# Conditional Measurements with Silicon Photomultipliers
## 3 Results
### The Effects of the OCT on the Photon-Number Statistics of the TWB
As \(\mu\to\infty\), the multi-thermal distribution of TWB converges to a Poissonian one, whereas the detected-event distribution in Equation (17) tends to  
\[\text{P}_{\mu\to\infty}(k_{i})=\exp\left(-\frac{\langle\hat{k}_{i}\rangle}{1+ \varepsilon}\right)\sum_{l=0}^{\lfloor\frac{k_{i}}{2}\rfloor}\binom{k_{i}-l}{ l}\frac{1}{(k_{i}-l)!}\bigg{(}\frac{\varepsilon}{1-\varepsilon}\bigg{)}^{l} \bigg{(}\frac{1-\varepsilon}{1+\varepsilon}\langle\hat{k}_{i}\rangle\bigg{)} ^{k_{i}-l}. \tag{18}\]  
Note that here we replaced the parameter \(\lambda\) with \(\langle\hat{k}_{i}\rangle\) through Equation (10). At a first glance to Figure 2, we note that the OCT gives rise to an asymmetry in the detected-event distribution: the detection probability of even events enhances proportionally to \(\varepsilon\) as the detection probability of odd events declines. Moreover, this effect is smoothed as the detected-event \(k_{i}\) increases. A further inspection of our OCT model may help to understand why. Let \(m\) be the number of photons detected with probability \(\eta\). As mentioned above, according to our OCT model, the outcome is a number \(k\) such that \(m\leq k\leq 2m\). If \(m\) is odd, then \(m/2\) of the possible values for \(k\) are odd and \(m/2\) are even, but, if \(m\) is even, \((m+1)/2\) of the possible values for \(k\) are even while still just \(m/2\) are odd. This is basically due to the fact that \(2m\), the superior bound to \(k\), is always even. However, as \(m\) increases, such a difference between even and odd detected-photon numbers becomes negligible compared to \(k\). This effect is especially apparent if we look at Equation (17) in the single-mode case. As mentioned above, in such a situation the detected-event distribution reduces to a linear combination of Fibonacci polynomials. This family of polynomials can be defined as [31]  
\[\mathcal{F}_{n}(x)\equiv\frac{1}{2^{n}}\frac{\left(\sqrt{x^{2}+4}+x\right)^{n} +(-1)^{n+1}\Big{(}\sqrt{x^{2}+4}-x\Big{)}^{n}}{\sqrt{x^{2}+4}} \tag{19}\]for given \(n\in\mathbb{N}\). The index of the polynomials in the single-mode detected-event distribution is \(n=k_{i}+1\), so that we get larger contributions as \(k_{i}\) is even and smaller otherwise. In Figures 4 and 5 we emphasize the most obvious effect of the OCT on the statistics of detected photons, i.e., compared to the case where no OCT affects the measurement, the probability of detecting smaller numbers of events is depleted, while, on the contrary, the larger values of \(k\) are more likely to be revealed. An expected effect of the OCT which, rather than a consequence, is the very definition of it. Note that here we focus on experimental values of \(\varepsilon\), which are typically small (\(\varepsilon<0.1\)) due to the recent technological improvements mentioned above. The plots show the ratio between the difference \(\Delta\text{p}\equiv\text{p}(k_{i})-p_{0}\) and \(p_{0}\), where \(p_{0}\equiv\text{p}(k_{i})|_{\varepsilon=0}\). Again, we explore the single-mode case in  
Figure 3: Detected-event distribution of the idler beam from Equation (18) in the multi-mode limit case (\(\mu\to\infty\)) as a function of the number of detected events \(k_{i}\) and of the OCT probability \(\varepsilon\). We set the quantum efficiency \(\eta\) to \(0.17\), while the mean photon number \(N\) to \(10\). These choices, together with a selected value of \(\varepsilon\), yield the corresponding mean value \(\langle k_{i}\rangle=(1+\varepsilon)\eta N\). The plot shows the evolution of a Poissonian distribution due to the OCT.  
Figure 2: Detected-event distribution of the idler beam from Equation (17) in the single-mode case (\(\mu=1\)) as a function of the number of detected events \(k_{i}\) and of the OCT probability \(\varepsilon\). We set the quantum efficiency \(\eta\) to \(0.17\), while the mean photon number \(N\) to \(10\). These choices, together with a selected value of \(\varepsilon\), yield the corresponding mean value \(\langle k_{i}\rangle=(1+\varepsilon)\eta N\).