# Conditional Measurements with Silicon Photomultipliers
## 3 Results  
### The Effects of the OCT on the Photon-Number Statistics of the TWB  
Here we exploit the model developed in Section 2.1 to investigate the effects of the OCT on the detection of light and, in particular, on the statistics of a multi-mode mesoscopic TWB. The topic has been already widely investigated [13; 14; 16; 17; 19]. Still, we are anyway going through this point in order to test our model and use it to provide new insights on the OCT effects implied by this description. From the inspection of Equation (8), we find that the joint probability of detecting \(k_{i}\) events on the idler and \(k_{s}\) on the signal is given by  
\[ \begin{split}\mathrm{P}(k_{s},k_{i})=&(1-\lambda^{2 })^{\mu}\bigg{(}\frac{\varepsilon}{1-\varepsilon}\bigg{)}^{k_{s}+k_{i}}\sum_{m _{i}=\left\lceil\frac{k_{s}}{2}\right\rceil}^{k_{s}}\sum_{m_{i}=\left\lceil \frac{k_{s}}{2}\right\rceil}^{k_{i}}\binom{m_{s}}{k_{s}-m_{s}}\bigg{(}\frac{m _{i}}{k_{i}-m_{i}}\bigg{)}\\ &\bigg{(}\frac{(1-\varepsilon)^{2}}{\varepsilon}\frac{\eta}{1- \eta}\bigg{)}^{m_{s}+m_{i}}\sum_{n=\max(m_{n},m_{i})}^{\infty}\binom{n+\mu-1}{ n}\binom{n}{m_{s}}\binom{n}{m_{i}}\big{(}\lambda(1-\eta)\big{)}^{2n}\end{split}  \tag{16}\]  
which is the extension of the joint probability retrieved in [4], where just the effect of a limited quantum efficiency is considered. Figure 1: Setup of the experiment described in [17] and addressed in Section 4 to provide experimental evidence to the model presented here. See the text for details. If we consider the marginal distribution from Equation (16) for the idler beam, we find  
\[ \begin{split}\text{P}(k_{i})=&\frac{(1-\lambda^{2})^{ \mu}((1-\varepsilon)\eta\lambda^{2})^{k_{i}}}{(1-\lambda^{2}(1-\eta))^{k_{i}+ \mu}}\sum_{l=0}^{\lfloor\frac{k_{i}}{2}\rfloor}\binom{k_{i}+\mu-1-l}{\mu-1} \binom{k_{i}-l}{l}\bigg{(}\frac{1-\lambda^{2}(1-\eta)}{(1-\varepsilon)^{2}\eta \lambda^{2}}\varepsilon\bigg{)}^{l}\\ =&\frac{(1-\lambda^{2})^{\mu}((1-\varepsilon)\eta \lambda^{2})^{k_{i}}}{(1-\lambda^{2}(1-\eta))^{k_{i}+\mu}}\binom{k_{i}+\mu-1}{ \mu-1}.\\ &{}_{2}\text{F}_{1}\bigg{(}-\frac{k_{i}-1}{2},-\frac{k_{i}}{2};-( k_{i}+\mu-1);-4\varepsilon\frac{1-\lambda^{2}(1-\eta)}{(1-\varepsilon)^{2}\eta \lambda^{2}}\bigg{)}\end{split}  \tag{17}\]  
where \(\lfloor\cdot\rfloor\) is the floor function and \({}_{2}\text{F}_{1}(a,b;c;x)\) is the ordinary hypergeometric function. It can be shown that Equation (17) can be obtained from Equation (9) as well by replacing \(P_{n}\) with the photon-number distribution of a multi-mode TWB state [6; 24]. We also remark that, as we showed in [21], in the single-mode case (i.e., \(\mu=1\)) Equation (17) reduces to a linear combination of Fibonacci polynomials, which should be kept in mind for the considerations that follow. As a first remark, we stress that our model for the OCT, as outlined in Section 2.1, accounts for first-order events only. In the following paragraph, we briefly explore the implications of our simplified model for arbitrary values of \(\varepsilon\). Then we move back to the realistic case related to our experiment. We show the transformation of the detected-photon number statistics of TWBs due to the OCT in Figure 2 for the single-mode case and in Figure 3 for the multi-mode one. In both figures, we set the quantum efficiency \(\eta=0.17\) and the mean photon number \(N=10\) (see Equation (2)), which are experimentally reasonable values as long as SiPMs are employed for detection (see Section 4) and the photon-number regime is mesoscopic (see Section 2). For what concerns the multi-mode case, we have considered the limit \(\mu\to\infty\), since, again, this case is comparable with the number of modes estimated in our experiments, where \(\mu\sim 2000\)[17].