This study tackles the critical challenge of inter-satellite laser power attenuation to the nanowatt (nW) level in the Taiji Program. We systematically validate the feasibility of weak-light phase-locking technology under extreme low-light conditions through ground-based experiments. Further, the key noise mechanisms that limit phase-locking precision are analyzed. The research objectives include the development of a mathematical model of the phase-locked loop suitable for long-baseline interferometry and the optimization of control parameters to address dynamic Doppler frequency shifts ranging across 5?25 MHz. Further, the contributions of shot noise, carrier-to-noise ratio (CNR) noise, phasemeter noise, and environmental noise are quantified, and the stability boundaries of phase locking under low CNR conditions, specifically at -86 dB-Hz is examined. By bridging the technological gap in ultra-weak-light phase locking in China, this study provides both theoretical and experimental foundations necessary for achieving picometer-level displacement measurements in the Taiji Program. This advances space-based gravitational wave detection from theoretical design to engineering implementation. Furthermore, the findings offer valuable insights into noise suppression strategies and system optimization.

The research team developed a modular, ground-based weak-light phase-locking verification system to simulate the inter-satellite low-light conditions. A master laser (Mephisto 500NEFC, wavelength: 1064 nm) was attenuated to 0.49 nW using a variable optical attenuator, simulating the weak signal from a distant satellite. Simultaneously, a slave laser served as a local oscillator, delivering a power of 20 μW. The two beams were coupled into a four-interferometer setup (fused silica substrate) within a vacuum chamber (10 Pa pressure) via a polarizing beam splitter. The interferometry unit employed a balanced detection scheme, where signals were captured through high-sensitivity avalanche photodiodes (Thorlabs APD430M/C, responsivity: 12 A/W, noise equivalent power (NEP): 0.45 pW/Hz^1/2) and converted into electrical signals using transimpedance amplifiers. The closed-loop control unit, implemented on a field programmable gate array platform (Moku:Pro), integrated a 24-bit resolution phasemeter, dual-loop PID controllers (PZT loop bandwidth: 100 kHz, TEC loop bandwidth: 1 Hz), and a fourth-order Butterworth low-pass filter with a 50 kHz cutoff. Real-time CNR monitoring was performed using a spectrum analyzer. Further, an LTPDA Toolbox was used for the frequency-domain decomposition of residual phase errors across a range of 0.1 mHz to 100 kHz. Theoretical models were developed to represent key noise sources, including shot noise Sshot=hc/λP and CNR noise SCNR=10C/N0/10. These models were compared against experimental data to systematically analyze the noise contributions.

Under weak-light conditions (0.49 nW, CNR=-86 dB-Hz), the system achieves a residual phase error of 6×10^-5 rad/Hz^1/2 (>0.02 Hz), approaching the theoretical CNR noise limit of 5.01×10^-5 rad/Hz^1/2 and corresponding to an equivalent displacement noise of 10 pm/Hz^1/2. Under strong-light conditions (1.7 μW), the residual phase error decreases to 4.5×10^-5 rad/Hz^1/2, reaching the phasemeter’s noise floor, which is limited by a 10 bit ADC quantization and the 1.25 GSa/s clock jitter. Noise analysis reveals that the CNR noise dominates the high-frequency range (>0.1 Hz) under low CNR conditions, contributing over 60% of the total noise and exhibiting a 1/P attenuation trend as the weak-light power increases. Low-frequency residuals (<0.1 Hz) are significantly affected by thermal drift, with the interferometer’s Invar base (thermal expansion coefficient of 1.2×10^-6 K^-1) facilitating a phase drift of 2×10^-4 rad/Hz^1/2 at 1 mHz. Further, mid-frequency laser frequency jitter is suppressed by 40 dB via the dual-loop PID control system, demonstrating strong broadband disturbance rejection. Compared to LISA’s analog phase-locked loop (17 nW, 5×10^-6 rad/Hz^1/2) and Tsinghua University’s digital approach (9 pW, 5.2×10^-4 rad/Hz^1/2), this study demonstrates superior high-frequency performance at the nW level. However, the low-frequency stability must be further improved through active thermal control (±0.01 ℃).

This study provides conclusive validation of the weak-light phase-locking technology for the Taiji Program, achieving a precision of 10 pm/Hz^1/2 at 0.49 nW. To the best of our knowledge, this is the first demonstration of CNR noise-dominated performance in China and the results successfully satisfy the Taiji mission’s displacement measurement requirement of 8 pm/Hz^1/2. The research delineates the full-bandwidth noise distribution, identifying CNR noise as the dominant factor at high frequencies (>0.1 Hz), thermal drift as the primary influence at low frequencies (<0.1 Hz), and phasemeter electronic noise as the limiting factor under strong-light conditions. The proposed optimization strategies include low-noise APDs (NEP <0.1 pW/Hz^1/2), photon-counting techniques for weak-light detection, and adaptive thermal control systems. The experimental results align with the theoretical models (<15% deviation), thereby confirming the reliability of the framework for space system simulations. Future work will focus on suppressing coupled multi-degree-of-freedom noise, such as vibration-thermal cross-sensitivity, and validating the system’s adaptability to in-orbit environmental conditions, including radiation hardening and microgravity effects. These advancements aim to accelerate the transition of the Taiji Program from laboratory research to practical deep-space applications.