Open Access
Add to BibSonomy
Abstract
We theoretically and experimentally investigate the Rydberg electromagnetically induced transparency (EIT) and Autler-Townes (AT) splitting of 87Rb vapor under the combined influence of a magnetic field and a microwave field. In the presence of static magnetic field, the effect of the microwave field leads to the dressing and splitting of each mF state, resulting in multiple spectral peaks in the EIT-AT spectrum. A simplified analytical formula was developed to explain the EIT-AT spectrum in a static magnetic field, and the theoretical calculations agree qualitatively with experimental results. The Rydberg atom microwave electric field sensor performance was enhanced by making use of the splitting interval between the two maximum absolute mF states separated by the static magnetic field, which was attributed to the stronger Clebsch-Gordon coefficients between the extreme mF states and the frequency detuning of the microwave electric field under the static magnetic field. The traceable measurement limit of weak electric field by EIT-AT splitting method was extended by an order of magnitude, which is promising for precise microwave electric field measurement.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Rydberg atom, highly excited atom with one or more electrons excited to states with high principal quantum number [1], exhibits unique properties including large radius, long lifetime, and high sensitivity to external fields [2]. These features make Rydberg atoms perfect candidates for precise measurements of external fields. M Tanasittikosol et al. first demonstrated microwave dressing of Rydberg dark states in cold $^{87}$Rb atoms [3], their research provided the foundational concept and experimental framework for utilizing Rydberg atoms in measuring microwave electric field strength. Experimentally realized in 2012 by Sedback et al. [4], Rydberg atom-based microwave electric field measurement method primarily utilizes the Rydberg atoms’ high sensitivity to external fields and the phenomena of electromagnetically induced transparency (EIT) and AC Stark shift [4–6]. Specifically, EIT refers to the appearance of a narrow transparency peak in the absorption spectrum when a coupling light is applied to a three-level system [7]. Under the resonant dressing effect of the microwave field on Rydberg-to-Rydberg transitions, the AC Stark shift of Rydberg state levels leads to the symmetric splitting of EIT, known as Autler-Townes (AT) splitting. When scanning the frequency of the probe or coupling light, the interval of EIT-AT splitting has the following relationship with the microwave electric field strength [5]:
Currently, extensive research has been conducted on EIT in the presence of magnetic field, providing insight for using magnetic field as a way to improve EIT-AT methods’ measurement limits. Rydberg atoms exhibit a wide range of spectral information and potential applications when subjected to a magnetic field, which has attracted significant attention [16,23–36]. Shanxia Bao et al. [23] and Fengdong Jia et al. [24] demonstrated the frequency locking of the coupling light using Rydberg EIT under magnetic field modulation. Shanxia Bao et al. [25,26], Yongmei Xue et al. [27], L. Ma et al. [28], Linjie Zhang et al. [29], and J. B. Naber et al. [30] have investigated the linear and nonlinear variations of Rydberg energy levels under different static magnetic field strengths, along with the corresponding Rydberg EIT spectra. H. Cheng et al. [31], H. J. Su et al. [32], Z.S. Xua et al. [33] and Zi-Shan Xu et al. [34] have contributed to optimizing and improving the intensity and resolution of Rydberg EIT spectra using a static magnetic field. In another study, Fendong Jia et al. [35] enhanced the signal-to-noise ratio of the readout region by modulating the Rydberg EIT spectrum with an alternating magnetic field, and further combined it with an atomic mixer to achieve phase-sensitive detection [16]. Additionally, D. J. Whiting et al. [36] successfully measured the hyperfine structure constant of the excited state using EIT under a magnetic field. By combining the findings from the above research, it is anticipated that the EIT-AT spectrum will exhibit even more intricate structures under the combined influence of a magnetic field and a microwave electric field. This will provide valuable insights into understanding the alterations in level structures caused by external field effects, and potentially offer new insights for precise measurements related to spectral splitting intervals. However,no systematic study has been conducted on the EIT-AT splitting spectrum behavior of Rb Rydberg atoms under a static magnetic field to the best of our knowledge.
In this paper, we theoretically and experimentally investigate the Rydberg EIT and EIT-AT splitting of $^{87}$Rb vapor under the combined influence of a magnetic field and a microwave field and further studied the enhancement of the Rydberg atom microwave field sensor performance by the static magnetic field. The structure of this paper is as follows: In Sec. 2, we develop a simplified analytical formula for the EIT-AT spectrum under the static magnetic field and the microwave field following the model proposed by H. J. Su et al. [32]. In Sec. 3, we introduce the experimental setup. In Sec. 4, we present the experimental results of EIT and EIT-AT under a static magnetic field and compare them with the theoretical calculations provided in Sec. 2. We also demonstrate the potential of the static magnetic field in enhancing the measurement of weak microwave fields using the EIT-AT splitting method. We concludes the paper in Sec. 5.
2. Theoretical model
In this study, we investigate the energy levels and optical fields as illustrated in Fig. 1. The degenerate energy levels of the four-level system in the absence of a magnetic field is depicted in Fig. 1(a): ground state $|g> (5S_{1/2})$, excited state $|e> (5P_{3/2})$, two Rydberg states $|r_1> (46D_{5/2})$ and $|r_2> (47P_{3/2})$. The Zeeman splitting of the hyperfine structure in the presence of a magnetic field is shown in Fig. 1(b), where the degeneracy of the Zeeman sublevels is broken by the magnetic field. We choose the basis of F and $m_F$ to describe the system, where $F=I+J$ and $m_F=m_I+m_J$. $I$ is the nuclear spin angular momentum, which is 3/2 for $^{87}$Rb. The arrows in Fig. 1(b) indicate possible transitions between different levels, with labels denoting distinct transition paths.
