Rydberg atom EIT-Stark spectrum analysis method under action of low-frequency electric field

Abstract

The invention discloses a Rydberg atom EIT-Stark spectrum analysis method under the action of a low-frequency electric field. The method comprises the following steps: collecting an EIT-Stark spectrum signal under the action of the low-frequency electric field; performing smooth envelope extraction on the EIT-Stark spectral signal under the action of the low-frequency electric field to obtain an envelope signal; performing multi-scale wavelet analysis on the envelope signal, and extracting a transmission peak position to obtain a Stark frequency shift amount; and inverting the amplitude of the low-frequency electric field in combination with the Stark effect theory. According to the method, the influence of environment noise and measurement system noise on a measurement result is effectively avoided, the Stark frequency shift amount can be intuitively extracted, and low-frequency electric field measurement with wide bandwidth and large dynamic range can be accurately and reliably carried out.

Description

Technical Field

[0001] The present invention relates to the field of laser technology, and in particular to a Rydberg atom EIT-Stark spectrum analysis method under the action of a low-frequency electric field. Background Art

[0002] Under the action of low-frequency electric fields, Rydberg atoms still possess their inherent characteristics and advantages. However, their spectral characteristics are different from the significant frequency shift characteristics exhibited by DC-Stark under extremely low frequencies and AC-Stark under radio frequencies. Instead, they exhibit a complex "comprehensive response," making it difficult to intuitively extract the Stark frequency shift. Further research is needed on the response mechanism and spectral peak morphology characteristics to analyze the electric field information in the spectrum. In addition, environmental noise and measurement system noise can also affect the EIT-Stark spectral characteristics, increasing the difficulty of spectral information analysis and electric field inversion. These two major issues significantly limit the wide-bandwidth, large-dynamic-range electric field measurements of Rydberg atoms.

[0003] The EIT transmission peak exhibits periodic frequency shifts under kHz-level electric fields, but these are difficult to clearly observe. Actual spectra observed under low-frequency electric fields are composed of numerous pulse signals, lacking clear peak splitting and frequency shifts. Direct Gaussian fitting or wavelet analysis of such spectra is ineffective. To analyze the EIT-Stark spectrum of Rydberg atoms under kHz-level low-frequency electric fields, Michael J. Lim et al. at Rowan University in the United States used numerical integration to quantify the average spectral line shape under the action of the electric field. They then used Gaussian fitting to effectively extract the frequency shift characteristics of the EIT transmission peak, thereby enabling the analysis of electric field information in spectra under low-frequency electric fields. However, this analysis method is susceptible to environmental noise, and the feature extraction process introduces manual intervention by the researcher, making it suitable for laboratory research but not for engineering applications. Summary of the Invention

[0004] In view of the above-mentioned deficiencies in the prior art, the present invention provides a Rydberg atom EIT-Stark spectrum analysis method under the action of a low-frequency electric field, which solves the problem that the prior art is greatly affected by environmental noise and limits the Rydberg atom bandwidth.

[0005] In order to achieve the above-mentioned object of the invention, the technical solution adopted by the present invention is: a method for analyzing the EIT-Stark spectrum of Rydberg atoms under the action of a low-frequency electric field, characterized by comprising: Collect EIT-Stark spectrum signals under the action of low-frequency electric field; The smooth envelope of the EIT-Stark spectrum signal under the action of low-frequency electric field is extracted to obtain the envelope signal; Perform multi-scale wavelet analysis on the envelope signal to extract the transmission peak position and obtain the Stark frequency shift; Based on the Stark frequency shift and combined with the Stark effect theory, the amplitude of the low-frequency electric field is inverted.

[0006] Furthermore, the specific method for extracting the smooth envelope of the EIT-Stark spectrum signal under the action of the low-frequency electric field includes: Extract the regional maximum within a fixed region length from the EIT-Stark spectrum signal under the action of a low-frequency electric field and obtain the extreme point; Use the cubic smoothing spline interpolation function to fit the extreme points and obtain the upper envelope; Perform segmented optimization on the upper envelope; The parameters of the cubic smoothing spline interpolation function are adjusted and refitted to obtain the optimal envelope signal.

