## 2 Materials and Methods
This means that the calculation of both \(g^{2}\) and \(\Gamma\) as a function of the mean value of the light is useful to extract information about the values of \(\mu_{f}\) and \(\mu_{s}\) by fitting the relations \(g^{2}(m)\) and \(\Gamma(m)\) with Equations (7) and (10), respectively. In order to practically investigate such properties, we realized the experimental setup shown in \[P_{\mu}(m)=\binom{m+\mu-1}{\mu-1}\bigg{(}1+\frac{\langle m\rangle}{\mu} \bigg{)}^{-\mu}\bigg{(}1+\frac{\mu}{\langle m\rangle}\bigg{)}^{-m} \tag{2}\]3  
The ps-pulses (at 1047 nm) of a Nd:YLF laser regeneratively amplified at 500 Hz and frequency doubled at 523 nm were focused on a rotating ground-glass disk (GD1), having scattering centers of the order of 4 \(\upmu\)m. A portion, roughly corresponding to a single speckle, of the generated speckle field was then selected by a pin-hole (PH1) placed in  
\[P_{\mu}(m)=\binom{m+\mu-1}{\mu-1}\bigg{(}1+\frac{\langle m\rangle}{\mu} \bigg{)}^{-\mu}\bigg{(}1+\frac{\mu}{\langle m\rangle}\bigg{)}^{-m} \tag{2}\]4 **(a)** Sketch of the experimental setup. M: mirror; ND: neutral density filter; L\({}_{1}\): 200-mm-focal length lens; L\({}_{2}\): 100-mm-focal length lens; GD1 and GD2: rotating ground-glass disks; PH1: first pin-hole; PH2: 150-\(\upmu\)m-diameter pin-hole; HWP: half-wave plate; PBS: polarizing cube beam splitter; L: achromatic doublet; MF: multi-mode optical fiber; HPD: hybrid photodetector. **(b)** Typical single-shot image of the speckle field in correspondence of the position of PH1. **(c)** Picture of the first pin-hole PH1. \[P_{\mu}(m)=\binom{m+\mu-1}{\mu-1}\bigg{(}1+\frac{\langle m\rangle}{\mu} \bigg{)}^{-\mu}\bigg{(}1+\frac{\mu}{\langle m\rangle}\bigg{)}^{-m} \tag{2}\]5 Cross-correlation coefficient \(\Gamma\) as a function of the mean number of detected photons at each of the outputs of the BS for different values of \(\mu_{f}\) and \(\mu_{s}\). From top to bottom, red curve: \(\mu_{f}=1\) and \(\mu_{s}=1\); purple curve: \(\mu_{f}=2\) and \(\mu_{s}=2\); magenta curve: \(\mu_{f}=5\) and \(\mu_{s}=5\); dashed black curve: \(\mu=1\) for single-mode thermal state of light. far field and focused on a second ground-glass disk (GD2), characterized by scattering centers with the same size as those of GD1. Both disks were rotated in such a way that for the duration (\(\sim\)5 ps) of a single pulse, they could be considered still, while they occupied different positions when different pulses were emitted. A portion of the generated speckled-speckle field (roughly corresponding to a single speckle) was then selected by a second pin-hole (PH2), whose diameter was equal to 150 \(\upmu\)m. The light exiting the pin-hole was then divided at a polarizing cube BS placed behind a half-wave plate in order to finely control the balancing between the two arms. The two outputs were then focused by two achromatic doublets (L) into multi-mode fibers (MF) with a core diameter of 600 \(\upmu\)m that delivered the light to two commercial PNR detectors operated at room temperature. More specifically, we used a pair of hybrid photodetectors (HPDs, mod. R10467U-40, Hamamatsu Photonics), whose output was amplified, synchronously integrated, and digitized. We performed measurements at different mean numbers of photons, attenuating the light by means of a variable neutral density filter (ND) placed in front of the first disk. We saved 100,000 acquisitions for each attenuation choice.