Fig. 1. (a) Degenerate four-level system energy level without magnetic field. (b) Non-degenerate energy level diagram of a four-level system under magnetic field. The arrows indicate possible transitions between different levels, with labels denoting distinct transition paths. The probe light is represented by the red, the coupling light by the blue, and the microwave field by the yellow.
We first consider the three-level system $|g>$, $|e>$, and $|r_1>$. The influence of the magnetic field on the EIT spectrum of this three-level system can be described by equating the Zeeman effect to the change in the optical field detuning [32]. In Ref. [32], Hsuan-Jui Su et al. solved the optical Bloch equations and the Maxwell-Schrodinger equation (MSE) in a ladder-type three-level Rb EIT model. They obtained an analytical solution for the transmittance of the probe light as a function of the detuning of the coupling light under the influence of the magnetic field [32]:
In the presence of a magnetic field $B$, the total detuning is given by:
Here, $B$ represents the magnetic field strength. $\Delta _g,\Delta _e,\Delta _{r_1}$ are the Zeeman shift of the ground state, excited state, and the Rydberg state respectively. $g_{F,g},g_{F,e},g_{J,r_1}$ represent the Landé g-factors of the ground state $5S_{1/2}$, excited state $5P_{3/2}$, and the Rydberg state $46D_{5/2}$, which are 1/2, 2/3, and 6/5 in this study. The Doppler shift is simplified as the detuning: $\Delta _D = (\Delta _e-\Delta _g)\frac {\lambda _p-\lambda _c}{\lambda _c}$. In our experiment, we lock the frequency of the coupling light and scan the frequency of the probe light to obtain the EIT-AT splitting spectra. While the probe light detuning is much smaller than the Doppler broadening, there exists a relationship $\Delta _c = \frac {\omega _c}{\omega _p}\Delta _p$ [37]. Therefore, we obtain the relation between the probe light transmittance and the probe light detuning: $\delta = B\times (\Delta _g-\Delta _e+\Delta _{r_1}+\Delta _D)+\frac {\omega _c}{\omega _p}\Delta _p$.Based on the work in Ref. [32] and the splitting behavior of EIT under a detuned microwave field [15], we construct a simplified analytical formula for the EIT-AT spectrum under the combined influence of the static magnetic field and the microwave field. Specifically, due to the dressing effect of the microwave field, each $m_F$ state which corresponds to one EIT transparency peak will splits into two branches. We use detuning $\Delta _S$ to represent the splitting interval of the spectrum with respect to EIT and obtain the total detuning of the system as: $\delta = B\times (\Delta _g-\Delta _e+\Delta _{r_1}+\Delta _D)+\frac {\omega _c}{\omega _p}\Delta _p+\Delta _S$.
Energy levels will shift under magnetic field, causing the originally resonant microwave electric field to detune. This leads to an asymmetry in the EIT-AT splitting doublet compared to the EIT position [15]. Representing the smaller splitting interval by $\Delta _{S_1}$ and the larger one by $\Delta _{S_2}$, combining the the EIT-AT splitting interval expression in off-resonance conditions $\delta _{MW}=\sqrt {\delta ^2_{MW}+\Omega ^2_{MW}}$ [15], we are able to derive the EIT-AT splitting interval expression under both magnetic field and microwave electric field as $\Delta _{S_1}=\frac {1}{2}\cdot (\sqrt {\delta ^2_{MW}+\Omega ^2_{MW}}-\delta _{MW})$ and $\Delta _{S_2}=\frac {1}{2}\cdot (\sqrt {\delta ^2_{MW}+\Omega ^2_{MW}}+\delta _{MW})$, where $\Omega _{MW}$ is the resonant Rabi frequency of the microwave field, $\delta _{MW}$ represents the microwave detuning under the magnetic field, which can be calculated by Eq. (6)–(8).
Combining the expression for the transition probability in Ref. [38], we derive the strengths of the two peaks in the EIT-AT spectrum $f_1=\frac {1}{2}\cdot \frac {\Omega ^2_{MW}}{\Omega ^2_{MW}+\delta ^2_{MW}}$ and $f_2=1-f_1$. By applying $\Delta _S$ to represent the EIT-AT splitting interval and adjusting the values of $\Omega _C$ for different transition paths, we can obtain the splitting behavior of EIT branches with involved $m_F$ states and quantitatively calculate their maximum splitting interval.
3. Experimental setup and methods
The experimental setup for EIT and EIT-AT spectrum in a Rb vapor cell under a magnetic field is shown in Fig. 2. The experiments were conducted under room temperature. The probe light, provided by a DL100 laser from Toptica, has a wavelength of $\thicksim$780 nm, and corresponds to the transition from $^{87}$Rb $5S_{1/2}$ to $5P_{3/2}$ (shown as red arrows in Fig. 1(b)). The coupling light, provided by a TA-SHG-Pro laser from Toptica, has a wavelength of $\thicksim$480 nm and corresponds to the transition from $^{87}$Rb $5P_{3/2}$ to $46D_{5/2}$ (shown as blue arrows in Fig. 1(b)). The probe and coupling lights were linearly polarized and collinearly counterpropagated through the Rb vapor cell.
