## 2 Theoretical Analysis  
An SRO device, which is pumped by a fundamental wave (\(\omega_{p}\)), is shown in Figure 1. The two-color quantum correlation between the signal (\(\omega_{s}\)) and idler (\(\omega_{i}\)) can be observed when the SRO is operated above the pump threshold. A type-I phase matched periodical poled nonlinear crystal is employed in the cavity. The signal is resonant in a ring cavity. The idler and pump single pass through the nonlinear crystal. \(\alpha_{p,in}\) is the input mode operator of pump. \(\alpha_{s,in}\) and \(\beta_{s,in}\) are the coupled vacuum mode operator introduced by the coupling mirror of the signal and other intra-cavity losses, respectively. \(\alpha_{p}\) (\(\alpha_{s}\), \(\alpha_{i}\)) is the pump (signal, idler) mode operator in the cavity. \(\alpha_{p,out}\) (\(\alpha_{s,out}\), \(\alpha_{i,out}\)) is the output mode operator of pump (signal, idler). The fluctuation characteristics of the SRO emission can be computed simply using a semi-classical approach [35; 36]. The quantum [PERSON] equations of motion for the intra-cavity modes can be expressed by  
\[\left\{ \begin{array}{rcl}\tau\dot{\alpha}_{s}+(\gamma+i\varphi)\alpha_{s}&=& 2\chi\alpha_{p}\alpha_{i}^{*}+\sqrt{2\gamma_{s}}\alpha_{s,in}+\sqrt{2\mu_{s}} \beta_{s,in}\\ \alpha_{i}&=&\chi\alpha_{s}^{*}\alpha_{p}\\ \alpha_{p}&=&\alpha_{p,in}-\chi\alpha_{s}\alpha_{i}\end{array} \right., \tag{1}\]  
where \(\tau\) is the cavity round-trip time of the signal. \(\gamma\) is the total intra-cavity losses, \(\gamma=\gamma_{s}+\mu_{s}\), \(\gamma_{s}\) is the intra-cavity losses due to the output coupler transmission, \(\gamma_{s}=t_{s}{}^{2}/2\), \(t_{s}\) is the transmission of the output coupler of the signal, and \(\mu_{s}\) is the other intra-cavity losses include crystal absorption, surface scattering, and imperfections of mirrors. \(\chi\) is the second-order nonlinear coefficient of the nonlinear crystal. \(\varphi\) is the detuning between the round-trip phase of the signal and the cavity resonance. Generally, \(\varphi\) is assumed to be small. The input-output relations of down-conversion modes are given by  
\[\left\{ \begin{array}{rcl}\alpha_{s,out}=\sqrt{2\gamma_{s}}\alpha_{s}-\alpha _{s,in}\\ \alpha_{i,out}=\sqrt{2\gamma_{i}}\alpha_{i}\end{array} \right., \tag{2}\]  
where \(\gamma_{i}\) is the intra-cavity losses of the idler due to the output coupler transmission. The stationary mean fields \(\overline{\alpha}_{j}\) (\(j=p,s,i\)) can be calculated considering vacuum input \(\overline{\alpha}_{s,in}=\overline{\beta}_{s,in}=0\). \(\overline{\alpha}_{p,in}\) is the mean field of the pump. The stationary state equation for the intra-cavity modes can be written as  
\[\left\{ \begin{array}{rcl}(\gamma+i\varphi)\overline{\alpha}_{s}&=&2\chi \overline{\alpha}_{p}\overline{\alpha}_{i}^{*}\\ \overline{\alpha}_{i}&=&\chi\overline{\alpha}_{s}\overline{\alpha}_{p}\\ \overline{\alpha}_{p}&=&\overline{\alpha}_{p,in}-\chi\overline{\alpha}_{s} \overline{\alpha}_{i}\end{array} \right.. \tag{3}\]  
The pump threshold of SRO is \(\alpha_{p,th}=\left(\gamma^{2}+\varphi^{2}\right)^{1/4}/\sqrt{2}\chi\), which can be obtained using Equation (3). The pump parameter is defined as \(\sigma=\alpha_{p,in}/\alpha_{p,th}\). The mean fields of signal and idler in the cavity can be written as  
\[\left\{ \begin{array}{rcl}\overline{\alpha}_{s}=\left(\sigma-1\right)^{1/2} /\chi\\ \overline{\alpha}_{i}=\left(\sigma-1\right)^{1/2}\left(\gamma^{2}+\varphi^{2} \right)^{1/4}/\sqrt{2}\chi\end{array} \right.. \tag{4}\]  
In order to obtain the fluctuation dynamics, we linearize Equation (1) for small fluctuations \(\delta\alpha_{j}\) (\(j=p,s\), \(i\)) around the mean field value of \(\overline{\alpha}_{j}\), and the fluctuation equations of the signal and idler in the cavity can be obtained as  
\[\left\{