Theoretical study on Stark effect of Rydberg atom in super low frequency electric field measurement

Abstract

Super low frequency electric field measurements are crucial in analysing electromagnetic compatibility, assessing equipment status, and other related fields. Rydberg atom‐based super low frequency electric field measurements are performed by observing the Stark shift in the spectrum of the Rydberg state. In a specific range of field strength (E < E_avoid, where E_avoid is the threshold to avoid crossing electric fields), the Rydberg atomic spectrum experiences a quadratic frequency shift in relation to the field strength, with the coefficient being determined by the atomic polarisability α. The authors establish a dynamic equation for the interaction between the external electric field and the atomic system, and present the Stark structure diagram of the Caesium Rydberg atom. The mathematical formulae for α and E_avoid in different Rydberg states are also obtained: α = A × (n*)⁶ + B × (n*)⁷ and E_avoid = C/(n*)⁵ + D/(n*)⁷, where A(B) = 2.2503 × 10⁻⁹(7.49,948 × 10⁻¹¹) and C(D) = 1.68,868 × 10⁸(2.45,991 × 10⁹). The error of α and E_avoid compared with the experimental values does not exceed 8% and is even lower in the low Rydberg states. Accurately calculating the values of α and E_avoid is crucial in incorporating the Rydberg atom quantum coherence effect into super low frequency electric field measurements in new power systems.

INTRODUCTION

The development of new power systems urgently requires advanced intelligent measurement and sensing technology. As an important electrical parameter of the power system, the electric field signals contain information about the operation of lines and electrical equipment. The electric field measurement is widely used in electrical equipment insulation status assessment [1, 2], power system electromagnetic environment assessment [3, 4], lightning warning [5, 6] and other occasions. Traditional electric field sensors based on electrical and optical principles face insurmountable limitations in the optimisation of measurement sensitivity, accuracy, operating bandwidth, and miniaturisation. Electric field sensing technology based on Rydberg atomic spectral characteristics has received widespread attention in recent years. In principle, the quantum coherence effect between atoms and external fields utilises the fixed properties of the atoms themselves for measurement, and has a self-calibration function. Compared with traditional electric field measurement technology, it has obvious advantages in measurement accuracy and reliability.

At present, Rydberg atoms are mostly used for electric field quantum measurements. Rydberg atoms refer to a type of atom whose outermost electrons are excited to a radius far away from the actual orbit of the atom. They have many peculiar properties. They have the characteristics of extremely large polarisability (∼n⁷), large transition dipole moment (∼n²), and long radiation lifetime (∼n⁴) [7]. The quantum system of Rydberg atoms can have a long coherence time and is extremely susceptible to manipulation and reaction with external fields. In recent years, the development of lasers and laser spectroscopy technology has made it easier to prepare and manipulate Rydberg atoms, The application of electric field measurement based on the electromagnetically induced transparency effect (EIT) of Rydberg atoms has quickly become a research hotspot. The EIT effect is a quantum coherent optical effect manifested by the interaction between laser and atoms [8, 9]. Specifically, a weak probe laser with specific resonance frequency is introduced to excite the atoms from the ground state to excited state, then another strongly coupling laser is resonant with the excited state to the Rydberg state, the quantum interference destruction effect between the two lasers and the atomic medium significantly changes the laser absorption properties of the atoms. This leads to the formation of a ‘dark state’ in which the probe laser is not absorbed, resulting in an EIT effect. Reading the spectral signal of the detection laser based on the EIT effect can directly obtain the rules inside the atom, becoming a new method to study the interaction between Rydberg atoms and external electric field [10].

