## 1 Introduction  
\"In 1909, [PERSON] realized that electromagnetic fluctuations vary depending on whether the energy is carried by waves or particles.\" [1]. The fluctuations of the information are also information. The noise produced by individual photons in the detectors depends on your location. That means if a detector is positioned towards the possible path of photons that do not have a precise location, the noise produced in the photon count around the average will be different from the noise produced when all photons are directed unambiguously to the detector. That implies the difference in noise between detecting photons in the wave and particle states. Although our quantum intuition is an approximation of reality, the concept of quantum particles still depends on the high probability that we associate with a single path. On the other hand, a quantum wave can go on different paths simultaneously. When a photon in the state of a quantum particle enters a 50:50 beam splitter, its outgoing state is now a quantum wave with two trajectories, transmitted and reflected. According to experience, the photon will be detected in the detector placed on the transmitted port or in the other on the reflected port [2]. The detectors we currently use detect quantum particles, but we do not have detectors of waves, although we can detect quantum waves indirectly. The test of Bell's inequality, in some way, is an indirect proof of the existence of these quantum waves [3]. In [4], it is proposed as a conjecture that quantum chaotic behavior comes from the balanced superposition of each photon's wave and particle states. To make these measurements, we use the Fano factor to measure quantum noise, as used in ballistic and nuclear billiards [1, 5, 6, 7], in which the quantum chaos is associated with a Fano factor \(F=1/4\). In [4], we talked about the superposition of two states of the same photon, one state with zero shot noise (\(F=0\)) and the other state with individual random noise (\(F=1/2\)). The chaotic behavior of individual photons is seen as an average of noise between those of a wave and a particle. It is imperative to understand that the randomness of individual photons, such as those crossing a 50:50 beam splitter with a Fano factor \(F=1/2\), is distinct from the randomness of a coherent state photon beam with \(F=1\). The latter is associated with shot noise, and this distinction is crucial for a clear understanding of our research. The importance of this conjecture about quantum chaos does not require the behavior of a particle bouncing off a nonregular billiard but rather the superposition of two of its possible states. In that context, we want to know whether quantum or classical chaos can be obtained from the superposition of individual photon states with different noise states and in what proportions. For example, single-photon states with coherent states with varying averages of photons. In this article, we present a theoretical study on the mixture of quantum and classical states of light to extend the understanding of the possibility of indirectly measuring the wave nature of photons through the noise they generate. We assume that the radiation field of a coherent \(|\alpha\rangle\) state has properties like those of a classical field [8; 9; 10], which can coherently mix with a quantum field, i.e., its density function contains non-diagonal matrix elements that are non-zero. In that sense, when we talk about a mixture, we are referring to a coherent mixture, which allows interference between the two states entering the beam splitter. We start with the experimental results of the Fano factor obtained from the [PERSON] experiment [4]. We used this research to study the superposition of photons that generate noise associated with chaotic fluctuations. We organize this article as follows: in the first section, we describe the various statistics behind shot noise and obtaining quantum chaos as a superposition of wave and particle states.