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

1. Introduction

Ultra-cold atoms confined within an optical lattice have played a pivotal role in the development of modern atomic clocks [1,2,3,4], the advancement of quantum techniques [5,6,7,8,9], and the exploration of fundamental physics [10,11,12,13]. Accurate determination of the atom number in the lattice is crucial to fully harness the potential of these applications. For instance, precise knowledge of the atom number is essential for studying many-body interactions [14], thereby reducing systematic uncertainties in optical lattice clocks (OLCs) [1,2,3,4].

Commonly employed methods for atom number measurement include fluorescence detection and absorption imaging [15,16,17]. However, the fluorescence detection method is typically limited by uncertainties arising from the effective solid angle and probe light intensity, resulting in a measurement uncertainty greater than 15% [15]. On the other hand, the absorption imaging technique offers a lower measurement uncertainty below 10%, relying on knowledge of the atomic sample shape and technical noise levels [16]. Recently, advancements in synchronized frequency comparison based on in situ measurements have effectively canceled out interrogation laser noise, improving measurement stability [18,19]. The comparison stability is constrained by atomic detection noises, including quantum projection noise (QPN), photon shot noise, and technical noise [20,21]. This progress has spurred the development of a new method to measure the atom number based on QPN. The previous method of measuring atom numbers from atomic detection noise required the neglect of technical noise [22]. This prerequisite prevented the utilization of atom number measurement based on the QPN.

In this paper, we present a method to distinguish the QPN noise, photon shot noise and technical noise in a Dick-noise-free OLC [18,19,23,24,25]. By conducting three separate stability measurements with modulations of the atom number (N₀) or the photon count (γ₀) 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 N₀ by differentiating QPN from other sources of noise. To further enhance precision, we use numerical simulations to investigate how modulation parameters influence the measurement uncertainty of N₀ and how measurement precision evolves as N₀ increases.

2. Methods

Assuming a transition probability of 0.5 and disregarding the detection laser noise, which is typically much less significant than other factors in an OLC [21],the clock stability σ_a at τ=1 can be represented by [23,26]

$${\sigma }_{\mathrm{a}}^2={\sigma }_{\mathrm{QPN}}^2+{\sigma }_{\mathrm{Shot}}^2+{\sigma }_{\mathrm{Det}}^2.$$

(1)

In Eq. (1), ${\sigma }_{\mathrm{QPN}}^2=T_0/4S_0^2{N}_0$ corresponds to the variance contributed by the QPN, with T₀ representing the clock cycle time and S₀ denoting the frequency-sensitive slope of the spectrum at half-height points. ${\sigma }_{\mathrm{Shot}}^2=T_0/4S_0^2{N}_0{\gamma }_0$ refers to the variance originating from the photon shot noise, while ${\sigma }_{\mathrm{Det}}^2=T_0{\delta }_{\mathrm{N}}^2/2S_0^2{N}_0^2$ represents the contribution of the technical noise. Here, δ_N stands for the rms fluctuation of atoms detected by the photoelectric detector. As the three types of noise exhibit distinct dependencies on N₀ and γ₀, it becomes possible to differentiate them effectively by modulating N₀ and γ₀.

By controlling the detection laser intensity or its duration (T_det), γ₀ can be manipulated. Specifically, by adjusting T_det to T_det /α while keeping other parameters constant as defined in Eq. (1), the value of γ₀ will change to γ₀/α. Consequently, the overall clock stability can be represented as

$${\sigma }_{\mathrm{b}}^2={\sigma }_{\mathrm{Q}\mathrm{P}\mathrm{N}}^2+\alpha {\sigma }_{\mathrm{S}\mathrm{h}\mathrm{o}\mathrm{t}}^2+{\sigma }_{\mathrm{D}\mathrm{e}\mathrm{t}}^2.$$

(2)

In the same manner, by changing N₀ to N₀/β, we can separate the technical noise from other noises, and the overall clock stability is denoted by

$${\sigma }_{\mathrm{c}}^2=\beta ({\sigma }_{\mathrm{Q}\mathrm{P}\mathrm{N}}^2+{\sigma }_{\mathrm{S}\mathrm{h}\mathrm{o}\mathrm{t}}^2)+{\beta }^2{\sigma }_{\mathrm{D}\mathrm{e}\mathrm{t}}^2.$$

(3)

Combining Eqs. (1)~(3), the contributions of different noise sources can be determined by solving

$$\left[\begin{array}{lll}1& 1& 1\\ 1& \alpha & 1\\ \beta & \beta & {\beta }^2\end{array}\right]\left[\begin{array}{l}{\sigma }_{\mathrm{Q}\mathrm{P}\mathrm{N}}^2\\ {\sigma }_{\mathrm{S}\mathrm{h}\mathrm{o}\mathrm{t}}^2\\ {\sigma }_{\mathrm{D}\mathrm{e}\mathrm{t}}^2\end{array}\right]=\left[\begin{array}{l}{\sigma }_{\mathrm{a}}^2\\ {\sigma }_{\mathrm{b}}^2\\ {\sigma }_{\mathrm{c}}^2\end{array}\right].$$

(4)

Once the value of σ_QPN is obtained using Eq. (4), it becomes possible to determine the absolute atom number in an OLC.

