## 2 Theoretical Analysis
Considering the intensity noise of the LO and the CMRR of photodetectors, the measured value of squeezing level can be expressed by  
\[V_{s}^{\prime}=\frac{V_{s}^{-}(\Omega)(1+G)^{2}+V_{LO}(1-G)^{2}}{(1+G)^{2}+V_ {LO}(1-G)^{2}}, \tag{4}\]  
According to the definition of CMRR, the value of the CMRR can be expressed as [19]  
\[R_{CMRR}=10\lg\frac{P_{com}}{\Delta\times P_{com}}=20\lg\frac{1+G}{2|1-G|}, \tag{5}\]  
where \(P_{com}\) is the measured power from one photodetector when the other one is blocked, and \(\Delta\times P_{com}\) is the measured power when both photodetectors are illuminated. \(G\) can be given as [19]  
\[G=\frac{2\times 10^{R_{CMRR}/20}-1}{2\times 10^{R_{CMRR}/20}+1}, \tag{6}\]  
Another influence issue on the measured squeezing level is the phase jitter effect. The relative phase between the LO and squeezed vacuum fields determines the measured quadrature of squeezed vacuum sates. The jitter of the phase relation caused by residual high frequency phase modulations will reduce the level of measured squeezing. Assuming that the relative phase between the LO and squeezed vacuum field has a normal distribution with a jitter of \(\delta\varphi\), the measured squeezing level can be written by [16]  
\[V_{s}^{-}(\Omega)\big{|}_{\delta\varphi}=V_{s}^{+}(\Omega)\cos^{2}\delta \varphi+V_{s}^{-}(\Omega)\sin^{2}\delta\varphi, \tag{7}\]  
Figure 1 shows the theoretical calculated normalized noise power of squeezed vacuum states at the analysis frequency of 5 kHz with respect to different values of \(V_{LO}\), \(R_{CMRR}\) and \(\delta\varphi\) considering an actual squeezed vacuum states generation and measurement system with the experimental parameters of \(\eta_{\rm{esc}}=0.95\), \(\eta_{\rm{det}}=0.96\), \(\eta_{\rm{prop}}=0.94\), \(\eta_{\rm{hom}}=0.98\), \(P=100\) mW, \(P_{th}=350\) mW, \(T=0.13\), \(L=0.012\), and \(l=33\) mm. Solid Curve (i), which is calculated by Equation (1), is the squeezing level without considering the influence of the unbalance factor \(G\) and the intensity noise of the LO. It can be seen that the theoretical calculated squeezing level is 6.25 dB that is not dependent on the CMRR of the detectors. Solid Curves (ii)-(iv) are the theoretical calculated squeezing levels using Equation (2) with the same experimental parameters and considering the influence of the unbalance factor \(G\) and \(V_{LO}\) exceeding the SNL 0 dB, 35 dB and 50 dB, respectively. According to Curve (ii), the value of the squeezing level is still influenced by the CMRR of the detectors when its value is less than 20 dB, even if the intensity noise of the LO reaches the SNL. From Curves (iii) and (iv), it can be seen that the critical values of the CMRR of the detectors, at which the squeezing level will not be influenced by the intensity noise of the LO, are 54 dB and 70 dB when the intensity noise of the LO is 35 dB and 50 dB above the SNL, respectively. Dashed Curves (a)-(d) are the theoretical calculated squeezing levels using Equation (2) except that the variance of the generated squeezed \(V_{s}^{-}(\Omega)|_{\delta\varphi}\) is substituted for \(V_{s}^{-}(\Omega)\) in Equation (7) considering the influence of a phase jitter of \(\delta\varphi=2^{\circ}\) on the measured squeezing level. Two-degree phase jitter is a typical value in the common experimental setup used to generate squeezed vacuum states. According to dashed Curve (a), the theoretical calculated squeezing level is reduced from 6.25 dB to 6.05 dB caused by \(2^{\circ}\) phase jitter without considering the influence of the unbalance factor \(G\) and the intensity noise of the LO. Dashed Curves (b)-(d) are the theoretical calculated squeezing levels considering the influence of \(2^{\circ}\) phase jitter, the unbalance factor \(G\) and \(V_{LO}\) exceeding the SNL 0 dB, 35 dB and 50 dB, respectively. The phase jitter effect becomes an important issue with the increasing of squeezing level.