# Noise Differentiation and Atom Number Measurement in Optical Lattice Clocks by Analyzing Clock Stabilities with Various Parameters

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

_{0}) or the photon count (γ

_{0}) detected per atom by the photoelectric detector, it becomes possible to extract the contributions of each noise responsible for clock stability. This method allows us to accurately determine the value of N

_{0}by differentiating QPN from other sources of noise. To further enhance precision, we use numerical simulations to investigate how modulation parameters influence the estimated uncertainty of N

_{0}and how measurement precision evolves as N

_{0}increases.

## 2. Methods

#### 2.1. Theory

_{a}at τ = 1 can be represented by [23,26]

_{0}representing the clock cycle time, and S

_{0}= 0.6π × T

_{p}denoting the frequency-sensitive slope of the spectrum at half-height point, where T

_{p}is the interrogation time for the clock transition in each clock cycle. ${\sigma}_{S\mathrm{hot}}^{2}={T}_{0}/4{S}_{0}^{2}{N}_{0}{\gamma}_{0}$ refers to the variance originating from the photon shot noise, while ${\sigma}_{D\mathrm{et}}^{2}={T}_{0}{\delta}_{\mathrm{N}}^{2}/2{S}_{0}^{2}{N}_{0}^{2}$ represents the contribution of the technical noise. Here, δ

_{N}stands for the electronic detection noise, including the contribution of the amplifier, digitizer, dark current noise, and so on. As the three types of noise exhibit distinct dependencies on N

_{0}and γ

_{0}; it becomes possible to differentiate them effectively by modulating N

_{0}and γ

_{0}.

_{det}), γ

_{0}can be manipulated. Specifically, by adjusting T

_{det}to T

_{det}/α while keeping other parameters constant as defined in Equation (1), the value of γ

_{0}will change to γ

_{0}/α. Consequently, the overall clock stability can be represented as

_{0}to N

_{0}/β, we can separate the technical noise from other noises, and the overall clock stability is denoted by

_{QPN}is obtained using Equation (4), it becomes possible to determine the absolute atom number in an OLC.

#### 2.2. Numerical Simulation Method

_{p}, T

_{0}, N

_{0}, γ

_{0}, δ

_{N}, α, and β, and calculate S

_{0}, σ

_{a}, σ

_{b}, and σ

_{c}. The noise-induced frequency fluctuation is represented by the discrete normal random numbers with the standard deviation of σ

_{a}for case 1, σ

_{b}for case 2, and σ

_{c}for case 3. In step 2, the clock comparison process between two clocks is executed with a simulated time of approximately 2.2 h for each case. The total simulated time is obtained by multiplying the total clock cycle number by 3T

_{0}, where the reason of the use of 3T

_{0}is that three measurements are required in real-world scenarios to determine the values of σ

_{a}, σ

_{b}, and σ

_{c}. In this work, the Dick effect is cancelled by setting the clock laser noise to be zero [27]. The cancellation of the Dick effect can be realized in experiment by synchronous frequency comparison between two clocks [23,28], or by using the in situ imaging technique to compare two regions of cold ensembles in a clock [18,19,22,25]. Regarding the simulation aspect of the in-loop clock operation, the excitation fractions at half-height points are determined by adjusting the center frequency of the Rabi spectrum by ±0.4/T

_{p}. Subsequently, the corresponding generated discrete normal random numbers from step 1 are added to the determined excitation fraction, and the frequency corrections of each clock are calculated. These corrections are then utilized to update the center frequency of the Rabi spectrum. In step 3, upon completion of all clock cycles, the comparison stability is derived by computing the Allan variance of the frequency difference between the two clocks. It is worth noting that the Allan variance of a single clock should be divided by 2 [28]. Subsequently, the parameters of σ

_{QPN}, σ

_{Shot}, and σ

_{Det}can be determined and the value of N

_{0}is determined by ${T}_{0}/4{\mathrm{S}}_{0}^{2}{\sigma}_{\mathrm{Q}\mathrm{P}\mathrm{N}}^{2}$.

_{0}at 1 s and utilize Rabi detection with a fixed interrogation time of T

_{p}= 0.1 s, which can be easily implemented in the experiment. It is important to note that the specific choice of T

_{p}has minimal impact on the numerical results obtained. The collision between atoms influences the atomic-density-dependent S

_{0}[13,14]. However, in our simulation, where T

_{p}equals 0.1 s and the maximum atom number is lower than 6000, the system operates in the weak interaction region. Consequently, the variation in S

_{0}can be reasonably disregarded as the atom number changes [13,14]. Nevertheless, if the collision effects become prominent in a regime characterized by high density and strong interactions, the value of S

_{0}will be reduced due to density broadening. This reduction in S

_{0}indicates a higher level of instability for the clock.

## 3. Results and Discussion

_{0}= 1 s, N

_{0}= 500, γ

_{0}= 1, δ

_{N}= 3 [23], α = 0.2, β = 0.1. The excellent agreement observed in Figure 2 between the numerical and theoretical results confirms the correctness of our code and the validity of our method.

