## 4 Experimental Results and Discussion
This is mainly related to temperature fluctuations (10 mK in this paper), vacuum performance (10\({}^{-3}\) Pa in this paper), etc. The entire interferometer is mainly composed of quartz glass and Super-Invar (4)32). The linear expansion coefficients of quartz glass and Super-Invar (4)32) are approximately \(5.5\times 10^{-7}/\mathrm{K}\) and \(9\times 10^{-7}/\mathrm{K}\). With a temperature variation of 10 mK, the linear expansion of quartz glass is approximately \(5.5\times 10^{-9}\), and for Super-Invar (4)32), it is roughly \(9\times 10^{-9}\). In practical terms, this translates to a change of approximately 550-900 pm within a 100 mm range. It is noteworthy that these linear expansions surpass the required target. Simultaneously, temperature fluctuations can alter the refractive indices of certain optical components, primarily the beam splitters (BSs), resulting in changes in the optical path due to refraction when the beams pass through the BSs. Based on the existing analysis, achieving a temperature control precision of 10 \(\mathrm{\upmu K}\) is deemed necessary to attain picometer-level  
Figure 8: (**a**) Time-domain results of the optical path difference. (**b**) Frequency-domain results of the optical path difference. measurement accuracy. This necessitates the implementation of measures in subsequent work to mitigate the impact of temperature fluctuations. These measures include adopting a combination of active and passive temperature-control methods to achieve higher-precision temperature control, as well as using materials with lower thermal expansion coefficients in the construction of the interferometer. The higher the vacuum level, the better the test results. The vacuum level primarily influences variations in the refractive index of the medium within the vacuum chamber, consequently affecting measurement accuracy. Due to the elusive movement of medium molecules in the three-dimensional vacuum environment, it becomes challenging to capture and describe the optical path difference caused by them through theoretical calculations. Currently, according to empirical knowledge, a vacuum level of \(10^{-5}\) Pa is typically required to achieve picometer-level accuracy at low frequencies. However, the vacuum chamber utilized in this paper can only attain a vacuum level of \(10^{-3}\) Pa, thereby impacting the overall measurement accuracy. To enhance the measurement accuracy of the interferometer, it is necessary to improve the vacuum performance of the vacuum chamber in subsequent experiments. However, the OPD caused by the steering mirror can be obtained through an indirect analogical approach. According to equation 8, the calculated optical path difference has a relationship with the angle jitter. When the angle jitter is very small, neglecting the second-order term, it can be approximated that there is a proportional relationship between angle jitter and optical path difference. Preliminary investigations have established a first-order relationship between voltage and angle, implying an analogous first-order relationship between voltage and optical path difference. Voltage signals of different amplitudes are given to the steering mirror to obtain different OPDs. Then, the relationship between the driving voltage amplitude and the OPD can be obtained. Based on the voltage at the steady state of the steering mirror, the OPD at the steady state of the steering mirror could be calculated. Since the size of the OPD in the time domain is greatly affected by the experimental environment, it is difficult to obtain real results. The linear characteristics of the frequency domain can be used for analysis. By providing the steering mirror with voltage signals of the same frequency but different amplitudes, we can obtain varying optical path differences at that frequency. Through the numerical fitting of the optical path difference and voltage values, the coupling relationship between the driving voltage and optical path difference can be determined.