## 2 Theory and Simulation  
To maximize common-mode rejection [10; 22; 23], we adopted a highly symmetric heterodyne interferometry to construct the interferometer. Figure 1 illustrates the layout of the entire interferometer. The system consists of a frequency modulation bench and an optical bench. The frequency modulation bench included a laser, acousto-optic modulators (AOMs), a signal generator, etc., which were used to generate two laser beams with a frequency difference of 1.6 MHz. The optical bench was situated within a multi-layer thermal shielding system placed inside a vacuum chamber. The two modulated laser beams pass through beam splitters, the PAAM and the reference mirror, respectively, and finally combine at the beam splitters to form a heterodyne beat optical signal. The heterodyne signals are converted into electrical signals \(\mathrm{I_{PD1}}\) and \(\mathrm{I_{PD2}}\) by the detectors, and the phase difference in the optical path can be obtained by a phase meter and through the phase conversion formula (1), where \(\Delta_{\varphi}\) is the phase difference in radians and \(\lambda\) is the wavelength of the laser. \[\mathrm{L=}\frac{\Delta_{\varphi}}{2\pi}\lambda \tag{1}\]  
From the interference measurement path, the optical paths for the two photodetectors can be calculated as:  
\[\mathrm{L_{PD1}}=\mathrm{L1+L2+L3-L2-L4-L6}=\mathrm{L1+L3-L4-L6} \tag{2}\]  
\[\mathrm{L_{PD2}}=\mathrm{L1+L3+L5-L4-L5-L6}=\mathrm{L1+L3-L4-L6} \tag{3}\]  
Thus, the optical path difference is:  
\[\Delta\mathrm{L=L_{PD1}-L_{PD2}=0} \tag{4}\]  
To reduce the background noise level of the measurement system, we have theoretically adopted an equal-arm interferometry scheme, thereby mitigating the impact of the laser frequency noise through common-mode rejection [22]. However, to mitigate the generation of stray light and the associated noise caused by the multiple reflections and transmissions of the laser within the beam splitter (BS), the front and rear surfaces of the BS were designed with a wedge angle of 0.5\({}^{\circ}\). Considering the effects of light transmission through the BS and the wedge angle, simulations of the interferometer were conducted using IFOCAD (1.7) [24]. The layout of the optical components was optimized to achieve a residual arm length difference of 1 nm to ensure that the optical path lengths of the reference and measurement arms were precisely matched. The simulation results are presented in  
Figure 1: Layout of the interferometer with heat shields. (**a**) Layout diagram of the interferometer. (**b**) Three-dimensional model of the interferometer. The vacuum chamber included the PAAM, photodetectors (PD), fiber couplers (FC), beam splitters (BS), a mirror (M), etc. Figure 1b, with the specific optical path lengths for the reference and measurement arms detailed below:  
\[{\rm L_{FD1}}^{\prime}={\rm L_{FC1-BS3}}+{\rm L_{BS}}+{\rm L_{BS3^{\prime}-PAAM}}+{ \rm L_{PAAM-BS2^{\prime}}}-{\rm L_{FC2-BS1}}-{\rm L_{BS1-BS2}}-{\rm L_{BS}} \tag{5}\]  
\[{\rm L_{FD2}}^{\prime}={\rm L_{FC1-BS3}}+{\rm L_{BS3-BS4^{\prime}}}-{\rm L_{FC2 -BS1}}-{\rm L_{BS}}-{\rm L_{BS1^{\prime}-M}}-{L_{M-BS4}}-{\rm L_{BS}} \tag{6}\]  
In the formula, \({\rm BS1^{\prime}}\), \({\rm BS2^{\prime}}\), \({\rm BS3^{\prime}}\), and \({\rm BS4^{\prime}}\) denote the positions where the laser beam exits after passing through beam splitters \({\rm BS1}\), \({\rm BS2}\), \({\rm BS3}\), and \({\rm BS4}\), respectively. \({\rm L_{BS}}\) represents the geometrical length of the laser within the beam splitter. The IFOCAD simulation results show that the arm length difference \(\Delta{\rm L_{designed}}\) is 10.58 pm. Although the interferometer was designed to have an equal-arm configuration, the actual installation errors introduced a residual arm length difference (\(\Delta{\rm L_{residual}}\
eq~{}0\)), resulting in some optical path noise. The temperature fluctuations introduce noise through the thermal expansion of the optical substrate, and the thermal expansion and thermo-optic effects of the optical components [18; 19; 20].