_{0}by studying the standard deviation of 50 independent simulations at different combinations of α and β, as shown in Figure 3a for N

_{0}= 500 and Figure 3b for N

_{0}= 2000. The white regions in both figures indicate errors larger than 500 for Figure 3a and 2000 for Figure 3b, which occurs when α or β approaches 1. In such cases, the modulation amplitudes of the parameters become close to zero, indicating that the different noises cannot be distinguished. Similar uncertainty distributions are observed for both cases. The smallest uncertainty is achieved at α = 5.71 and β = 0.1 for N

_{0}= 500, and α = 3.84 and β = 0.27 for N

_{0}= 2000.

_{0}using our approach. Figure 4 demonstrates good agreement between numerical and theoretical results, as N

_{0}exceeds approximately 200. However, it should be noted that larger uncertainties are observed when N

_{0}is lower than 200, due to larger noise and reduced stability. The larger noise can correspond to stronger excitation fraction fluctuation, which may cause deviations in the half-height points of the Rabi spectrum more frequently. Although the frequency–sensitivity slope S

_{0}is not constant for the Rabi spectrum, we use a constant S

_{0}to infer N

_{0}, leading to deviations for small N

_{0}.

_{0}simulated results using the Rabi and triangular spectra (shown in Figure 5), respectively. The values of S

_{0}remain unaffected by the frequency detuning on both the left and right sides in the case of a triangular spectrum. As a result, the stability calculation utilizing Equation (1) will not be influenced by fluctuations in the excitation fraction. Figure 4 demonstrates good agreement between theory and numerical result as the triangular spectrum is used, which is the powerful evidence of our hypothesis. Nevertheless, to achieve the triangular spectrum is challenging, this phenomenon suggests that our method can effectively work when N

_{0}exceeds about 200.

_{0}in every clock cycle, where the added noise followed a normal distribution

**N**~(2000, 2000σ

_{f}), where σ

_{f}represents the fractional fluctuation of the atom number and can be ranged from 0 to 1. Figure 7a shows the atom number as a function of simulated time, while Figure 7b presents the numerical results of the extracted averaging atom number and corresponding relative uncertainty as a function of σ

_{f}. Surprisingly, we found that the estimated uncertainty is almost independent of σ

_{f}when the value of σ

_{f}is smaller than approximately 14%. However, for larger values of σ

_{f}, the estimated uncertainty rapidly increases, and the determined atom number deviates from the theoretical value of 2000. Figure 7b indicates that our method is effective, as long as the standard deviation of atom number fluctuation is controlled below 0.14 times the averaging atom number.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Comparisons of numerical and theoretical results of stabilities. The Allan variances of frequency fluctuations at three-group parameters. The points represent the numerical results and dashed lines indicate the corresponding theoretical results. All error bars represent the 1σ standard error.

**Figure 3.**Numerical results of estimated uncertainties at different modulation parameters of α and β. (

**a**) The standard deviation of 50 independent simulations at N

_{0}= 500. (

**b**) The case of N

_{0}= 2000.

**Figure 4.**Numerical results of the determined atom number as a function of set value. Points are the results of single simulation, and the relative uncertainties are calculated by dividing error bars (the standard deviation of 50 numerical simulations) by corresponding set values. The dashed line indicates the theoretical values. As the atom number is smaller than 40, strong excitation fraction fluctuation leads to lock-lose, indicating the importance of maintaining a sufficiently high atom number to ensure stable and accurate measurements.

**Figure 5.**Numerical results of the determined atom number as a function of set value using the Rabi spectrum (circles) and triangular spectrum (triangles), respectively. Points are the results of single simulation, and the error bars indicate the standard deviation of 50 numerical simulations. The dashed line indicates the theoretical values. The inset shows the spectra of Rabi (solid line) and triangular spectrum (dotted line), respectively, wherein δ

_{F}denotes frequency detuning and P

_{e}represents excitation fraction.

**Figure 6.**Relative uncertainty of the atom number measurement as a function of time consumption. The relative uncertainty is obtained by dividing the standard deviation of 50 numerical simulations by the atom number set at 2000. The red solid lines indicate the linear fitting with a fixed slope of −0.5.

**Figure 7.**The atom number fluctuation and estimated uncertainty as a function of σ

_{f}. (

**a**) The fluctuation of atom number at σ

_{f}= 5%, 10% and 20%, respectively. (

**b**) Measurements of the atom number (averaging value of 50 simulations) and corresponding relative uncertainty (the standard deviation of 50 simulations) as a function of σ

_{f}. The dashed line shows the set averaging number of atom of 2000.

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**MDPI and ACS Style**

Zhao, G.; Guo, F.; Lu, X.; Chang, H.
Noise Differentiation and Atom Number Measurement in Optical Lattice Clocks by Analyzing Clock Stabilities with Various Parameters. *Appl. Sci.* **2024**, *14*, 1758.
https://doi.org/10.3390/app14051758

**AMA Style**

Zhao G, Guo F, Lu X, Chang H.
Noise Differentiation and Atom Number Measurement in Optical Lattice Clocks by Analyzing Clock Stabilities with Various Parameters. *Applied Sciences*. 2024; 14(5):1758.
https://doi.org/10.3390/app14051758

**Chicago/Turabian Style**

Zhao, Guodong, Feng Guo, Xiaotong Lu, and Hong Chang.
2024. "Noise Differentiation and Atom Number Measurement in Optical Lattice Clocks by Analyzing Clock Stabilities with Various Parameters" *Applied Sciences* 14, no. 5: 1758.
https://doi.org/10.3390/app14051758