unit impulse. Following [@Erkmen2008Unified], we will take both the pseudothermal and SPDC sources to produce signal and reference beams that are in zero-mean, jointly Gaussian states, which are then completely characterized by their nonzero autocorrelation and cross-correlation functions. All the autocorrelations will be assumed to have the same Gaussian-Schell model form, $$\begin{aligned}
\label{eq:AutoCorr}
{\langle {\hat{E}}^\dagger_K&({\bm\rho}_1,t_1){\hat{E}}_K({\bm\rho}_2,t_2) \rangle} \nonumber\\[.05in]
& =\frac{2P}{\pi a_0^2}e^{-\frac{|{\bm\rho}_1|^2+|{\bm\rho}_2|^2}{a_0^2}} e^{-\frac{|{\bm\rho}_1-{\bm\rho}_2|^2}{2\rho_0^2}}e^{-\frac{(t_1-t_2)^2}{2 T_0^2}},\end{aligned}$$ for $K = S,R$, where $P$ is the photon flux, $a_0$ is the source’s $e^{-2}$ intensity radius, $\rho_0$ is its coherence length, and $T_0$ is its coherence time. The distinction between the Gaussian-state models for the pseudothermal and SPDC sources lies in their signal-reference cross correlations, as we will now explain.

The pseudothermal signal and reference beams start as continuous-wave (cw) laser light, that is passed through a rotating ground-glass diffuser to make it spatially incoherent. Signal and reference beams are then obtained by 50-50 beam splitting. The signal and reference thus have no phase-sensitive cross correlation, and the maximum phase-insensitive cross correlation allowed by classical physics given their autocorrelation functions, i.e., $$\begin{aligned}
\label{eq:CCorrPS}
{\langle {\hat{E}}_S&({\bm\rho}_1,t_1){\hat{E}}_R({\bm\rho}_2,t_2) \rangle}_C=0, \\[.05in]
\begin{split}\label{eq:CCorrPIS}
{\langle {\hat{E}}^\dagger_S&({\bm\rho}_1,t_1){\hat{E}}_R({\bm\rho}_2,t_2) \rangle}_C \\[.05in]
& =\frac{2P}{\pi a_0^2}e^{-\frac{|{\bm\rho}_1|^2+|{\bm\rho}_2|^2}{a_0^2}} e^{-\frac{|{\bm\rho}_1-{\bm\rho}_2|^2}{2\rho_0^2}}e^{-\frac{(t_1-t_2)^2}{2 T_0^2}},
\end{split}\end{aligned}$$ where the subscript $C$ indicates that these are classical-state 