Fig. 2. Experimental setup. The $\thicksim$780 nm probe lights and $\thicksim$480 nm coupling lights were set linearly polarized and collinearly counterpropagated through the Rb vapor cell. The static magnetic field was generated by a pair of rectangular Helmholtz coils (50 cm $\times$ 50 cm) placed perpendicular to the axis of the Rb cell and the microwave electric field was provided by a signal source (8340 B, HP Corporation) connected to a Horn Antenna placed by the side of the Rb cell. HWP: half wave plate; PBS: polarizing beam splitter.
A uniform and static magnetic field was used to study the EIT-AT splitting and perform the measurement of MW fields. To ensure a stable uniform magnetic field, we dedicated our efforts to two key aspects. Firstly, within the confines of feasible experimental conditions, we employed a 50 cm $\times$ 50 cm square Helmholtz coil to guarantee a uniform and steady magnetic field within the region of the Rb cell (approximately 5 cm in size). Secondly, we implemented a PI current feedback loop to precisely regulate the current in the magnetic field. The magnetic strength B at the location of the vapor cell was measured by a Gauss meter as a reference beforehand. Then the accurate magnetic field strength experienced by atoms was calibrated by the linear correlation between the Rydberg EIT splitting interval and the magnetic field strength. The microwave electric field was provided by a signal source (8340 B, HP Corporation) connected to a Horn Antenna placed by the side of the Rb cell, with a frequency of 22.067233 GHz, corresponding to the transition from $^{87}$Rb $46D_{5/2}$ to $47P_{3/2}$ (shown as yellow arrows in Fig. 1(b)).
4. Results and discussion
4.1 Experimental spectra and theoretical analysis of Rydberg EIT splitting under static magnetic field
To investigate the magnetic field’s influence on Rydberg EIT, we varied the current in Helmholtz coils to obtain a static magnetic field range of 0-10 G. We employed a frequency-locked coupling light while scanning the probe light within the range $\Delta _p=\pm 2\pi \times 30$ MHz to acquire the spectrum of the $5S_{1/2}-5P_{3/2}-46D_{5/2}$ Rydberg transition under static magnetic field. The experimental spectrum is shown in Fig. 3(a). The experimental results revealed that the Rydberg EIT spectrum split symmetrically into four peaks as the magnetic field increased. The outer two peaks exhibited larger splitting intervals, indicating their higher sensitivity to the static magnetic field, while the inner two peaks displayed smaller splitting intervals, suggesting a lower sensitivity. These distinctions are due to the transitions of different Zeeman sublevels, which can be determined by referencing the energy level diagram in Fig. 1 and the experimental results. Specifically, when subjected to a static magnetic field ranging from 0 to 10 G, Zeeman sublevels with the same absolute value of $m_F$ exhibited symmetric energy level shifts, leading to the appearance of symmetrical peaks in the EIT spectrum. Our theoretical analysis revealed that the two outermost peaks in Fig. 3(a) corresponded to transitions labeled as 1 and 2 in Fig. 1 with CGC = 1, while the relatively magnetic-field-insensitive two middle peaks corresponded to transitions labeled as 3 and 4 in Fig. 1 with CGC = 0.32.
Fig. 3. The experimental and calculated result of EIT spectrum under a static magnetic field of 0-10 G. (a) The experimental conditions of the EIT spectrum are as follows: The probe light has an optical power of approximately 60 $\mu W$ with a spot size of $\thicksim$400 $\mu m$ in radius, while the probe light has an optical power of approximately 50 mW with a spot size of $\thicksim$450 $\mu m$ in radius. The interval between the outmost EIT spectra are denoted as $\Delta f_{\rm EIT}$ in Sec. 4.3. (b) The theoretical calculations of the EIT spectrum under a magnetic field of 0-10 G are as follows. The calculation parameters used are: $\alpha =0.257$, $\gamma =2\pi \times 3.3\times 10^6$ MHz, and $\Omega _c=2\pi \times 1.2\times 10^6$ MHz (for CGC = 1). The two side peaks correspond to the transition paths 1 and 2 in Fig. 2(b) for CGC = 1, while the two middle peaks correspond to the transition paths 3 and 4 in Fig. 2(b) for CGC = 0.32. The theoretical calculations qualitatively agree well with the experimental results. Note that, the data in (a) and (b) are normalized using EIT and EIT-AT transmittance under zero magnetic field as the reference standard. This adjustment is rooted in our focus on extracting positional information from spectral peaks, which is more distinctly discernible through relative changes in transmittance.
By applying the theoretical model shown in Sec. 2, we are able to fit and calculate the experimental EIT spectra obtained under the magnetic field. The results are shown in Fig. 3(b). By comparing Fig. 3(a) and (b), we can see a high level of agreement between the experimental spectrum and theoretical calculations in terms of peak positions, thereby confirming the suitability of the energy level model. However, slight differences are observed in peak strengths and linewidths. The differences and explanations are as follows. Firstly, theoretical calculations predict symmetric peak strengths for the peaks corresponding to excitation route with Zeeman sublevels of the same absolute value of $m_F$, while the experimental EIT spectra display asymmetric peak strengths. This difference is caused by the imperfect polarization of the probe and coupling light after passing through the atomic vapor cell. Secondly, the theoretical calculations yield narrower linewidths compared to the experimental observations. This disagreement arise because the analytical solution in the theoretical calculations only considered the peak position variation caused by the Doppler effect, without accounting for the residual Doppler effect-induced broadening due to the dismatch of $\lambda _p$ and $\lambda _c$. Still, the theoretical calculations based on the analytical solution has accurately reproduced the peak positions, satisfying the requirements of focusing on the EIT-AT splitting interval.