[0007] Furthermore, the cubic smoothing spline interpolation function is the optimal spline curve function obtained by minimizing the objective function. The expression of the objective function is:

[0008] in, is the smoothing factor, N is the number of extreme points, t represents the extreme point index, are the coordinates of the extreme points, is the cubic smoothing spline interpolation function, z is the minimum horizontal coordinate of the extreme point, y is the maximum horizontal coordinate of the extreme point, for The second derivative of .

[0009] Furthermore, the specific method for segmented optimization of the upper envelope is as follows: Calculate the standard deviation of each data segment, the expression is:

[0010] in, is the standard deviation, n is the signal length, i is the signal data segment index, is the first i The discrete data values ​​of the data segments, for The average value of according to The relationship between the data segment and the preset standard deviation threshold determines the signal category of the data segment, where the signal category includes background signal and valid signal; For the data segment determined to be background signal, the smoothing factor Set to a value close to 0.

[0011] Furthermore, the specific method for adjusting the parameters of the cubic smoothing spline interpolation function is: Calculate the kurtosis, unsmoothness and integrated absolute error of the upper envelope respectively; For the data segment of the valid signal, a smoothing factor is selected such that the kurtosis is greater than the set value, the unsmoothness is less than the set value, and the integral absolute error is less than the set value; The calculation expression of kurtosis is:

[0012] in, represents the kurtosis, are discrete data values ​​in the envelope signal, for The mean of The calculation expression of roughness is:

[0013] in, is the roughness, is the second-order difference of the envelope signal data; The expression of the integrated absolute error is:

[0014] in, is the integrated absolute error, is the envelope signal, is the original EIT-Stark spectrum signal.

[0015] Furthermore, the envelope signal is subjected to multi-scale wavelet analysis, and the specific method for extracting the transmission peak position is as follows: Select the wavelet scale, and its expression is:

[0016] in, is the wavelet scale, is a constant; Perform multi-scale wavelet analysis on the envelope signal to obtain the wavelet coefficients of the envelope signal at different scales; Extract the local maximum of the wavelet coefficients in each scale dimension through a sliding window; Connect the extreme points on adjacent scales to construct ridge lines; The wavelet coefficient threshold is set by an adaptive threshold selection strategy based on data distribution; The ridge lines that are greater than the wavelet coefficient threshold are retained to obtain the optimized ridge lines; The maximum wavelet coefficient amplitude of a single ridge line in the optimized ridge line at a specific scale is analyzed to determine the centroid position of the transmission peak.

[0017] Furthermore, the adaptive threshold selection strategy based on data distribution is specifically: using the 97% quantile of the wavelet coefficient as the dynamic threshold.

[0018] Furthermore, the calculation expression of the amplitude of the low-frequency electric field is:

[0019] in, is the amplitude of the low-frequency electric field, Rydberg state The polarizability, is the Stark frequency shift under the action of the electric field.

[0020] The beneficial effects of the present invention are: 1. Using an extreme value-smoothing spline envelope extraction algorithm and a segmented optimization approach, we configured different smoothing parameters for different noise levels in different data segments, minimizing the impact of spurious signals on subsequent peak-finding operations. Furthermore, by introducing a targeted parameter evaluation method to assist in smoothing parameter selection, we achieved both signal feature preservation and optimal control of the degree of smoothing.

[0021] 2. Through optimized multi-scale wavelet transform feature extraction, the wavelet coefficients are screened based on statistical characteristics, and the position of the center of mass of the spectral peak is estimated by using the value of the maximum wavelet coefficient corresponding to a certain scale on a single ridge line. This ensures the accuracy of peak finding, avoids the errors introduced by manual peak marking in traditional methods, and improves the accuracy and reliability of spectral analysis. It is of great significance for the measurement of electric field (or voltage) parameters at the kHz level. BRIEF DESCRIPTION OF THE DRAWINGS

[0022] Figure 1 A flow chart of a method for analyzing the EIT-Stark spectrum of Rydberg atoms under a low-frequency electric field provided in an embodiment; Figure 2 This is a schematic diagram of envelope signal extraction results; Figure 3 is the two-dimensional wavelet coefficient image of the EIT-Stark spectrum envelope; Figure 4 This is the ridge extraction result based on local maximum detection; Figure 5 This is the result image of the optimized ridge extraction; Figure 6 Schematic diagram of the centroid position of the transmission peak; Figure 7 Schematic diagram of the response frequency range. DETAILED DESCRIPTION

[0023] The specific embodiments of the present invention are described below to facilitate understanding of the present invention by those skilled in the art. However, it should be clear that the present invention is not limited to the scope of the specific embodiments. For those skilled in the art, as long as various changes are within the spirit and scope of the present invention as defined and determined by the appended claims, these changes are obvious, and all inventions and creations utilizing the concepts of the present invention are protected.