3. Results and Discussion

To validate our approach and identify the optimal modulation parameters (α and β) for minimizing measurement uncertainty, we employ numerical simulations. This simulation involves three main steps. In step 1, all the required parameters are initialized and the values of σ_a~σ_c are calculated. In step 2, the clock comparison process between two clocks is executed with a total measurement time of approximately 2.2 hours, where the Dick effect is cancelled by setting the clock laser noise to be zero [27]. The cancellation of the Dick effect is realized in experiment by synchronous frequency comparison between two clocks [23,28] or using the in situ imaging technique to compare two regions of cold ensembles in a clock [18,19,22,25]. Three cases with noise amplitudes of σ_a, σ_b, and σ_c are simultaneously simulated. The noise-induced frequency fluctuation is generated by multiplying σ_a ~σ_c with normally distributed random numbers. During the frequency correction, the noise is added to the measured frequency errors. Consequently, the comparison stability can be obtained for the Allan variance of the frequency difference between the two clocks. The Allan variance of a single clock should divide the comparison stability by 2 [28]. In this work, we utilize Rabi detection with an interrogation time of T_p =0.1 s, which can be easily implemented in the experiment. It should be noted that the specific choice of T_p has minimal impact on the numerical results obtained. The frequency correction can be expressed as (P_eR- P_eL)/2S₀, where P_eR and P_eL represent the excitation fractions on the right and left sides of the spectrum, respectively. In step 3, the stabilities are calculated and the parameters of σ_QPN, σ_Shot and σ_Det are extracted. Subsequently, the value of N₀ is determined by $T_0/4S_0^2{\sigma }_{\mathrm{Q}\mathrm{P}\mathrm{N}}^2$.

To verify the accuracy of our numerical calculation code, we compare the stabilities obtained from numerical simulations at three sets of parameters to the theoretical results shown in Figure 1. The parameters used for this comparison are T₀=1 s, N₀=500, γ₀=1, δ_N=3 [23], α=0.2, β=0.1. The excellent agreement observed in Figure 1 between the numerical and theoretical results confirms the correctness of our code and the validity of our method.

To investigate the influence of α and β on the measurement uncertainty of N₀, we study the standard deviation of 50 independent simulations at different combinations of α and β, as shown in Figure 2(a) for N₀=500 and Figure 2(b) for N₀=2000. Similar uncertainty distributions are observed for both cases. The smallest uncertainty is achieved at α=5.71 and β=0.1 for N₀=500, and α=3.84 and β=0.27 for N₀=2000.

We also found that the maximum difference in measurement uncertainty within the ranges of α=0.1~0.44 and β=1.97~6.73 is below 4% at the current total measurement time of 6.7 h. Therefore, we choose the parameters of α=3.84 and β=0.27 to numerically simulate the measurements of N₀ using our approach. Figure 3 demonstrates good agreement between numerical and theoretical results as N₀ exceeds about 200. However, it should be noted that larger measurement uncertainties are observed when N₀ 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 S₀ is not constant for the Rabi spectrum, we use a constant S₀ to infer N₀, leading to deviations for small N₀.

We verify our inference by conducting a comparison of N₀ measurement results using the Rabi and triangular spectra (shown in Figure 4), respectively. Benefitting from the frequency-detuning independent S₀, the triangular spectrum will not be affected by the excitation fraction fluctuations. 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 function when N₀ exceeds about 200.

Exploring the relationship between total time consumption and measurement uncertainty using our approach presents an intriguing avenue for further study, given their crucial role in experiments. Figure 5 shows the relationship between relative uncertainty and total measurement time. It is evident that as the time increases, the uncertainties decrease following a slope of -0.5. This observation aligns with the fact that clock-comparison instability also decreases at the same slope of -0.5 with increasing averaging time. Through linear regression analysis, we determined that achieving a 1% uncertainty (one order of smaller than the typical uncertainty of absorption imaging [15,16,17]), requires a measurement time of approximately 20 hours.

In the real world, the number of atoms trapped in the lattice may vary from shot-to-shot clock cycles, making it important to study the relationship between the fluctuation amplitude of atom number and measurement uncertainty using our method. Both factors are critical in experiments. To induce atom number fluctuations, we added discrete random integers to the set value of N₀ 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 6(a) shows the atom number as a function of measurement time, while Figure 6(b) 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 measurement uncertainty is almost independent of σ_f when the value of σ_f is smaller than about 14%. However, for larger values of σ_f, the measurement uncertainty rapidly increases, and the determined atom number deviates from the theoretical value of 2000. Figure 6(b) 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

In conclusion, our proposed method provides a valuable tool for accurately measuring the atom number in an OLC system where clock stability is limited by quantum projection noise, photon shot noise, and technical noise. The numerical results indicate that our approach can achieve a measurement uncertainty below 4% across a wide range of atom numbers with a total measurement time of 6.7 h, and this uncertainty can be further reduced by increasing the measurement time. We also investigated how the fluctuation of atom number affects the measurement results. The numerical results indicate that the standard deviation of atom number fluctuation should be controlled to be below 0.14 times the average atom number. This work can advance research in many-body interaction [14,19], nondestructive detection [29], and entanglement of atoms [30], as the precise measurement of atom number enables more precise control and manipulation of atomic systems.