## 2 Quantum Chaos as a Superposition of Wave-like and Particle-like Behavior on Single Photons
The transmission and reflection coefficients are \(t\) and \(r\) for input port 1, and \(t^{\prime}\) and \(r^{\prime}\) for input port 0.  
the density matrix of the resulting state can be expressed as follows:  
\[\hat{\rho}_{23}=\hat{D}_{2}(t^{\prime}\alpha)\hat{D}_{3}(r^{\prime}\alpha)\hat{ \rho}_{23}\hat{D}_{3}^{\dagger}(r^{\prime}\alpha)\hat{D}_{2}^{\dagger}(t^{ \prime}\alpha) \tag{7}\]  
where  
\[\hat{\beta}_{23}=(r\hat{a}_{2}^{\intercal}+t\hat{a}_{3}^{\intercal})|0\rangle_ {2}|0\rangle_{3}\langle 0|_{3}\langle 0|_{2}(r^{*}\hat{a}_{2}+t^{*}\hat{a}_{3}) \tag{8}\]  
As our research focuses on analyzing the noise at output port 3, the density operator is represented as:  
\[\hat{\rho}_{3}=Tr_{2}(\hat{\rho}_{23}) \tag{9}\]  
By taking the trace over port 2, we obtain:  
\[\hat{\rho}_{3}=|r|^{2}|r^{\prime}\alpha\rangle_{3}\langle r^{\prime}\alpha|+|t |^{2}\hat{D}(r^{\prime}\alpha)|1\rangle_{3}\langle 1|\hat{D}^{\dagger}(r^{ \prime}\alpha) \tag{10}\]  
By employing the identity [20], \(\hat{a}\hat{D}(r^{\prime}\alpha)=\hat{D}(r^{\prime}\alpha)(\hat{a}+r^{\prime }\alpha)\) and its conjugated equivalent. This operator allows us to determine the mean number of photons and the variance from the mean count in port 3. \[\langle\hat{n}_{3}\rangle=tr(\hat{n}_{3}\hat{\rho}_{3}) \tag{11}\] \[\langle(\Delta\hat{n}_{3})^{2}\rangle=\langle\hat{n}_{3}^{2}\rangle -\langle\hat{n}_{3}\rangle^{2} \tag{12}\]  
where \(\langle\hat{n}_{3}^{2}\rangle=Tr(\hat{n}_{3}^{2}\hat{\rho}_{3})\). Thus,  
\[\langle\hat{n}_{3}\rangle=|t|^{2}+|r^{\prime}\alpha|^{2} \tag{13}\] \[=\hat{D}_{2}(t^{\prime}\alpha)\hat{D}_{3}(r^{\prime}\alpha)(r\hat {a}_{2}^{*}+\hat{a}_{3}^{*})|0\rangle_{2}|0\rangle_{3} \tag{6}\]0  
Finally,  
\[=\hat{D}_{2}(t^{\prime}\alpha)\hat{D}_{3}(r^{\prime}\alpha)(r\hat {a}_{2}^{*}+\hat{a}_{3}^{*})|0\rangle_{2}|0\rangle_{3} \tag{6}\]1  
Figure 3 shows the photon average curves of these mixtures. When the reflection probability \(|r|^{2}\) is nearly zero, indicating the absence of a coherent state in the mixture exiting through port 3, the photon count is \(n=1\) with \(F_{3}=0\). As the proportion of the coherent state reflected towards output 3 increases, the average photon number grows to \(|\alpha|^{2}\). For the magenta full-box graph with \(|\alpha|^{2}=0\), the photon count decreases from 1 to 0. In the scenario where the averaging results in a one-photon average in both the coherent state and the Fock state, the average number of photons consistently remains \(\hat{n}=1\). The other graphs are also linear in terms of the increase in the average number of photons depending on the probability that the coherent state leaves the port 3. Figure 4 illustrates the Fano factor distributions for various combinations of single-photon states and coherent states with \(|\alpha|^{2}\) values of 0, 1, 2, 3, and 4. For \(|\alpha|=0\), the transmitted single photon exhibits a Fano factor between \(F_{3}=0\) and \(F_{3}=1\). This plot (represented by filled magenta boxes) resembles the Fano factor observed in the experiment [4] involving single photons, where the quantum chaos value \(F_{3}=1/4\) corresponds to a transmission probability of \(|t|^{2}=0.75\). For the other graphs, represented by green filled circles for \(|\alpha|=1\), cyan circles for \(|\alpha|=2\), orange boxes for \(|\alpha|=3\), and yellow stars for \(|\alpha|=4\), it is evident that the maximum noise level indicated by the Fano factor approaches \(F_{3}=2\). This suggests a proximity to photon thermal noise with bunching. In addition, it becomes clear that there is an upper limit for the Fano factor when the average photon number in the coherent state is very large. This limit is \(F_{3}=3\).