In 2020, Jia Suotang team's superheterodyne receiver model based on the Rydberg atomic electric field increased the measurement sensitivity to 55 nV⋅cm⁻¹ Hz^−1/2, and the minimum measurable field strength is about 400 pV/cm [11]. However, under the action of super low frequency electric field in power systems, the frequency of the electric field is not enough to cause the Rydberg energy level to transition and cause Autler–Townes splitting. Instead, the energy level of the Rydberg atoms shifts, causing the frequency shift of the spectrum, which is the so-called Stark effect [12, 13]. When valence electrons approach the ion nucleus, they are influenced by the coupling between the ion nucleus and the spin-orbit. This can be seen in the Stark diagram of heavier alkali metal atoms, such as Rb, where under the influence of an external field, adjacent energy levels will exhibit an avoidance of crossing phenomenon [14–16]. Before this avoidance occurs, the Stark energy and field strength show a quadratic relationship with the coefficient of atomic polarisability. Therefore, accurately determining the avoidance crossing electric field and calculating atomic polarisability are crucial in achieving precise power frequency electric field measurements using Rydberg atoms. Wang et al. studied the Stark effect of caesium Rydberg atoms near an electric field of 6.0 V/cm, and observed the avoid crossing phenomenon between the nS state and the (n-4) manifold [15]. Furthermore, dipolar Rydberg-atom gas are prepared by adiabatic passage through an avoided crossing [16]. In 2022, the American National Institute of Standards and Technology studied the initial cross-electric field avoidance of Rydberg atoms in the 28S_1/2, 40S_1/2 and 47S_1/2 states, and used experiments to prove that nS_1/2 state Rydberg atoms were exposed to DC and power frequency electric fields. When the field to be measured is smaller than the initial avoid crossing electric field, the frequency shift amount of the spectrum shows a quadratic relationship with the electric field, and the coefficient is determined by the atomic polarisability α [17]. Recently, Cui et al. prepared the Rydberg state of n = 20 and measured the power frequency electric field of 0∼2800 V/m [18], The group also studied the EIT effect of radio frequency modulation in the Rydberg atomic three-level system, using the odd-order sidebands of the 53S_1/2 state EIT spectrum to oscillate with the power frequency field to achieve the amplitude of the power frequency field measurement [19–21]. On the one hand, the atomic polarisability can be calculated through the numerical calculation method based on Coulomb approximation (CA) [22], and the B-spline basis set based on Dirac–Fock Core-Polarisation [23]. On the other hand, it can also be obtained directly through experimental measurements [24, 25].

Currently, what is widely used is to excite atoms in a fixed Rydberg state for low frequency electric field measurement. This does not take into account the optimal field strength measurement range corresponding to different Rydberg states. However, for 0∼kV/m in new power systems, it is inevitably impossible to measure the entire large-scale electric field through a Rydberg state. Therefore, it is necessary to study the structural properties of Rydberg atoms in large-scale super low frequency electric field measurements in order to perform accurate electric field measurements in Rydberg atom electric field sensing.

This paper first introduces the polarisation and penetration effects of the interaction between the outer electrons of the caesium atom and the nuclear charge, so the Coulomb potential is corrected, and then the Numerov algorithm is used to obtain the low l state (S, P, D, F state). Accurate atomic wave function, further diagonalise the energy matrix under the action of external field to obtain the Stark structure diagram of different Rydberg states. Compared with existing methods, this paper directly determines the E_avoid and α sizes corresponding to each Rydberg state from the Stark structure diagram, and further fits the relationship between α and E_avoid and the principal quantum number n. The relationship is obtained, and the mathematical formulae of α and E_avoid under different Rydberg states are obtained, thereby obtaining the optimal electric field measurement range corresponding to different Rydberg states. Before E_avoid, the frequency shift of the Rydberg atomic spectrum had a quadratic relationship with the field strength, and its coefficient is determined by the polarisability α of the atom. These two parameters are the key to accurate measurement of low-frequency electric fields in new power systems in the future.

THEORETICAL METHOD

Alkali Rydberg atoms wavefunction under the corrected Coulomb potential

The Schrödinger equation for alkali Rydberg atoms in zero field (in atomic units) [7]. 1 $$\left[-\frac{1}{2}{\nabla }^{2}+V(r)\right]\psi (r,\theta ,\phi )=W\psi (r,\theta ,\phi )$$ where V(r) is the improved and corrected Coulomb potential [26], W is the energy in zero field, ψ(r,θ,ϕ) is the wave function in zero field, and ∇² represents the Laplacian operator.

Electronic states and energies of alkali Rydberg atoms resemble those of hydrogen, but differ because of the interaction between the valence electrons and the atomic core. The valence electrons of an atomic state with a smaller angular quantum number l will penetrate into the atomic core, and the nuclear charge is no longer fully screened, the Coulomb potential alone cannot fully describe this effect. Therefore, this paper employs V(r) in place of the Coulomb potential and separates the wavefunction ψ(r,θ,ϕ) = R(r)Y(θ,ϕ), where Y(θ,ϕ) represents the spherical harmonics depending only on the angles θ and ϕ, and independent of V(r). Substituting this into Equation (1) yields the radial equation for R(r) 2 $$\frac{1}{R}\frac{d}{dr}\left({r}^{2}\frac{dR}{dr}\right)-2{r}^{2}[V(r)-W]=l(l+1)$$ where l is the angular quantum number. By introducing X = r^1/2R, x = lnr, the equation is transformed into a standard form of ordinary differential equation suitable for solving using the Numerov algorithm 3 $$\frac{{d}^{2}X}{d{x}^{2}}=g(x)X$$ where g(x) = 2e^2x(V(x) ‒ W) + (l+1/2)², and by setting r_j = r_sexp(‒jh) with h as the step size, obtaining the calculation formula for the radial wave function 4 $${X}_{i+1}=\frac{{X}_{i-1}\left({g}_{i-1}-12/{h}^{2}\right)+{X}_{i}\left(10{g}_{i}+24/{h}^{2}\right)}{12/{h}^{2}-{g}_{i+1}}$$

