## 4 Quantum Chaos Mixing Coherent States with Fock States  
From Figure 5, we observe that the values for which we have \(F_{3}=1/4\) for a chaotic mixture have a behavior that decays depending on the magnitude of the proportion of the coherent state. For the case in which \(|\alpha|^{2}=0\), we recover the value of \(F_{3}=1/4\), for \(|r^{\prime}|^{2}=0.25\) or \(|t|^{2}=0.75\). As the average number of photons in the coherent state increases, the proportion of the coherent state required to reach a stage of chaotic noise becomes smaller and smaller. It is important to mention that the interaction between the coherent states and the individual states of the photons creates a sum of the noises associated with each contribution. In other words, the non-diagonal or coherence terms in the mixture generate noise that is greater than the sum of the separate noises of the individual photons and the photons in the coherent state. For the state with a coherent state with one photon on average, a reflection probability of \(P_{r}=0.065\) is required, which implies that the Fock photon entering through port 1 behaves like a particle approximately 93.5% of the time it crosses the beam splitter. The coherent state with its 6.5% behaves like a wave, a classical wave compared to the quantum wave mentioned in Section 2. In another hand, there is a gap between the states with \(|\alpha|^{2}=0\) and \(|\alpha|^{2}>1\) in the noise \(F>1\) for \(|r^{\prime}|^{2}>1/3\). Above this coherent state reflection probability, there is no noise suppression, so all quantum behavior must be located below \(|r^{\prime}|^{2}=1/3\). As an example of an extreme interaction, the noise \(F\) generated by the interaction of an individual photon with a coherent state with \(\hat{n}=1000\), see Figure 6. A coherent state whose energy can be 1000 times greater than the individual photon interacts as a function of the energy difference. Clearly, the limit of the Fano factor is \(F_{3}=3\), which is reached quickly. The proportion of the mixture that provides a Fano factor with chaotic noise consists of particle states in a proportion close to one and coherent states in a very small proportion because the intensity of the interaction of the Fock state with the coherent state depends on the energy difference between both states. If the energy difference is  
Figure 5: Reflection probability \(P_{R}=|r^{\prime}|^{2}\) for which \(F_{3}=1/4\) in function of the \(|\alpha|^{2}\) for the cases in Figure 4. We have included some points \(|\alpha|^{2}=1/2,1/4\) to have a smooth curve. large, then the interaction will be weak, and therefore, the associated noise will be smaller. For this reason, the maximum noise shifts to the left (lower \(r^{\prime}\)) depending on the average number of photons in the coherent state. However, Fano noise decays linearly to \(F_{3}=1\) as the probability of reflection increases. It may be of interest to know why the noise \(F\) has as its limit the value \(F_{3}=3\), regardless of the value of the average number of photons that have the coherent state. We can say that the knee bend of \(F\) versus \(|r^{\prime}|^{2}\) for an arbitrary number of photons in the coherent state is close to the maximum interaction between the two states, i.e., in the area where the energy is comparable. In any case, the mixture of states that we propose here has a limit equivalent to the noise that would produce quantum states of three photons \(|3\rangle\) mixed with the vacuum. The Fano factor obtained from the statistical mixture expressed by the density function [12] describes the competition between the quantum character of the state \(|1\rangle\) and the state \(|\alpha\rangle\) that we can consider classical. In this way, the statistics of the photons that can be detected at the output port 3 can have the three characteristic limits of light sources. Sub-poisson, superpoisson, and poisson for different values of \(|r^{\prime}|^{2}\), since the value \(\sigma/\sqrt{\tilde{n}}=\sqrt{F}\) is an indicator of this behavior.