experimental results of microwave magnetic field (blue squares) with m = 2 rad are shown in Fig. 4. The blue error bars indicate the uncertainty with about 10% for the microwave magnetic field. It is clear that the equivalent calculated and experimental results exhibited nearly consistent gradient trend in their respective power range. In our experiment, we used a vacuum vapor cell instead of the superior vapor cell coated with paraffin for the magnetic field measurement, the higher microwave power was needed to ensure a good signal-to-noise ratio of the Rabi resonance signal. The amplitude of the modulated signal m also needs to be optimized to determine the measuring power range satisfying the small signal approximation condition. Note that the equivalent calculated and experimental results had a good agreement in determined microwave power range.
Although we proved that the fitting calculation method is feasible in a certain microwave power range. To make measurement power range between the microwave electric field and magnetic field closer or overlap, we can use a DC magnetic field combined with a hybrid vapor cell to achieve the higher microwave resonant frequency for magnetic field [62]. While higher resonant frequency correspond to lower Rydberg states or smaller microwave transition dipole moments. Rydberg atoms in lower states become less sensitive to electric field, which will extend the measureable power range of the microwave electric field closer to that of microwave magnetic field. In addition, we will try measure microwave electric and magnetic fields in the superior vapor cell coated with paraffin or buffer gas vapor cell. The signal-to-noise ratio of Rabi resonance lineshape using the above two cells will be higher, this may reduce the power measurement range of the microwave magnetic field. Both the characteristics of the microwave waveguide and the vapor cell may cause that the microwave electromagnetic component cannot exactly satisfy the ideal proportional relationship. We will investigate the above process in further experiments.
5. Conclusions
We demonstrated an SI-traceable atoms-based technique for the measurement of microwave electric and magnetic fields using the same probe laser in a single rubidium vapor cell. Under the frequency tuning of a DC magnetic field, a weak probe laser first acts as a magnetic probe via atomic Rabi resonance to measure the microwave magnetic field consistent with the resonant frequency of the electric field. When a strong coupling laser in a counter-propagating geometry further excited atoms to the Rydberg state, the same probe laser was also used as the electric probe to measure the same frequency microwave electric field by the resonant microwave dressed EIT-ATS. We presented the comparisons between microwave magnetic fields obtained with experimental measurements of the microwave magnetic fields and those obtained by fitting inversion to the measured microwave electric field in the same microwave power range. The deviation between them were observed and the influenced factors are analyzed. This study was an important step toward the measurement of quantum-based microwave multiple parameters.
Funding
National Key Research and Development Program of China (2021YFF0603704); National Natural Science Foundation of China (61975194, 62071443).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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