## 4 Results and Discussion
### Experimental Adjustment of \(P_{\text{tel}}\) and \(f_{\text{upper}}\)  
According to Equation (5), the value of \(P_{\text{receive}}\) is positively proportional to \(P_{\text{tel}}\). The optimal value of \(P_{\text{local}}\) is directly influenced by \(f_{\text{upper}}\). To describe the effect of the values of \(P_{\text{tel}}\) and \(f_{\text{upper}}\) on \(P_{\text{receive}}\) and \(P_{\text{local}}\) in a more intuitive manner, we conducted the following experiments: (1) vary the value of \(P_{\text{tel}}\) in the range [2 W, 3 W] with an interval of 0.1 W and observe the variations of \(P_{\text{receive}}\) and total readout displacement noise; and (2) vary the value of \(f_{\text{upper}}\) in the range [20 MHz, 30 MHz] with an interval of 1 MHz and observe the variations of the optimal \(P_{\text{local}}\) value and minimum total readout displacement noise \(\delta x^{\text{total}}\). The results are shown in Figures 5 and 6. It should be noted that in the first experiment, other parameters such as \(P_{\text{local}}\), \(m\), and \(f_{\text{upper}}\) were set according to the first optimization results shown in Table 3, while in the second experiment, we set \(P_{\text{tel}}=2\) W.  
Figure 4: The sensitivity curve of the Taiji program detection limit (red) and after optimization (blue).  
Figure 5: Variation of \(P_{\text{receive}}\) and the total readout displacement noise as a function of \(P_{\text{tel}}\). The x-, left-hand y-, and right-hand y-axes show \(P_{\text{tel}}\), \(P_{\text{receive}}\), and the total readout displacement noise, respectively.  
In Figure 5, as \(P_{\rm tel}\) increases, \(P_{\rm receive}\) increases linearly, and \(\delta x^{\rm total}\) decreases approximately linearly. In Equation (4) and Step 5 of the optimization algorithm proposed in this study, the variation of \(P_{\rm receive}\) may cause a change in the optimal \(P_{\rm local}\) value. However, in practice, the variation of \(P_{\rm receive}\) is so small that it barely affects the optimal value of \(P_{\rm local}\). Figure 6 shows the variations of \(P_{\rm local}\) and \(\delta x^{\rm total}\) with \(f_{\rm upper}\). The optimal value of \(P_{\rm local}\) increases as \(f_{\rm upper}\) increases, while \(\delta x^{\rm total}\) first decreases and then increases. When \(f_{\rm upper}\) = 27 MHz, the optimal \(P_{\rm local}\) values is 2.175 mW, and \(\delta x^{\rm total}\) has its smallest value of 4.2078 pm\(/\sqrt{\rm Hz}\).  
In actual mission operations, a larger value of \(P_{\rm receive}\) can not only reduce \(\delta x^{\rm total}\), but also reduce the difficulty of weak-light phase-locked loops to some extent [27], which means that the power of \(P_{\rm tel}\) must be increased. However, increasing \(P_{\rm tel}\) undoubtedly increases the difficulty of the design of the devices associated with this system, and therefore, a trade-off needs to be made considering the practical applications of this system. In addition, different upper bounds of the beat-note frequency correspond to different optimal values of \(P_{\rm local}\), and different minimum total readout displacement noise values. The value of \(P_{\rm local}\) can be set by referring to Figure 6.