[0024] like Figure 1 As shown, in one embodiment of the present invention, a method for analyzing the EIT-Stark spectrum of Rydberg atoms under the action of a low-frequency electric field includes the following steps: S1, collect EIT-Stark spectrum signals under the action of low-frequency electric field; S2, performing smooth envelope extraction on the EIT-Stark spectrum signal under the action of the low-frequency electric field to obtain an envelope signal; S2-1. Extract the regional maximum within a fixed region length from the EIT-Stark spectral signal under the action of a low-frequency electric field, obtain the extreme point, and use it as the key position for subsequent cubic smoothing spline calculation, so that the envelope can wrap the top of the signal as accurately as possible, retaining the morphological characteristics of the approximate projection peak.

[0025] S2-2, use the cubic smoothing spline interpolation function to fit the extreme points to obtain the upper envelope; The cubic smoothing spline interpolation function is the optimal spline curve function obtained by minimizing the objective function. The expression of the objective function is:

[0026] in, is the smoothing factor, N is the number of extreme points, t represents the extreme point index, are the coordinates of the extreme points, is the cubic smoothing spline interpolation function, z is the minimum horizontal coordinate of the extreme point, y is the maximum horizontal coordinate of the extreme point, for The fitting term (the first term) is the residual sum of squares between the original value and the fitted value, reflecting the goodness of fit; the penalty term (the second term) is the curvature integral of the penalty curve, emphasizing overall smoothness. Therefore, when When it is close to 1, the fitting curve will pass through as many original data points as possible, resulting in a rougher curve. When it is close to 0, the fitting curve will be closer to linear, resulting in a smoother curve.

[0027] S2-3, perform segmented optimization on the upper envelope; In the effective signal, it is necessary to avoid over-smoothing and retain the signal's morphological characteristics similar to the transmission peak; in the background signal, the signal needs to be smoothed as much as possible to avoid the influence of stray signals on subsequent peak-finding operations. There are significant differences in data distribution between the effective signal and the background signal. The signal and background signal are quantified and distinguished by calculating the standard deviation of each data segment. The expression is:

[0028] in, is the standard deviation, n is the signal length, i is the signal data segment index, is the first i The discrete data values ​​of the data segments, for The experiment shows that the standard deviation of the background signal and the effective signal is significantly different, and this significant difference also provides a clear basis for setting the standard deviation threshold.

[0029] In this embodiment, 1 / 10 of the signal length is used as the length of the data segment, and the signal category of the data segment is determined based on the relationship between the calculated standard deviation of each data segment and the preset standard deviation threshold. The signal categories include background signal and valid signal. For the data segment determined to be background signal, the smoothing factor Set to a value close to 0.

[0030] S2-4, adjusting the parameters of the cubic smoothing spline interpolation function and re-fitting to obtain the optimal envelope signal; The present invention introduces three evaluation parameters to assist in optimizing the parameters of the cubic smoothing spline interpolation function, namely kurtosis, unsmoothness and integrated absolute error; The calculation expression of kurtosis is:

[0031] in, represents the kurtosis, are discrete data values ​​in the envelope signal, for The larger the kurtosis value is, the sharper the envelope signal is.

[0032] The calculation expression of roughness is:

[0033] in, is the roughness, is the second-order difference of the envelope signal data. A higher roughness value indicates more residual high-frequency oscillation components in the envelope signal, and a poor smoothing effect.

[0034] The expression of the integrated absolute error is:

[0035] in, is the integrated absolute error, is the envelope signal, is the original EIT-Stark spectrum signal. A larger value of the calculated integral absolute error means a larger error between the envelope signal and the original signal, which means that the envelope signal is less closely aligned with the original signal.

[0036] Based on the above three parameters, the smoothing factor is traversed and optimized. For the data segment of the valid signal, a smoothing factor is selected that satisfies the conditions that the kurtosis is greater than the set value, the unsmoothness is less than the set value, and the integral absolute error is less than the set value, so as to achieve the reasonable control of the retention of signal characteristics and the smoothing process.

