cross correlations. The coherence time is directly related to how fast the ground-glass rotates, the coherence length is typically on the order of a few wavelengths, and the intensity radius is set by the radius of the initial laser beam.

A type-II phase-matched, collinear, cw SPDC source emits orthogonally-polarized, co-propagating signal and idler beams, which are separated by a polarizing beam splitter to become the signal and reference beams for ghost imaging. They are in a maximally-entangled, jointly-Gaussian state with no phase-insensitive cross correlation, and the maximum phase-sensitive cross correlation permitted by quantum theory given their autocorrelation functions, namely [@Erkmen2009SNR]

$$\begin{aligned}
\label{eq:QCorrPIS}
{\langle {\hat{E}}^\dagger_S({\bm\rho}_1,t_1)&{\hat{E}}_R({\bm\rho}_2,t_2) \rangle}_Q = 0, \\[.05in]
\begin{split}\label{eq:QCorrPS}
{\langle {\hat{E}}_S({\bm\rho}_1,t_1)&{\hat{E}}_R({\bm\rho}_2,t_2) \rangle}_Q = \frac{2P}{\pi a_0^2} e^{-\frac{|{\bm\rho}_1|^2+|{\bm\rho}_2|^2}{a_0^2}} \!\!
\left[e^{-\frac{|{\bm\rho}_1-{\bm\rho}_2|^2}{2\rho_0^2}}e^{-\frac{(t_1-t_2)^2}{2T_0^2}} + i \left(\frac{2}{\pi}\right)^\frac{1}{4} \!\sqrt{\frac{a_0^2}{P T_0 \rho_0^2}}e^{-\frac{|{\bm\rho}_1-{\bm\rho}_2|^2}{\rho_0^2}}e^{-\frac{(t_1-t_2)^2}{T_0^2}}  \right],
\end{split}\end{aligned}$$

where the subscript $Q$ indicates that these are quantum (nonclassical) cross correlations.

Propagation through Turbulence
------------------------------

The signal and reference beams in Fig. \[fig:setup\] first propagate $L\,$m over spatially-separated paths from their shared source location [@footnote1]. The reference beam is then measured by a high-spatial resolution CCD camera, while the signal field reflects off the target and travels $L\,$m to the bucket detector. We will assume statistically-independent turbulence on all three paths: the reference 