We performed a quantitative analysis of the spectral position information presented in Fig. 4 to accurately calibrate the magnetic field strength experienced by the atoms. Due to the limited EIT linewidth resolution observed in the experimental results, precise fitting of the peak positions for the two intermediate lines is not feasible. As a result, we calibrate the magnetic field experienced by the atoms by utilizing the two outer lines, which correspond to the $m_F=\pm \,4$ states. By fitting the peak positions corresponding to the same $m_F$ state excitation configuration under the 0-10 G static magnetic field in the experiment, we establish the relationship between the experimental splitting interval of the two outer lines and the magnetic field, as depicted by the black empty squares in Fig. 4. These points exhibit a linear correlation with the magnitude of the magnetic field. Notably, when the magnetic field is less than 2 G, the splitting interval cannot be resolved because the splitting interval caused by the magnetic field is smaller compared to the EIT linewidth. By performing a linear fit on the spacing between the outer two lines, we derive the following expression: $\Delta f=(4.067\pm 0.036)\cdot B-(0.641\pm 0.243)$, which enables the absolute calibration of the magnetic field. The blue solid line in Fig. 4 represents the results obtained using the calibrated magnetic field and the theoretical analytical solution for the three-level system, exhibiting good agreement with the experimental findings. It is worth noting that for small magnetic field values, the calculated spectral intervals cannot be distinguished, which is consistent with the experimental observations. Through the investigation of the $^{87}$Rb EIT splitting behavior under a magnetic field, we not only identify the magnetic sublevels and excitation channels involved in Rydberg EIT, but also provide an absolute calibration of the magnetic field experienced by the atoms. This serves as a solid foundation for further exploration of the EIT-AT splitting behavior under the influence of microwave fields.
Fig. 4. We calibrate the experimental results using theoretical calculations to determine the relationship between the EIT splitting interval and the magnetic field strength. The blue solid line represents the theoretical calculation results with the expression $\Delta f=(4.067\pm 0.036)\cdot B-(0.641\pm 0.243)$. The black empty squares represent the splitting intervals of the outer two spectral lines from Fig. 3(a) after magnetic field calibration. The calculation results are in good agreement with the experimental findings.
We will discuss the choice of magnetic field strength and its applicable range from two perspectives. Firstly, the choice of a magnetic field ranging from 0 to 10 G stems from the constraints posed by our experimental conditions. Specifically, using a higher magnetic field strength will require a greater current flow through the Helmholtz coils. Consequently, this heightened current will lead to a notable rise in the temperature of the Helmholtz coils during the experiment, thereby influencing the ambient environmental temperature and experimental results. The Helmholtz coils employed in the experiment comprises 50 turns. Achieving a magnetic field strength of 10 G at the center of the Helmholtz coils corresponds to a coil current of approximately 5.3 A. Meanwhile, the equilibrium temperature of the coils reaches 70 degrees Celsius, without exerting any impact on the surrounding equipment or measurement results. Consequently, we have settled upon 10 G as the upper limit for the magnetic field strength in our experiment. Secondly, within the magnetic field range of 0-10 G, we have observed a linear correlation between the interval of EIT splitting and the magnitude of the magnetic field, as shown in Fig. 4. This linear relationship enables precise calibration of the actual magnetic field intensity experienced by the atom. Moreover, during this linear regime, the analysis and processing of the EIT-AT splitting spectrum are comparatively straightforward. However, theoretically, the conclusions drawn from our method are not confined solely to the 0-10 G magnetic field range; the available magnetic field range can be broader. Even if the magnetic field becomes significant to the extent that the interval between EIT splitting and magnetic field strength is no longer linear, we can still achieve magnetic field calibration and perform data analysis by fitting the interval of EIT splitting with the magnetic field strength.
4.2 Experimental spectra and theoretical analysis of Rydberg EIT-AT splitting under static magnetic field
We conducted measurements on the Rydberg EIT-AT spectrum under a combination effect of static magnetic field and microwave field. The magnetic field can be fixed at 0 G to 10 G when measuring the EIT-AT spectrum with microwave field strength ranging from 0 to 30 mV/cm. Figure 5(a) illustrates the EIT-AT splitting behavior under a static magnetic field at 10 G with different microwave field strengths. It is important to point out that we maintained the microwave field frequency the same as the resonance frequency with the Rb atoms at zero magnetic field throughout the experiment. The results in Fig. 5(a) experimentally demonstrate that after the degeneracy broken caused by the magnetic field, each EIT spectrum splits as the microwave field strength increases. The outermost EIT spectral lines exhibit asymmetrical EIT-AT splitting in terms of their positions and strengths. The position asymmetry arises from the magnetic field-induced shifting of Zeeman sublevels with nonzero $m_F$, causing the microwave field frequency to deviate from resonance with the Rydberg levels. This behavior is consistent with the previously reported characteristics of Rydberg EIT-AT splitting under non-resonant microwave field frequencies [15]. In contrast, the two middle spectral lines experience insignificant splitting due to the insensitivity of their excited states to the magnetic field and weak or zero Rydberg-pair states interactions [39]. Consequently, no further theoretical analysis was conducted for these lines.