The calculation is performed using a numerical integration method from the outer region inward. The calculation is stopped when the radial wave function becomes divergent or oscillatory. The starting point r₁ = 2n(n + 15), with a step size of h = 0.01. The final expression for X is determined, and the radial wave function R(r) and the transition matrix element r ⟨W₁,l₁|r|W₂,l₂⟩ have the following form [27]. 5 $\begin{align*}R(r)=X/{r}^{1/2}\\ \left\langle {W}_{1},{l}_{1}\vert r\vert {W}_{2},{l}_{2}\right\rangle =\sum\limits _{i}{X}_{ii}{X}_{2i}{r}_{i}^{2}{\left(\sum\limits _{i}{X}_{ii}^{2}{r}_{1i}^{2}\sum\limits _{j}{X}_{2j}^{2}{r}_{2j}^{2}\right)}^{\tfrac{1}{2}}\end{align*}$

Stark effect of Rydberg atom under the influence of the electric field

The Hamiltonian for the atomic system in an external field consists of two components [28]. 6 $$\hat{H}={\hat{H}}^{0}+{\hat{H}}^{E}$$ where $${\hat{H}}^{0}=-1/2{\nabla }^{2}+V(r)$$ is the Hamiltonian in the absence of an electric field, and $${\hat{H}}^{E}=e\overrightarrow{r}\cdot \overrightarrow{E}=erE\,\cos \,\theta $$ represents the interaction between the super low frequency external field E and the atom.

The Schrödinger equations for the zero field and external field cases are as follows: 7 $\begin{align*}{\hat{H}}^{0}\vert {\varphi }_{k}\rangle ={\varepsilon }_{k}^{0}\vert {\varphi }_{k}\rangle \\ \left({\hat{H}}^{0}+{\hat{H}}^{E}\right)\vert {\psi }_{i}\rangle ={\mathcal{E}}_{i}\vert {\psi }_{i}\rangle \end{align*}$ where $${\varepsilon }_{k}^{0}$$ and |φ_k⟩ represent the eigenenergies and wavefunctions at zero field, while ε_i and |ψ_i⟩ represent the eigenenergies and wavefunctions in the presence of an external field.

Expressing the atomic wavefunction in the presence of an external field as a linear combination of K zero field wavefunctions, denoted as $$\vert {\psi }_{i}\rangle =\sum \nolimits_{k=1}^{K}{a}_{ik}\vert {\varphi }_{k}\rangle $$, with a_ik as the wavefunction coefficients, substituting this into the second term of Equation (7), and multiplying both sides by ⟨φ_m| (with m ranging from 1 to K) get the following equation, 8 $$\sum\limits _{k=1}^{N}{a}_{ik}\left({\mathcal{E}}_{k}^{0}{\delta }_{mk}+\left\langle {\varphi }_{m}\vert {\hat{H}}^{E}\vert {\varphi }_{k}\right\rangle \right)={\mathcal{E}}_{i}{a}_{im}$$

Written in the form of a matrix equation 9 $$\left(\begin{array}{@{}ccc@{}}{\varepsilon }_{11}^{0}+{H}_{11}^{E}& \text{\ldots }& {H}_{1K}^{E}\\ {\vdots}& {\ddots}& {\vdots}\\ {H}_{K1}^{E}& \text{\ldots }& {\varepsilon }_{K}^{0}+{H}_{KK}^{E}\end{array}\right)\left(\begin{array}{@{}c@{}}{a}_{i1}\\ {\vdots}\\ {a}_{iK}\end{array}\right)={\varepsilon }_{i}\left(\begin{array}{@{}c@{}}{a}_{i1}\\ {\vdots}\\ {a}_{iK}\end{array}\right)$$ where $${H}_{mk}^{E}=\left\langle {\varphi }_{m}\vert \right.{\hat{H}}^{E}\vert {\varphi }_{k}\rangle $$, diagonalising the above matrix equation will yield the eigenenergies of the Stark structure diagram.

