\hspace{-.1in}{\langle {\hat{E}}^\dagger_R({\bm\rho}'_1,\tau_1) {\hat{E}}^\dagger_S({\bm\rho}'_2,\tau_2) \rangle} {\langle {\hat{E}}_R({\bm\rho}''_1,\tau_1) {\hat{E}}_S({\bm\rho}''_2,\tau_2) \rangle}+ \nonumber \\
&& \hspace*{-.1in}{\langle {\hat{E}}^\dagger_R({\bm\rho}'_1,\tau_1){\hat{E}}_R({\bm\rho}''_1,\tau_1) \rangle} {\langle {\hat{E}}^\dagger_S({\bm\rho}'_2,\tau_2){\hat{E}}_S({\bm\rho}''_2,\tau_2) \rangle} + \nonumber \\
&& \hspace*{-.1in} {\langle {\hat{E}}^\dagger_R({\bm\rho}'_1,\tau_1){\hat{E}}_S({\bm\rho}''_2,\tau_2) \rangle} {\langle {\hat{E}}^\dagger_S({\bm\rho}'_2,\tau_2){\hat{E}}_R({\bm\rho}''_1,\tau_1) \rangle}.
\label{eq:MomentFactor}\end{aligned}$$

Until this point, our analysis applies equally well to the pseudothermal and SPDC ghost imagers. That equivalence disappears, however, when we evaluate the second-order moments in Eq.  using Eqs. – for the pseudothermal ghost imager, and using Eqs. , , and for the SPDC ghost imager. The results we obtain are $$\begin{aligned}
\lefteqn{{\langle \hat{C}({\bm\rho}_p) \rangle}_C = \frac{q^2\eta^2 A_p A_b}{L^2}\left(\frac{2P}{\pi a_L^2}\right)^2} \nonumber \\
&\times& \int\!d{\bm\rho}\, {\mathcal{T}}({\bm\rho})\left[1+\frac{ e^{-|{\bm\rho}-{\bm\rho}_p|^2/\alpha\rho_L^2}}{\alpha}\right],
\label{eq:CAvgMean}\end{aligned}$$ for the pseudothermal (classical) imager, and $$\begin{aligned}
\lefteqn{{\langle \hat{C}({\bm\rho}_p) \rangle}_Q= \frac{q^2\eta^2A_p A_b}{L^2} \left(\frac{2P}{\pi a_L^2}\right)^2\int \! d{\bm\rho}\, {\mathcal{T}}(-{\bm\rho})} \nonumber \\[.05in]
&\times& \left[ 1+ \frac{\Omega_B T_0}{4}\frac{e^{-|{\bm\rho}-{\bm\rho}_p|^2/\alpha\rho_L^2}}{\alpha}\left(1+\frac{1}{4\sqrt{\pi}\mathcal{I}}\right)\right].
\label{eq:QAvgMean}\end{aligned}$$ for the SPDC (quantum) ghost imager, where $A_b$ is the area of $\mathcal{A}_b$, $\mathcal{I}\equiv P T_0 \rho_0^2/a_0^2 = PT_0\rho_L^2/a_L^2$ is the source brightness in photons per spatiotemporal mode, and $$\begin{aligned}
\alpha \equiv \frac{2\rho_R^2\rho_S^2+a_0^2(\rho_R^2 +\rho_S^2)}{2\rho_R^2\rho_S^2}\end{aligned}$$ is a turbulence-induced performance-degradation factor. We see that both average correlation functions are the sum of the same featureless background term $$\begin{aligned}
\label{eq:C0}
C_0=\frac{q^2\eta^2 A_p A_b}{L^2}\left(\frac{2P}{\pi a_L^2}\right)^2 \int\!d{\bm\rho}\, {\mathcal{T}}({\bm\rho}),\end{aligned}$$ and different image-bearing terms: $$\begin{aligned}
C_{1,C}&({\bm\rho}_p)=\frac{q^2\eta^2 A_p A_b}{\alpha L^2}\left(\frac{2P}{\pi a_L^2}\right)^2 \int\!d{\bm\rho}\, {\mathcal{T}}({\bm\rho}) e^{-\frac{|{\bm\rho}-{\bm\rho}_p|^2}{\alpha\rho_L^2}},\end{aligned}$$ for the classical-state 