Fig. 5. The experimental and calculated result of EIT-AT spectrum. (a) The experimental results of the EIT-AT split spectra under a static magnetic field of 10 G are presented as follows. For the experimental parameters, the probe light is characterized by a power of approximately 60 $\mu W$ and a beam size of approximately 400 $\mu m$ in radius. Similarly, the coupling light exhibits a power of approximately 50 mW and a beam size of approximately 450 $\mu m$ in radius. The interval between the asymmetry EIT-AT peak and the EIT peak is denoted as $\Delta _{S_1}$ and $\Delta _{S_2}$ respectively. The interval between the outmost EIT-AT spectra are defined as the new EIT-AT splitting interval $\Delta f'$. (b) Theoretical calculations of the EIT-AT split spectrum under the same 10 G static magnetic field are conducted with the following parameters: $\alpha =0.257$, $\gamma =2\pi \times 3.3\times 10^6$ MHz, and $\Omega _c=2\pi \times 1.2\times 10^6$ MHZ (for CGC = 1), and $\Omega _{MW}=2\pi \times 1.35\times 10^6$ MHz (for CGC = 1). The theoretical calculations are in qualitative agreement with the experimental results displayed in Fig. 4(a).
We qualitatively calculated the EIT-AT spectrum shown in Fig. 5(b) based on the analytical solutions for the four-level system, which agrees with the experimental results well. This calculation further validates the selection of energy levels and the applicability of the chosen model. Similarly, when calculating the resonant frequency of the microwave field, we took into account the proportions between different hyperfine structure levels of the atoms at different Rydberg states. The calculated ratio of $m_F$ states corresponding to the outermost and innermost EIT lines is approximately 1.35:1. A narrower linewidth was acquired by theoretical calculation compared to the experimental result. The reason is that since the two middle lines are insensitive to both the magnetic and microwave fields, we only considered the splitting of the largest mF state, i.e. the two outermost EIT spectral lines in the theoretical calculation. The experimental results in Fig. 5(a) may involve additional transition paths, leading to a broader linewidth compared to the theoretical calculation. Nevertheless, this discrepancy already satisfies our requirements regarding the peak positions and splitting intervals of EIT-AT. Besides, the theoretical calculation indicates that the strengths of the spectral lines corresponding to the excited routes with the same absolute value of $m_F$ are symmetric while that are slightly asymmetric observed in experiments. The reason is the same as discussed in Section 4.1. Through our investigation of Rb atoms’ EIT-AT behavior under static magnetic field, we discovered that the interval of the outermost EIT spectra were no longer limited by the EIT linewidth. Therefore, the EIT-AT splitting can still be resolved under weak microwave fields, which provides insights for enhancing the EIT-AT splitting interval method in RF electric field measurement.
4.3 Enhancing the lower measurement limit utilizing EIT-AT splitting of the maximum $m_F$ state in a static magnetic field
Now we try to measure the strength of the microwave field using the outermost spectral intervals, as depicted in Fig. 5(a) and (b). There are three motivations behind this choice. Firstly, the outermost spectral lines demonstrate the highest intensity when subjected to EIT optical pumping. Secondly, the overlapping of intermediate spectral lines poses challenges in accurately determining and reading peak positions, particularly under strong microwave field. Lastly and most importantly, our focus in this study is on the expanding of EIT-AT intervals induced by static magnetic field. And the outermost spectral intervals indicate the possibility that Zeeman effect-induced EIT-AT splitting in the magnetic field can enlarge the EIT-AT intervals, thereby enabling the measurement of weaker microwave field strengths. By fitting the positions of the two outermost peaks in Fig. 5(a), we establish the relationship between the experimental splitting interval and the microwave field, as shown by the red empty circles in Fig. 6. For comparison, we also provide the relationship between the EIT-AT splitting interval and the microwave field at zero magnetic field, illustrated by the black empty squares in Fig. 6. Notably, a linear relationship between the EIT-AT splitting interval and the microwave field is observed when $E > 8.53 mV/cm$ at zero magnetic field. This suggests that we can determine the Rabi frequency of the microwave field and achieve absolute calibration of its strength in this range. However, for microwave fields below 8.53mV/cm, the relationship between the EIT-AT splitting interval and the microwave field deviates from linearity, and for E < 6.39 mV/cm, the EIT-AT splitting becomes indistinguishable, reaching the lower limit of microwave field strength measurement using the conventional EIT-AT splitting method. When a magnetic field of 10 G is applied, the redefined splitting interval $\Delta f'$, i.e. the interval between the outermost two spectral lines, persists and exhibits a monotonic relationship with the microwave field, as shown in Fig. 6. Notably, we discover that the minimum measurable microwave field is 0.31 $\pm$ 0.15 mV/cm, which extends the measurement limit by a factor of 20.6 compared to the EIT-AT measurement without static magnetic field in our work.
Fig. 6. The relationship between the EIT-AT intervals and the magnitude of the microwave field for 0 and 10 G static magnetic fields. The black hollow squares correspond to the EIT-AT splitting interval as a function of the microwave field at 0 G magnetic field, while the black solid line represents the linear fit of the EIT-AT splitting interval at resonance under 0 G magnetic field. The red hollow circles depict the spacing between the outermost two EIT-AT spectral lines obtained experimentally, and the red solid line represents the calculated result of the EIT-AT splitting interval under 10 G by Eq. (10).
