## 2 Quantum Chaos as a Superposition of Wave-like and Particle-like Behavior on Single Photons
Now, we are in a position to perform calculations for mixing time series with coherent states (Didactic interpretations of this mixture can be obtained with simulations [16]). Figure 1: Determination of the Fano factor for photons traversing a polarizing beam splitter (PBS). There are five significant values of the Fano factor. Initially, for \(F=0\), the fluctuations in the photon count are completely suppressed. The Fano factor value \(F=1/4\), corresponding to quantum chaos noise, occurs at a transmission probability of \(P_{T}=0.75\). The wave-state-driven counting noise manifests itself at \(F=1/2\), representing an equal likelihood of the photon being transmitted through or reflected by the PBS. Another critical state is the complementary quantum chaos noise \(F=3/4\), commonly referred to as blue noise. Lastly, in the limit where the transmission probability approaches zero, the Fano factor reaches \(F=1\), signifying shot noise [4]. Coherent Mixing States of Single Photons \(|1\rangle\) with Coherent States with Different Averages  
There are several methods to produce noise related to quantum chaos. Quantum chaos evidently necessitates both the particle's localization and its wave-like delocalization. At the semi-classical or classical level, different statistical combinations might also exhibit similar noise characteristics. For example, unlike Equation (1), a wave with a different structure could be utilized. An example would be a coherent state's wave passing through a beam splitter. Figure 2 illustrates the theoretical setup for combining a coherent state with a single-photon Fock state. Unlike in the experimental realization [4], we make calculations with photons in spatial modes. In this article, it is not necessary to consider states with defined polarization where a polarizing beam splitter is used. Since in this work, we consider a nonpolarizing beam splitter, the process is equivalent to [4] if we consider the transmission and reflection coefficients as parameters of the problem. The coherent state, represented by \(|\alpha\rangle\) with an average photon number of \(|\alpha|^{2}\), enters through port 0, while the quantum state \(|1\rangle\) enters through port 1. For the Fock state, the transmission and reflection coefficients are denoted by \(t\) and \(r\), respectively, while for the coherent state, the coefficients are \(t^{\prime}\) and \(r^{\prime}\). These coefficients adhere to the Stokes relations [17]. \[|r^{\prime}|=|r|,|t^{\prime}|=|t|,|r|^{2}+|t|^{2}=1,r^{*}t^{\prime}+r^{\prime }t^{*}=0 \tag{3}\]  
The states on ports 2 and 3 are given by this unitary transformation [18]:  
\[ \begin{pmatrix}\hat{a}_{2}\\ \hat{a}_{3}\end{pmatrix} = \begin{pmatrix}t^{\prime}&r\\ r^{\prime}&t\end{pmatrix}\begin{pmatrix}\hat{a}_{0}\\ \hat{a}_{1}\end{pmatrix}  \tag{4}\]  
To determine the noise in photon counting, especially the noise related to quantum chaos, we need to find the average number of photons from the state mixture and their variance. These measurements allow us to calculate the Fano factor [19]. The output state of the beam splitter is:  
\[|\alpha\rangle_{0}|1\rangle_{1}\rightarrow\epsilon^{a(r^{\prime} \hat{a}_{2}^{*}+r^{\prime}\hat{a}_{3}^{*})-\alpha^{*}(r^{\prime*}\hat{a}_{2}+ r^{\prime*}\hat{a}_{3})}(r\hat{a}_{2}^{*}+\hat{a}_{3}^{*})|0\rangle_{2}|0\rangle_{3} \tag{5}\] \[=\hat{D}_{2}(t^{\prime}\alpha)\hat{D}_{3}(r^{\prime}\alpha)(r\hat {a}_{2}^{*}+\hat{a}_{3}^{*})|0\rangle_{2}|0\rangle_{3} \tag{6}\]  
Figure 2: In theory, to mix single-photon states with coherent states, we use a beam splitter where the coherent state \(|\alpha\rangle\) and the Fock state \(|1\rangle\) enter through the input ports.