Optical clocks have developed rapidly in the past 20 years and have achieved a systematic uncertainty of 9.4×10^-19 and frequency stability of 4.8×10^-17 @1 s. Except for the generation of standard time and frequency, optical clocks have many important applications, such as verification of general relativity, measurement of possible variation of the fine structure constant with time, detection of ultralight bosonic dark matter, quantum simulation, and relativistic geodesy. As more and more systematic uncertainty of the optical clock enters the order of 10^-18, and the absolute frequency measurement accuracy of the optical clock is fundamentally limited by the systematic uncertainty of the ¹³³Cs fountain clock, it has been proposed to use optical frequency transition to redefine the second in the international system of units. The 27th General Conference of Weights and Measures (CGPM) officially passed a resolution: The 28th CGPM will be held in 2026 to discuss the choice of optical clock types for redefining the second, and the optical frequency transition will be formally used to define the second expected in 2030. One of the main methods of absolute frequency measurement of optical clocks is to trace the international atomic time through a satellite link, and measurement uncertainty of less than 3×10^-16 is a precondition for the redefinition of the second in the international system of units by the optical frequency transition. In this complex tracing link, the uncertainty caused by the measurement dead time of the hydrogen maser is one of the main sources of absolute frequency measurement uncertainty for most optical clocks. After removing the contribution of frequency drift of the hydrogen maser, the statistical uncertainty caused by the measurement dead time of the hydrogen maser can be obtained by numerical simulation. This method needs to know the relevant parameters of the noise model of the hydrogen maser accurately and then generate the relevant random noise sequences by software.

In this paper, the frequency comparison between the ⁸⁷Sr optical lattice clock and hydrogen maser is made by using an optical frequency comb. By calculating the stability of the frequency ratio and fitting with the function of y(τ)=A12τ-2+A22τ-1+A32τ0+A42τ1 (τ is the measurement time, and A_1–4 indicate the amplitudes of phase white noise, frequency white noise, frequency flicker noise, and random walk noise, respectively), the values of A_1–4 are obtained. Finally, according to the noise model of the hydrogen maser, we use the software to generate random noise series of 86400×5 s, 86400×10 s, and 86400×30 s, respectively, and calculate the statistical uncertainty of optical frequency measurement under different effective operating rates (the measurement dead time of the hydrogen maser) and total measurement durations.

The parameters of the noise model of the hydrogen maser are determined as A₁=2.21×10^-13 (τ/s)^-1, A₂=3.05×10^-13 (τ/s)^-0.5, A₃=6.01×10^-16, and A₄=4.49×10^-19 (τ/s)^0.5, respectively after about 10-day measurement with an effective operating rate of 89% [Fig. 2(b)]. The frequency difference caused by the measurement dead time of the hydrogen maser is simulated 100 times by using the method of generating random noise sequences (using the software of Stable32) according to the noise model. Three types of random noise sequences are generated with a total measurement time of 86400×5 s, 86400×10 s, and 86400×30 s, respectively. The difference in the mean frequency from the total mean over partial times is calculated for the specific case. Each case is repeated by 100 times, and the measurement uncertainty caused by the measurement dead time of the hydrogen maser is represented by the 1 times standard deviation of these results. Figure 4 shows the calculation results of the measurement uncertainty as a function of the effective operating rate. The measurement uncertainty due to the measurement dead time decreases with the increase in the effective operating rate, and when the effective operating rate is less than 10% or so, increasing the total measurement time can significantly reduce the measurement uncertainty.

In this study, the frequency stability of the hydrogen maser is measured by comparing the ⁸⁷Sr optical lattice clock (with an 89% effective operating ratio and a total measurement time of about 10 days) with the hydrogen maser for a long time. By fitting the data of the frequency stability of the hydrogen maser with the noise model function, the influence of each noise of the hydrogen maser is determined as 2.21×10^-13 (τ/s) ^-1 for phase white noise, 3.05×10^-13 (τ/s) ^-0.5 for frequency white noise, 6.01×10^-16 for frequency flicker noise, and 4.49×10^-19 (τ/s)^0.5 for random walk noise. The calculation results indicate that the measurement uncertainty caused by the measurement dead time of the hydrogen maser decreases with the increase in the effective operating rate, and when the effective operating rate is less than 10% or so, increasing the total measurement time can significantly reduce the uncertainty. This work can be widely used to measure the absolute frequency of optical clocks by tracing the international atomic time and provide an important reference for selecting the effective operating rate of the optical clock to reduce the measurement uncertainty caused by the measurement dead time of the hydrogen maser.