In the following analysis, we aim to develop an analytical formula that describes the relationship between the redefined EIT-AT splitting interval $\Delta f'$ under the magnetic field and the microwave field. Notably, the two outermost EIT lines in Fig. 4(a) exhibit an asymmetric EIT-AT splitting in terms of position and intensity, consistent with the behavior of Rydberg EIT-AT splitting reported in Ref. [15] when the microwave field frequency is off-resonance. Building upon the findings in Ref. [15], we establish a new analytical solution to describe the EIT-AT splitting interval exhibited by peaks with identical $\lvert m_F \rvert$ values under the magnetic field. Reference [40] provides a relationship between the EIT-AT splitting interval and the detuning:
where $\Delta f$ is the detuning, $\Delta f_0$ represents the EIT-AT splitting interval at resonance. The strong peak of the EIT-AT splitting is closer to the EIT peak without microwave field. We denote this smaller interval as $\Delta _{S_1}=\frac {1}{2}\cdot (\sqrt {\delta ^2_{MW}+(\beta \cdot \Delta f_0)^2}-\delta _{MW})$, where $\beta$ denotes the ratio of mF states corresponding to the outer and inner EIT spectral lines. This assumption is reasonable since the formula reduces to the resonance case when $\delta _{MW}$ approaches 0, and the microwave field no longer interacts with the Rydberg energy levels as $\delta _{MW}$ approaches infinity, resulting in no EIT splitting, and the formula returns to Eq. (9). Similarly, the interval between the strong peak of the right EIT-AT splitting and its corresponding EIT peak is denoted as $\Delta _{S_2}=\frac {1}{2}\cdot (\sqrt {\delta ^2_{MW}+(\beta \cdot \Delta f_0)^2}-\delta _{MW})$. Additionally, we consider the interval caused by the magnetic field in zero microwave field as $\Delta f_0$. Then the interval between the outermost two spectrum has the expression: where $\delta _{MW}$ is the detuning and can be calculated by Eq. (6)–(8), $\Delta f_0$ represents the EIT-AT splitting interval at resonance. The calculated $\beta$ value is $\beta \approx 1.35$ [38], indicating a favorable ratio constant greater than 1 for EIT-AT splitting measurements. $\Delta f_{\rm EIT}$ is the splitting interval under a magnetic field, as depicted in Fig. 3(b).Note that $\Delta f_0$ can be obtained by linear fitting of the EIT-AT splitting interval at 0 G, as shown by the solid black line in Fig. 6. Using Eq. (10), the calculated results for the outermost two spectral lines at 10 G are shown by the solid red line in Fig. 6. The calculated results demonstrate a good agreement with the experimental data. This confirms our experimental objective: the EIT-AT splitting induced by the Zeeman effect under a magnetic field can increase the EIT-AT interval, thereby characterizing weaker microwave electric field strengths. Moreover, this interval can be described by a simple analytical formula.
We will discuss the underlying physical nature of the static-magnetic-field-induced enhancement on the EIT-AT splitting interval measurement in the microwave electric fields weak-field regime. In the method for measuring microwave electric field strength using the Rydberg EIT-AT splitting method, the minimum disdinguishable interval is constrained by the Rydberg EIT linewidth [11]. Once the EIT-AT splitting interval falls below the extent of the Rydberg EIT linewidth, the EIT-AT splitting interval is no longer distinguishable and loses the simple analytical relationship with the intensity of the microwave electric field. In this study, the application of a static magnetic field enhances EIT-AT splitting resolution through two distinct mechanisms. Firstly, there is a gain to the interval achieved by the static-magnetic-field-induced microwave frequency detuning. Specifically, the shift in the Rydberg levels induced by the static magnetic field results in detuning the frequency of the microwave electric field when applied in resonance with the atomic Rydberg levels in the absence of an external field. This detuned microwave electric field contributes to an improvement in the interval and measurement sensitivity of EIT-AT splitting [15]. Secondly, the stronger Clebsch-Gordon coefficients between different $m_F$ states also contribute to an increased EIT-AT splitting interval. Therefore, compared to the absence of an external field, the EIT-AT splitting method under the influence of a static magnetic field extends the traceable measurement limit of weak microwave electric fields by an order of magnitude.
5. Conclusion
In this study, we investigate the splitting behavior of $^{87}$Rb Rydberg atoms’ EIT and EIT-AT spectrum in a static magnetic field. By introducing an equivalent detuning to describe the magnetic field effect, we derive a simple analytical solution that accurately explains the experimental observations. Under the influence of the magnetic field, the EIT peaks exhibit multiple spectral lines, with symmetric splitting observed in the same $m_F$ state. The calculated results are in excellent agreement with the experimental data. Through the analysis of the EIT splitting behavior of Rb atoms in a magnetic field, we not only identify the magnetic sublevels and excitation transition routes involved in EIT system, but also achieve absolute calibration of the magnetic field experienced by the atoms. Moreover, we investigate the split spectrum of EIT-AT under the combined effect of static magnetic field and microwave field. Our findings reveal a monotonic relationship between the interval of the two outermost peaks of EIT-AT splitting (corresponding to the largest absolute $m_F$ values) and the microwave field at 10 G. The minimum measurable microwave field is 0.31 mV/cm, which represents a 20.6-fold improvement in the lower limit of microwave field measurement compared to the EIT-AT measurement without static magnetic field in our work. More interestingly, we propose a simple analytical formula to explain the relationship between the splitting interval of these two spectral lines and the microwave electric field strength. The calculated results exhibit good agreement with the experimental data, thereby providing a simplified approach for investigating EIT-AT splitting in the presence of a magnetic field. Our work deepens our understanding of the Rydberg EIT-AT spectrum under the combined influence of a static magnetic field and a microwave field, and unveils its potential applications.
