## 2 Theoretical Analysis  
Squeezed vacuum states can be obtained when the OPO is operated below the threshold. The variance of the generated squeezed \(V_{s}^{-}(\Omega)\) and anti-squeezed \(V_{s}^{+}(\Omega)\) quadrature can be expressed by [16]  
\[V_{s}^{\pm}(\Omega)=1\pm 4\eta_{\text{{\em f}ec}}\eta_{\text{{\em det}}}\eta_{prop} \eta_{\text{hom}}\times\frac{\sqrt{P/P_{\text{th}}}}{\left(1\mp\sqrt{P/P_{ \text{th}}}\right)^{2}+4\Omega^{2}}, \tag{1}\]  
where \(\eta_{\text{{\em f}ec}}=T/(T+L)\) is the cavity escape efficiency, \(T\) is the transmittance of output coupler, \(L\) is the intracavity losses, \(\eta_{\text{det}}\) is the quantum efficiency of the photodetector, \(\eta_{prop}\) is the propagation efficiency, \(\eta_{\text{hom}}=V^{2}\) is the homodyne detection efficiency [18], \(V\) is the fringe visibility for the LO and a bright alignment beam in the BHD system, \(P\) is the pump power of the OPO, \(P_{\text{th}}\) is the threshold of the OPO, \(\Omega=2\pi f/\gamma\) is normalized frequency, \(f\) is the analysis frequency, \(\gamma=c(T+L)/l\) is the OPO cavity decay rate, \(c\) is the speed of light in the vacuum and \(l\) is round trip length of the cavity. The noise power of squeezed vacuum states can be detected by the BHD system. The measured field (\(\hat{a}=\alpha+\delta\hat{a}\)) and LO (\(\hat{b}=\beta+\delta\hat{b}\)) are combined at a 50/50 beam splitter and detected by a pair of photodetectors with quantum efficiency of \(\eta_{\text{det1}}\) and \(\eta_{\text{det2}}\), respectively. \(\delta\hat{a}\) and \(\delta\hat{b}\) are fluctuations around the mean field values of the measured field (\(\alpha\)) and LO (\(\beta\)), respectively. Assuming the mean field of the measured field \(\alpha\) is far smaller than \(\beta\), the items related to the mean field of the measured field can be ignored, and the photocurrent subtraction from the two photodetectors can be written by [19]  
\[I=\frac{\eta_{\text{det1}}}{2}\left(\beta\delta\hat{X}_{a}(\theta)+\beta\delta \hat{X}_{LO}(\theta)\right)+\frac{\eta_{\text{det1}}}{2}\beta^{2}-\frac{\eta _{\text{det2}}}{2}(-\beta\delta\hat{X}_{a}(\theta)+\beta\delta\hat{X}_{LO}( \theta))-\frac{\eta_{\text{det2}}}{2}\beta^{2}, \tag{2}\]  
where \(\delta\hat{X}_{LO}(\theta)\) is the quadrature fluctuation operator for the LO at relative phase (\(\theta\)) between the LO and measured field. \(\delta\hat{X}_{a}(\theta)\) is the quadrature fluctuation operator for the measured field. Because the intensity noise of the LO is far beyond the SNL at audio-band frequencies, the measured squeezing level of the squeezed vacuum states at audio-band frequencies is related to not only the losses above-mentioned in Equation (1), but also the intensity noise of the LO and the CMRR of photodetectors in the BHD. To improve the CMRR, a single electronic board design is usually employed to reduce the influence from the discrepancy of electronics components, where the photocurrents of both photodiodes are directly subtracted before any further electronic signal processing [20]. However, the discrepancy from parameters of both photodiodes, including their quantum efficiencies, equivalent capacitances, equivalent resistances, etc., is also a crucial factor to decrease the CMRR of photodetectors. If we define an unbalanced factor related to the CMRR of photodetectors, i.e. \(G=\sigma\eta_{\text{det2}}/\eta_{\text{det1}}\), the variances of the photocurrent subtraction can be written as [19]  
\[V(I)=\frac{\beta^{2}\eta_{\text{det1}}^{2}}{4}\Big{(}(1+G)^{2}V_{a}+(1-G)^{2} V_{LO}\Big{)}, \tag{3}\]where \(\sigma\) represents the unbalanced item induced from the discrepancy of electronics components, \(V_{LO}\) is the intensity noise of the LO, and \(V_{a}\) is the variances of the measured field. \(V_{a}=V_{s}^{-}(\Omega)\) when the measured field is the generated squeezed vacuum state.