CALCULATION RESULTS

According to the quadratic Stark theory, Rydberg atoms exhibit the following relationship with an external electric field [29]. 10 $${\Delta }=-\frac{1}{2}\alpha {E}^{2}$$ where ∆ represents the change in Stark energy induced by the external electric field E, and α is the atomic polarisability.

It can be seen from the above formula that in order to achieve accurate measurement of the external electric field, the accurate value of the polarisability must be obtained. According to the wave function theory of caesium atoms under modified Coulomb potential proposed in Section 2.1, Python program was written to solve the more accurate radial wave function R(r) of Rydberg atoms. Figure 1 shows in Section A of Part two, program in Python to solve the more accurate radial wave function R(r) of Rydberg atoms, which is the solution to Equation (5). The figure below shows the relationship between rR(r) and r (the unit is Bohr radius a₀) when the principal quantum number n = 25 and the angular quantum number l = 0, 1, 2 (only the results of solutions to three wave functions are listed, in fact, for each n, there are l = n-1 wavefunctions.), Furthermore, Equations (6–10) are combined to solve the Stark structure diagram of Rydberg atom under the action of external field.

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At present, most electric field measurements based on Rydberg atoms use two-photon excitation energy level schemes. According to the transition selection rule, the excited Rydberg state is the nS_1/2 state (S corresponds to the angular quantum number l = 0). Use the energy matrix diagonalisation theory in Section B of Part two combined with the radial wave function R(r) of the Rydberg atom to solve the Stark structure diagram of the Rydberg atom under the action of an external field. Figure 2 shows the Stark structure diagram of the caesium atom in the 25S_1/2 state and its adjacent states. The electric field range is 0∼75 V/cm. For the fan-shaped region of n = 21, the atomic energy levels are degenerated at zero field due to the approximately equal quantum defect numbers. Under the condition of external electric field, they gradually split and show the linear Stark effect. The low l state shows a non-linear Stark shift. Specifically, as shown by the red solid line in Figure 2a, the 25S_1/2 state (electric field strength ≤ E_avoid ≈ 36.4 V/cm), and Stark energy initially decreases in a quadratic form as the electric field intensity increases. Enlarge this part of the Stark structure diagram (grey rectangular shaded part) as shown in Figure 2b. Setting the Stark energy of the 25S_1/2 state to relative zero when E = 0. The Stark energy change in this range is fitted according to formula (10) to obtain the coefficient of the curve, that is, the polarisability of the atom α[25S_1/2] = 0.3209 MHz⋅cm²/V², fitting correlation coefficient R² = 0.9999. After the electric field intensity is greater than E_avoid, the linear Stark effect replaces the previous quadratic Stark effect, and there is a behaviour of avoiding cross with the adjacent energy levels 21F_5/2、7/2, as shown in Figure 2c.

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From the above analysis, it is evident that the Stark structure diagram provides information about α and E_avoid for different nS_1/2 states. In this paper, calculations of α and E_avoid have been performed based on the Stark structure diagrams for various nS_1/2 states within the principal quantum number range of n = 10∼70. Table 1 presents some of the calculated results.

TABLE 1 Atomic polarisability α [in MHz/(V/cm)²] and initial avoided crossing electric field E_avoid [in V/cm] for nS_1/2 states.

n	α	Eavoid
10	0.00012	43,360
11	0.00033	15,200
12	0.00074	6500
13	0.00147	3680
14	0.00302	1890
15	0.00538	1153
16	0.00913	721.7
17	0.0152	470.3
20	0.0582	157.3
21	0.0857	113.2
22	0.1238	85.8
23	0.1737	65.1
24	0.2374	49.9
25	0.3210	36.4
30	1.292	13.1
35	4.154	5.36
39	8.837	2.91
40	11.06	2.53
42	15.53	1.92
44	21.34	1.47
46	29.31	1.16
47	34.25	1.04
48	39.92	0.92
49	46.30	0.81
50	53.46	0.74
55	106.3	0.44
60	197.7	0.27
65	349.6	0.18
66	389.6	0.16
67	433.5	0.15
68	481.8	0.14
69	534.7	0.12
70	592.6	0.12