[0037] After the above envelope signal extraction, the extraction results are as follows Figure 2 As shown in the figure, it can be seen that the envelope signal successfully extracts the effective part of the original signal, and the amplitude of the background signal area is close to zero, which has a good effect on suppressing background noise, thereby achieving higher extraction accuracy.

[0038] After signal envelope extraction, the envelope signal exhibits a three-peak spectral peak morphology. To accurately quantify the frequency shift parameter ΔStark, an effective mutation detection algorithm is used to identify significant singular points in the signal, thereby accurately extracting the characteristic position parameters of the spectral peak.

[0039] S3, perform multi-scale wavelet analysis on the envelope signal, extract the transmission peak position, and obtain the Stark frequency shift; To address the problem of high redundancy in continuous wavelet transform operations, this paper discretizes and optimizes the wavelet scale, reducing the redundancy of the algorithm while retaining effective information. When the scale is too small, the resolution is too high, and almost no information can be extracted from the envelope signal extracted by the smoothing spline. When the wavelet coefficients are too large, the resolution is too high, and the wavelet coefficients will transmit excessive baseline fluctuation interference information. Therefore, based on the above analysis, the optimal wavelet scale is:

[0040] in, is the wavelet scale, This wavelet scale selection method focuses on the part that highlights the transmission peak information and reduces the redundancy of the calculation.

[0041] Perform multi-scale wavelet analysis on the envelope signal to obtain the continuous wavelet transform coefficients of the envelope signal at different scales, such as Figure 3 As shown; The mathematical model of wavelet transform is:

[0042] in, a is the wavelet scale, b is the wavelet translation, ψ ( t ) is the mother wavelet function, ψ a,b ( t ) is the wavelet basis function after scaling and translation, C ( a , b ) are wavelet coefficients, S ( t ) is the signal to be processed, i.e., the extracted envelope signal. To ensure morphological matching, time-frequency resolution, and signal phase characteristics, this embodiment selects the Mexican Hat Wavelet, which has a shape similar to the EIT transmission peak, to accurately capture the spectral peak characteristics. The Mexican Hat mother wavelet function is expressed as:

[0043] The local maximum of the wavelet coefficients at each scale dimension is extracted through a sliding window; then the extreme points at adjacent scales are connected to construct the ridge line, such as Figure 4 As shown in the figure, it can be seen that the initial ridge line extraction based on local maximum detection still has a lot of noise interference. To address this problem, the present invention combines the statistical distribution characteristics of the wavelet coefficients and sets the wavelet coefficient threshold through an adaptive threshold selection strategy based on data distribution. That is, the 97% quantile of the wavelet coefficient is used as the dynamic threshold, thereby further optimizing the feature extraction range and improving the positioning accuracy. After setting the threshold, the ridge lines greater than the wavelet coefficient threshold are retained to obtain the optimized ridge line, as shown in the figure. Figure 5 shown.

[0044] Finally, the maximum wavelet coefficient amplitude of a single ridge in the optimized ridge at a specific scale is analyzed to determine the centroid position of the transmission peak, such as Figure 6 shown.

[0045] S4. Invert the amplitude of the low-frequency electric field by combining the Stark effect theory.

[0046] The calculation expression of the amplitude of the low-frequency electric field is:

[0047] in, is the amplitude of the low-frequency electric field, Rydberg state The polarizability, is the Stark frequency shift under the action of the electric field.

[0048] In order to detect the frequency band adapted by the present invention, this embodiment locks the frequencies of the detection light and the coupled light to detect the periodic dynamic response of the electromagnetic induced transparency (EIT) signal intensity at a specific frequency point under AC electric field modulation, such as Figure 7 As shown in Table 1, it can be seen that the proposed method can reliably measure low-frequency electric fields within 500 kHz. Furthermore, to test the measurement accuracy of this method, a low-frequency electric field measurement experiment was conducted using parallel metal plates to apply an electric field, combined with the spectral analysis method proposed in this invention. The measurement results, shown in Table 1, all had measurement errors less than 0.55%, demonstrating high accuracy.

