## 2 Theoretical Analysis  
In order to improve the entanglement preparation system, we first analyze the factors that are crucial to the entanglement quality. For a NOPA, the measurable level of the quantum noise spectrum of the amplitude quadrature difference (phase quadrature sum) can be modeled by [29; 30]:  
\[V_{sq(Anti-sq)}=1+4\sqrt{P/P_{th}}\eta_{\text{tot}}[\frac{\sin^{2}(\theta+ \alpha_{\pm})}{\left(1-\sqrt{P/P_{th}}\right)^{2}+4(f/f_{c})^{2}}-\frac{\text{ COS}^{2}(\theta+\alpha_{\pm})}{\left(1+\sqrt{P/P_{th}}\right)^{2}+4(f/f_{c})^{2}}] \tag{1}\]  
where \(\alpha_{\pm}=0\) and \(\alpha_{\pm}=\pi/2\) correspond to the squeezing spectrum and anti-squeezing spectrum, respectively. \(P\) and \(P_{th}\) are the pump power and the oscillation threshold of the NOPA, respectively, \(\eta_{\text{tot}}\) is the total efficiency of the system, \(f\) is the analysis frequency, \(f_{c}\) is the decay rate of the optical cavity, and \(\theta\) is the root mean squared (RMS) phase fluctuation of the locking between the local oscillator (LO) and the signal field. From the above model, we find that the entanglement quality depends on four factors: the pump power level of the NOPA (\(P/P_{th}\)), the total efficiency of the system (\(\eta_{\text{tot}}\)), the normalized analysis frequency \(f/f_{c}\), and the fluctuation of the phase locking (\(\theta\)). The total efficiency of the system consists of five parts: \(\eta_{\text{tot}}=\eta_{\text{esc}}\ \times\ \eta_{\text{t}}\ \times\ \eta_{\text{vis}}\ \times\ \eta_{\text{pd}}\ \times\ \eta_{\text{en}}\). The quantity \(\eta_{\text{esc}}\) is the escape efficiency of the NOPA cavity, defined as \(\eta_{\text{esc}}=\frac{T}{T+L}\) and \(L\) present the amplitude transmittance of the output coupling mirror, and the intracavity round trip loss, respectively. The quantity \(\eta_{\text{t}}\) is the propagation efficiency, and \(\eta_{\text{vis}}\) is the interference efficiency with \(\eta_{\text{vis}}=v^{2}\), where \(v\) is the interference visibility of the homodyne detection, which is the quality of the spatial mode overlapping between the signal and the LO on the 50/50 beam splitter. The quantity \(\eta_{\text{pd}}\) is the quantum efficiency of the photodiodes, and \(\eta_{\text{en}}\) is the equivalent measurement efficiency that is induced by electronic noise. Without taking account the phase fluctuation, Equation (1) can be simplified to:  
\[V_{sq}=1-\eta_{\text{tot}}\frac{4\sqrt{P/P_{th}}}{\left(1+\sqrt{P/P_{th}} \right)^{2}+4(f/f_{c})^{2}} \tag{2}\]  
\[V_{Anti-sq}=1+\eta_{\text{tot}}\frac{4\sqrt{P/P_{th}}}{\left(1-\sqrt{P/P_{th} }\right)^{2}+4(f/f_{c})^{2}} \tag{3}\]  
Figure 1 shows the simulated quantum correlation spectrum as a function of the analysis frequencies at different total efficiencies. The higher the loss of the system, the lower the squeezing level one can obtain given a certain analysis frequency. Furthermore, the squeezing level decreases with the increase of the analysis frequency, due to the limitation of the cavity linewidth. The squeezing level reaches \(-9.2\) dB at 1 MHz with 10% total loss, while it reduces to \(-5\) dB at 1 MHz when the loss increases to 30%. There is an approximate 4.2 dB gain for the squeezing when the loss is reduced from 30% to 10%. The amount of detectable squeezing is not only limited by the optical loss, it is also limited by the fluctuations of the relative phase, which are always present in reality. Considering the phase jitter effects, the realistic squeezing \(V_{sq}\) and anti-squeezing \(V_{Anti-sq}\) are given by:  
\[V_{sq}^{M}=V_{sq}\cos^{2}\theta+V_{Anti-sq}\sin^{2}\theta \tag{4}\]  
\[V_{Anti-sq}^{M}=V_{sq}\sin^{2}\theta+V_{Anti-sq}\cos^{2}\theta \tag{5}\]  
Equation (4) indicates that the phase fluctuations inevitably transfer the anti-squeezing noise to the squeezing noise. Figure 2 shows that the phase fluctuations deteriorate the level of measured quantum squeezing, and the squeezing disappears when \(\theta\) grows to \(\sim\)7.5 degrees.