It can be seen from Table 1 that the atomic polarisability increases rapidly with the principal quantum number. It ranges from 0.00,012 MHz⋅cm²/V² for the 10S_1/2 state to 592.6 MHz⋅cm²/V² for the 70S_1/2 state, experiencing a change of 10⁵ orders of magnitude. The maximum measurable electric field for each nS_1/2 state is determined by its corresponding avoided crossing electric field. This provides strong support for achieving a wide range of electric field measurements. To validate the correctness of the calculated results, this paper further compares them with previous work. Figure 3a shows α of nS_1/2 states for different principal quantum numbers n. The red solid line represents the fitting curve based on the results of this paper. The figure also displays experimental results for n = 10–13 [30], n = 65, 67, and 70 [31], as well as theoretical results for n = 10–17, 39, 50 [23], and n = 20–70 [32]. The inset in Figure 3a magnifies the results for low n values. It can be observed that the calculated results in this paper are in good agreement with the experimental measurements by Wijngaarden et al., with errors of no more than 1%. However, after n exceeds 65, there is a larger deviation compared to the measurements by Bai et al., which may be attributed to significant errors in the transition matrix element calculations. Nevertheless, for high n values, the results based on B-spline functions and the alkali metal atomic model potential have a maximum deviation of less than 8% from the results obtained in this paper. Figure 3b illustrates the initial avoidance-crossing electric field for different n, which undergoes a rapid change in the range of n = 10–70. The results within the range of n = 24–28 are magnified and compared with previous results (inset in Figure 3b, green triangles [33]). The maximum deviation between the two sets of results is less than 6%.

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Due to the increasing number of wavefunction levels that need to be considered in the calculations as n increases, which is approximately ten times of n, the higher order matrices need to be diagonalised for high n calculations. In this paper, empirical formulae for α and E_avoid based on low n states have been derived, and they have the following relationships [34, 35]: 11 $\begin{align*}\alpha =A\times {({n}^{\ast })}^{6}+B\times {({n}^{\ast })}^{7}\\ {E}_{\text{avoid}}=\frac{C}{{n}^{\ast 5}}+\frac{D}{{n}^{\ast 7}}\end{align*}$ where α is measured in MHz·cm²/V², and E_avoid is measured in V/cm. n* represents the effective quantum number, with n* = n ‒ δ, δ is the quantum defect [36].

Using the least squares fitting method, the obtained results are A = (2.25,033 ± 0.01,704) × 10^‒9 and B = (7.49,948 ± 0.02,754) × 10⁻¹¹, with fitting correlation coefficient R² = 1. The results reported in ref. [30], A = 2.52 × 10⁻⁹ and B = 6.62 × 10⁻¹¹, were obtained within the range of n = 15–55, which is the main reason for the discrepancy between the two sets of values. As for E_avoid, there is currently no empirical formula specifically tailored for the initial avoidance-crossing electric field of Cs. In this paper, based on the calculated results within the range of n = 10–70, the fitting coefficients C = (1.68,868 ± 0.05,471) × 10⁸ and D = (2.45,991 ± 0.37,629) × 10⁹ were obtained with R² = 0.9989.

EXPERIMENTAL RESULTS

In the experiment, a two-photon driven ladder-type three-level system is established. The probe laser has a power of 50∼100 μW and is focused to 0.1 mm full width at half maximum (FWHM). The coupling laser has a power of 80∼150 mW and is focused to 0.5 mm FWHM. The atom is gradually excited to the 35S_1/2 state, the corresponding atomic polarisability α[35S_1/2] = 4.154 MHz⋅cm²/V², avoided crossing electric field E_avoid = 5.36 V/cm.

In addition, in order to eliminate the shielding effect of the vapour cell on the super low frequency electric field in the power system, we placed the electrode plates in a vapour cell with a length of 5 cm and a diameter of 2 cm, and the distance between the two electrode plates is d = 0.6 cm. As shown in the inset in Figure 4. The inside of the vapour cell is filled with Cs and reacts with the electric field between the electrode plates. The amplitude of the voltage U applied by the signal source is controlled between 0 and 6 V, and the normalised EIT spectra under different electric fields U/d are finally obtained, as shown in the figure below.