[0049] Table 1

[0050] In summary, the spectral analysis method based on the extreme value-smoothing spline envelope extraction algorithm and the optimized multi-scale wavelet transform algorithm provided by the present invention effectively avoids the influence of environmental noise and measurement system noise on the measurement results, can intuitively extract the Stark frequency shift, and realize accurate and reliable low-frequency electric field measurement with wide bandwidth and large dynamic range.

Claims

1. A method for analyzing the EIT-Stark spectrum of Rydberg atoms under the action of a low-frequency electric field, characterized in that: include: Collect EIT-Stark spectrum signals under the action of low-frequency electric field; The smooth envelope of the EIT-Stark spectrum signal under the action of low-frequency electric field is extracted to obtain the envelope signal; Perform multi-scale wavelet analysis on the envelope signal to extract the transmission peak position and obtain the Stark frequency shift; Based on the Stark frequency shift and combined with the Stark effect theory, the amplitude of the low-frequency electric field is inverted.

2. The method according to claim 1, characterized in that The specific method for extracting the smooth envelope of the EIT-Stark spectrum signal under the action of a low-frequency electric field includes: Extract the regional maximum within a fixed region length from the EIT-Stark spectrum signal under the action of a low-frequency electric field and obtain the extreme point; Use the cubic smoothing spline interpolation function to fit the extreme points and obtain the upper envelope; Perform segmented optimization on the upper envelope; The parameters of the cubic smoothing spline interpolation function are adjusted and refitted to obtain the optimal envelope signal.

3. The method according to claim 2, characterized in that The cubic smoothing spline interpolation function is the optimal spline curve function obtained by minimizing the objective function. The expression of the objective function is: in, is the smoothing factor, N is the number of extreme points, t represents the extreme point index, are the coordinates of the extreme points, is the cubic smoothing spline interpolation function, z is the minimum horizontal coordinate of the extreme point, y is the maximum horizontal coordinate of the extreme point, for The second derivative of .

4. The method according to claim 2, characterized in that The specific method for segmented optimization of the upper envelope is: Calculate the standard deviation of each data segment, the expression is: in, is the standard deviation, n is the signal length, i is the signal data segment index, is the first i The discrete data values ​​of the data segments, for The average value of according to The relationship between the data segment and the preset standard deviation threshold determines the signal category of the data segment, where the signal category includes background signal and valid signal; For the data segment determined to be background signal, the smoothing factor Set to a value close to 0.

5. The method according to claim 4, characterized in that The specific method of adjusting the parameters of the cubic smoothing spline interpolation function is: Calculate the kurtosis, unsmoothness and integrated absolute error of the upper envelope respectively; For the data segment of the valid signal, a smoothing factor is selected such that the kurtosis is greater than the set value, the unsmoothness is less than the set value, and the integral absolute error is less than the set value; The calculation expression of kurtosis is: in, represents the kurtosis, are discrete data values ​​in the envelope signal, for The mean of The calculation expression of roughness is: in, is the roughness, is the second-order difference of the envelope signal data; The expression of the integrated absolute error is: in, is the integrated absolute error, is the envelope signal, is the original EIT-Stark spectrum signal.

6. The method according to claim 4, characterized in that The specific method of performing multi-scale wavelet analysis on the envelope signal and extracting the transmission peak position is as follows: Select the wavelet scale, and its expression is: in, is the wavelet scale, is a constant; Perform multi-scale wavelet analysis on the envelope signal to obtain the wavelet coefficients of the envelope signal at different scales; Extract the local maximum of the wavelet coefficients in each scale dimension through a sliding window; Connect the extreme points on adjacent scales to construct ridge lines; The wavelet coefficient threshold is set by an adaptive threshold selection strategy based on data distribution; The ridge lines that are greater than the wavelet coefficient threshold are retained to obtain the optimized ridge lines; The maximum wavelet coefficient amplitude of a single ridge line in the optimized ridge line at a specific scale is analyzed to determine the centroid position of the transmission peak.

7. The method according to claim 6, characterized in that The adaptive threshold selection strategy based on data distribution is as follows: the 97% quantile of the wavelet coefficient is used as the dynamic threshold.

8. The method according to claim 6, characterized in that The calculation expression of the amplitude of the low-frequency electric field is: in, is the amplitude of the low-frequency electric field, Rydberg state The polarizability, is the Stark frequency shift under the action of the electric field.

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