Funding
Beijing Municipal Natural Science Foundation (1212014); National Natural Science Foundation of China (11604177, 11604334, U2031125); the Strategic Priority Research Program (grant no. XDB28000000) of the Chinese Academy of Sciences; Fundamental Research Funds for the Central Universities; National Key Research and Development Program of China (2017YFA0304900, 2017YFA0402300).
Acknowledgments
We are grateful for the fruitful discussions with Professor T. F. Gallagher at University of Virginia; Dr. Yuechun Jiao, Dr. Linjie Zhang, and Prof. Jie Ma at Shanxi University; Dr. Kaiyu Liao and Prof. Hui Yan at South China Normal University; and Dr. Fei Meng at the National Institute of Metrology, China.
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.
References
1. T. F. Gallagher, Rydberg Atoms (Cambridge University, 1994).
2. C. S. Adams, J. D. Pritchard, J. P. Shaffer, et al., “Rydberg atom quantum technologies,” J. Phys. B: At., Mol. Opt. Phys. 53(1), 012002 (2020). [CrossRef]
3. M. Tanasittikosol, J. D. Pritchard, D. Maxwell, et al., “Microwave dressing of rydberg dark states,” J. Phys. B: At., Mol. Opt. Phys. 44(18), 184020 (2011). [CrossRef]
4. J. A. Sedlacek, A. Schwettmann, H. Kübler, et al., “Microwave electrometry with rydberg atoms in a vapour cell using bright atomic resonances,” Nat. Phys. 8(11), 819–824 (2012). [CrossRef]
5. C. T. Fancher, D. R. Scherer, M. C. S. John, et al., “Rydberg atom electric field sensors for communications and sensing,” IEEE Trans. Quantum Eng. 2, 1–13 (2021). [CrossRef]
6. H. Fan, S. Kumar, J. Sedlacek, et al., “Atom based rf electric field sensing,” J. Phys. B: At., Mol. Opt. Phys. 48(20), 202001 (2015). [CrossRef]
7. J. P. Marangos, “Electromagnetically induced transparency,” J. Mod. Opt. 45(3), 471–503 (1998). [CrossRef]
8. C. L. Degen, F. Reinhard, P. Cappellaro, et al., “Quantum sensing,” Rev. Mod. Phys. 89(3), 035002 (2017). [CrossRef]
9. S. Kumar, H. Fan, H. Kübler, et al., “Atom-based sensing of weak radio frequency electric fields using homodyne readout,” Sci. Rep. 7(1), 42981 (2017). [CrossRef]
10. S. Kumar, H. Fan, H. Kübler, et al., “Rydberg-atom based radio-frequency electrometry using frequency modulation spectroscopy in room temperature vapor cells,” Opt. Express 25(8), 8625–8637 (2017). [CrossRef]
11. C. L. Holloway, M. T. Simons, J. A. Gordon, et al., “Electric field metrology for SI traceability: Systematic measurement uncertainties in electromagnetically induced transparency in atomic vapor,” J. Appl. Phys. 121(23), 233109 (2017). [CrossRef]
12. F. Zhou, F.-D. Jia, J. Mei, et al., “The effect of the doppler mismatch in microwave electrometry using rydberg electromagnetically induced transparency and autler–townes splitting,” J. Phys. B: At., Mol. Opt. Phys. 55(7), 075501 (2022). [CrossRef]
13. M. Jing, Y. Hu, J. Ma, et al., “Atomic superheterodyne receiver based on microwave-dressed rydberg spectroscopy,” Nat. Phys. 16(9), 911–915 (2020). [CrossRef]
14. M. Cai, S. You, S. Zhang, et al., “Sensitivity extension of atom-based amplitude-modulation microwave electrometry via high Rydberg states,” Appl. Phys. Lett. 122(16), 161103 (2023). [CrossRef]
15. M. T. Simons, J. A. Gordon, C. L. Holloway, et al., “Using frequency detuning to improve the sensitivity of electric field measurements via electromagnetically induced transparency and Autler-Townes splitting in Rydberg atoms,” Appl. Phys. Lett. 108(17), 174101 (2016). [CrossRef]
16. F.-D. Jia, H.-Y. Zhang, X.-B. Liu, et al., “Transfer phase of microwave to beat amplitude in a rydberg atom-based mixer by zeeman modulation,” J. Phys. B: At., Mol. Opt. Phys. 54(16), 165501 (2021). [CrossRef]
17. J. Yuan, T. Jin, L. Wang, et al., “Improvement of microwave electric field measurement sensitivity via dual-microwave-dressed electromagnetically induced transparency in rydberg atoms,” Laser Phys. Lett. 19(12), 125207 (2022). [CrossRef]
18. X. Liu, F. Jia, H. Zhang, et al., “Using amplitude modulation of the microwave field to improve the sensitivity of Rydberg-atom based microwave electrometry,” AIP Adv. 11(8), 085127 (2021). [CrossRef]
19. K.-Y. Liao, H.-T. Tu, S.-Z. Yang, et al., “Microwave electrometry via electromagnetically induced absorption in cold rydberg atoms,” Phys. Rev. A 101(5), 053432 (2020). [CrossRef]
20. F. Zhou, F. Jia, X. Liu, et al., “Improving the spectral resolution and measurement range of quantum microwave electrometry by cold rydberg atoms,” J. Phys. B: At., Mol. Opt. Phys. 56(2), 025501 (2023). [CrossRef]
21. F. R. L. Ripka, C. Lui, M. Schmidt, et al., “Rydberg atom-based radio frequency electrometry: hyperfine effects,” in Optical and Quantum Sensing and Precision Metrology II, J. ScheuerS. M. Shahriar, eds., International Society for Optics and Photonics (SPIE, 2022).