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It can be seen that when E = 0, the spectral frequency shift amount Δ = 0, the line width is about 20 MHz. When E = 5 V/cm, Δ is about 48.4 MHz, Δ and E satisfy a good quadratic relationship, accompanied by spectrum broadening and amplitude reduction, at this time, the line width is around 37.2 MHz, and the amplitude is reduced to 0.56. Continue to increase the field strength E to 6 V/cm (>E_avoid = 5.36 V/cm), the field strength is greater than E_avoid, and the spectrum is distorted. When E = 10 V/cm, the spectrum completely crosses E_avoid, and the EIT spectrum of the adjacent Rydberg state of 35S_1/2 appears.

In summary, when the electric field is small, the EIT spectral Stark shift and the electric field satisfy a good quadratic relationship but when the electric field is too large, the linewidth of the EIT spectrum broadens and the amplitude decreases. Therefore, for the measurement of super low frequency electric fields in power systems, if the field strength is too large, Rydberg atoms with smaller n should be excited (smaller n corresponds to smaller α), so that Δ will not be very large, and there will be no linewidth broadening and amplitude reduction. This study determines the Rydberg state of atoms that should be excited in different electric field ranges, which is helpful to realise a wide range of super low frequency electric field measurements in power systems.

CONCLUSION AND DISCUSSION

This paper has optimised the Stark structure of Cs in the super low frequency electric field by considering the polarisation and penetration effects of valence electrons in the low l state. In comparison to existing methods, this paper directly fits α and E_avoid from the Stark structure diagram. The accurate calculation of α directly affects the measurement precision of the electric field, while E_avoid determines the range of measurable E. This is a crucial step in connecting the Rydberg atom Stark effect with the precise measurement of super low frequency electric fields in power systems. The calculated results of this paper are basically consistent with experimental measurements. The error between the atomic polarisability and experimental results is within 1% for low n states and doesn't exceed 8% for higher n states. Similarly, for E_avoid, the error is within 6% for n = 24–28 compared to existing measurements. Additionally, based on these results, this paper uses the least squares fitting method to obtain two empirical formulae for the α and E_avoid of Cs. These empirical formulae can be used to calculate the polarisability and level-avoidance crossing positions for high-Rydberg states of Cs, particularly in the study of heavy alkali metal atom Rydberg state Stark level structures. At high n states, there is a large error between the fitting function obtained in this article and the experimental data, which requires more complete and accurate theoretical research in the future to make up for this flaw. Fortunately, for super low frequency strong electric field measurement in power systems, the value of n needs to be small enough. From the relationship given by Equation (10), it can be seen that when E is very large, if the frequency shift amount Δ of the EIT spectrum is very large, it will cause the spectrum to broaden and the laser peak to become lower. The atomic polarisation of Rydberg atoms in the low n state has The rate α is small enough so that the frequency shift Δ of the EIT spectrum will not be very large.

It should also be noted that Rydberg atoms are formed by laser excitation of ground state atoms to the Rydberg state, so different n uses different laser wavelengths. Currently, a two-photon scheme is generally used to form Rydberg atoms, as shown in the inset in Figure 5, for caesium atoms, the probe laser locked at the resonance frequency (corresponding to the wavelength of 852 nm) causes the caesium atom to transition from the ground state |6S_1/2> to the first excited state |6P_3/2>, accompanied by the atomic Resonance absorption. In addition, the wavelength of 509 nm corresponds to the centre frequency, and the frequency-sweeping input coupling laser causes the caesium atom to transition from |6P_3/2> to the Rydberg state |nS_1/2> (n = 10∼70, corresponding to different coupling laser frequencies). According to Bohr's theory, the energy difference ΔW between the two states corresponds to the wavelength of the laser, that is, ΔW = h⋅c/λ, where h is Planck's constant, c is the speed of light, and λ is the wavelength of the laser. Knowing |6P_3/2> and |nS_1/2>, the corresponding laser wavelength can be obtained. For caesium atoms, the relationship between λ and n is as shown in the figure below.

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At present, the tuning range of lasers is generally within 5 nm. As can be seen from Figure 4, the low n state Rydberg atoms required for ultra-low frequency electric field measurements can only be in a certain fixed nS_1/2 state (fixed α and E_avoid). Take measurements. The limitations of lasers prevent us from tuning in a wide range of Rydberg states, which requires more advanced technology and laser equipment in the future to meet the requirements.

CONFLICT OF INTEREST STATEMENT

The authors declare no conflicts of interest.

DATA AVAILABILITY STATEMENT

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

References

Chen, W., et al.: Review of optical fibre sensors for electrical equipment characteristic state parameters detection. High Volt. 4(4), 271–281 (2019). [DOI: https://dx.doi.org/10.1049/hve.2019.0157]