## 4 Discussion  
The results presented in the previous section confirm that the developed model is in excellent agreement with the data. In particular, we can see that the different quantities used to estimate the number of modes collected at the output of the second disk are essentially equivalent. This is quite expected since all of them are based on the moments of the distribution in Equation (4), even if the distribution of the number of photons contains all the moments of the statistics, whereas the autocorrelation function \(g^{2}\) and the cross-correlation \(\Gamma\) depend only on the first two moments. This difference may explain the small discrepancies between the obtained results. We verified that the number of modes is independent of the mean value of light, both considering the photon number distribution and the autocorrelation function. In fact, the experimental values of \(\mu_{s}\) extracted from the fitting procedures in both cases coincide within \(1\sigma\). As to the difference between the values obtained at the two BS outputs, we argue that it is due to a slight imbalance between the light in the two arms. This is also demonstrated by the calculation of \(\Gamma\), from which a value of \(\mu_{s}\) was obtained that is an average between the values extracted from the calculation of \(g^{2}\). As already specified in the Introduction, the preliminary characterization we carried on is important in itself for all those applications in the field of quantum information, in which  
Figure 7: (**a**) Cross-correlation coefficient \(\Gamma\) as a function of the mean number of detected photons obtained as the average between the two BS outputs. Black dots and error bars: experimental data; red curve: theoretical model in the case of a single-mode thermal state; gray curve: theoretical fitting function according to Equation (10), in which \(\mu_{f}=1.22\), while \(\mu_{s}\) is the only fitting parameter. The obtained value is: \(\mu_{s}=1.69\pm 0.06\). (**b**) Black dots and error bars: mean number of photons detected at one BS output as a function of the mean number of photons detected at the other output. Gray line: diagonal \(y=x\).  
investigating the statistics of light is sufficient. For instance, this kind of light can find applications in novel communication protocols, such as those presented in Refs. [40; 41]. The original scheme is based on the experimental quantification of nonclassicality of mesoscopic twin-beam states to transmit binary signals encoded in two single-mode pseudo-thermal states with different mean values. Interesting variations of this protocol could involve either the use of super-thermal states with different mean values or the alternate use of pseudo-thermal and super-thermal states with the same mean value. Moreover, this specific type of light can also be used in the context of imaging, where some other works exploiting super-thermal light have been recently published [31; 49]. At variance with these investigations, we note that we performed our experiment in the pulsed regime with PNR detectors able to operate in the mesoscopic intensity domain. Finally, we observe that for imaging applications, it might be important to treat even more populated states of light and use macroscopic detectors, such as CCD cameras. To this end, more intense light states and less opaque diffusers may be desirable. Work is in progress in this direction. As an alternative to the macroscopic regime, imaging protocols involving super-thermal states can be implemented in the mesoscopic one thanks to the use of arrays of PNR detectors, such as complementary metal-oxide-semiconductor (CMOS) cameras having each pixel endowed with PNR capability [42].