22. N. Prajapati, N. Bhusal, A. P. Rotunno, et al., “Sensitivity comparison of two-photon vs three-photon Rydberg electrometry,” J. Appl. Phys. 134(2), 023101 (2023). [CrossRef]
23. S. Bao, H. Zhang, J. Zhou, et al., “Tunable frequency stabilization to zeeman sublevel transitions between an intermediate state and rydberg states,” Laser Phys. 27(1), 015701 (2017). [CrossRef]
24. F. Jia, J. Zhang, L. Zhang, et al., “Frequency stabilization method for transition to a rydberg state using zeeman modulation,” Appl. Opt. 59(7), 2108–2113 (2020). [CrossRef]
25. S. Bao, W. Yang, H. Zhang, et al., “Splitting of an electromagnetically induced transparency window of a cascade system with 133Cs rydberg atoms in a static magnetic field,” J. Phys. Soc. Jpn. 84(10), 104301 (2015). [CrossRef]
26. S. Bao, H. Zhang, J. Zhou, et al., “Polarization spectra of zeeman sublevels in rydberg electromagnetically induced transparency,” Phys. Rev. A 94(4), 043822 (2016). [CrossRef]
27. Y. Xue, L. Hao, Y. Jiao, et al., “Rydberg electromagnetically induced transparency in a large hilbert space,” Phys. Rev. A 99(5), 053426 (2019). [CrossRef]
28. L. Ma, D. A. Anderson, G. Raithel, et al., “Paschen-back effects and rydberg-state diamagnetism in vapor-cell electromagnetically induced transparency,” Phys. Rev. A 95(6), 061804 (2017). [CrossRef]
29. L. J. Zhang, S. X. Bao, H. Zhang, et al., “Interplay between optical pumping and rydberg eit in magnetic fields,” Opt. Express 26(23), 29931–29944 (2018). [CrossRef]
30. J. B. Naber, A. Tauschinsky, H. B. van Linden van den Heuvell, et al., “Electromagnetically induced transparency with rydberg atoms across the breit-rabi regime,” SciPost Phys. 2(2), 015 (2017). [CrossRef]
31. H. Cheng, H. M. Wang, S. S. Zhang, et al., “High resolution electromagnetically induced transparency spectroscopy of rydberg 87Rb atom in a magnetic field,” Opt. Express 25(26), 33575–33587 (2017). [CrossRef]
32. H. J. Su, J. Y. Liou, I. C. Lin, et al., “Optimizing the rydberg eit spectrum in a thermal vapor,” Opt. Express 30(2), 1499–1510 (2022). [CrossRef]
33. Z. Xu, M. Cai, S. You, et al., “Optical-optical double resonance spectroscopy of Rb 5D3/2,5/2 in magnetic fields,” Spectrochim. Acta, Part B 193, 106453 (2022). [CrossRef]
34. Z.-S. Xu, H.-M. Wang, Z.-L. Ba, et al., “Transient electromagnetically induced transparency spectroscopy of 87Rb atoms in buffer gas,” Chin. Phys. B 31(7), 073201 (2022). [CrossRef]
35. F. Jia, Y. Yu, X. Liu, et al., “Dispersive microwave electrometry using zeeman frequency modulation spectroscopy of electromagnetically induced transparency in rydberg atoms,” Appl. Opt. 59(27), 8253–8258 (2020). [CrossRef]
36. D. J. Whiting, J. Keaveney, C. S. Adams, et al., “Direct measurement of excited-state dipole matrix elements using electromagnetically induced transparency in the hyperfine paschen-back regime,” Phys. Rev. A 93(4), 043854 (2016). [CrossRef]
37. M. Mack, F. Karlewski, H. Hattermann, et al., “Measurement of absolute transition frequencies of 87Rb to $\mathit {nS}$ and $\mathit {nD}$ rydberg states by means of electromagnetically induced transparency,” Phys. Rev. A 83(5), 052515 (2011). [CrossRef]
38. C. C. Tannoudji, G. Grynberg, J. Dupont-Roe, et al., Atom-photon interactions, (Wiley-VCH Verlag GmbH, 2008).
39. T. G. Walker and M. Saffman, “Consequences of zeeman degeneracy for the van der waals blockade between rydberg atoms,” Phys. Rev. A 77(3), 032723 (2008). [CrossRef]
40. P. R. Berman and V. S. Malinovsky, Principles of Laser Spectroscopy and Quantum Optics (Princeton University, 2011).
