Detailed considerations based on the theory of atom-field interaction show that the optical detectors are square law detectors, i.e. they respond to the square of the field amplitude $I(\boldsymbol r,t)=E^2(\boldsymbol r,t)$[intensity]
$$ \begin{aligned}&G^{(n)}\Big(\{\boldsymbol{r}it_i\}n;\{\boldsymbol{r}{i+n}t{i+n}\}n\Big)=\prod{i=1}^nE^{(-)}\left(\boldsymbol{r}it_i\right)E^{(+)}\left(\boldsymbol{r}{n+i}t_{n+i}\right)\\&\equiv G^{(n)}\big(\boldsymbol{r}1t_1,\ldots,\boldsymbol{r}nt_n;\boldsymbol{r}{n+1}t{n+1},\ldots,\boldsymbol{r}{2n}t{2n}\big).\end{aligned} $$
$$ G^{(n)}\left(\{\boldsymbol{r}{i}t{i}\}{n};\{\boldsymbol{r}{i}t_{i}\}{n}\right)=\left\langle\prod{i=1}^{n}I\left(\boldsymbol{r}{i}t{i}\right)\right\rangle\geq0. $$
This function determines the correlation between intensities at different space-time points.
first-order correlation → Young’s double-slit
Second-order correlation → the Hanbury Brown and Twiss (HBT) effect
Gaussian approximation
The observable statistical properties of the e.m. field in the quantum theory are determined by the correlation function
quantized coherent and Thermal Fields[quantum analogs of classcial stable and chaotic fields]
A measurement of $\Delta n^2$ in homodyned detection of a single-mode field is a measure of the variance in $\hat A_\phi$
$$ \begin{equation} \begin{aligned}\hat{A}_\phi=\frac{1}{\sqrt{2}}\left[\exp(\mathrm{i}\phi)\hat{a}+\exp(-\mathrm{i}\phi)\hat{a}^\dagger\right].\end{aligned}\end{equation} $$