# Quantum Features of Atom-Field Systems in the Framework of Deformed Fields
## 4 Interactions of a Two-Level Atom with a Field in the Framework of su\({}_{2}\)(2) Algebra  
We consider an atomic system relating to a single field state described as the superposition of DSCSs. Let \(|e\rangle\) and \(|g\rangle\) represent the upper state and lower state of the atom. The interaction Hamiltonian is expressed as  
\[\hat{H}_{I}=\hbar\mu\left(\int_{-}^{q}|e\rangle\langle g|+\int_{+}^{q}|g \rangle\langle e|\right), \tag{17}\]  
where \(\mu\) represents the atom-field coupling constant and \(\int_{-}^{q}\left(\int_{+}^{q}\right)\) is the lowering (raising) operator in the quantized field. We assume that the two-level atom is firstly made in the upper state and the quantized field in the deformed cat spin coherent states (DCSCSs) in the context of HPR. The principle of linear superposition is the core of quantum mechanics via the control of the features of the single states that make them less or more pronounced. Considering the initial state of the bipartite system as a product state of quantum subsystems \(|\psi(0)\rangle_{AF}=|\psi(0)\rangle_{A}\otimes|\psi(0)\rangle_{F}\). The wave function of the atom-field system is expended with respect to the states \(|g,\,n+1\rangle\) and \(|e,\,n\rangle\). The atom-field state at subsequent time \(t\) is given as  
\[|\psi(t)\rangle_{AF}=\sum_{n=0}^{\infty}\big{[}\mathbf{A}_{e,\,n}|e,\,n\rangle +\mathbf{A}_{g,\,n+1}|e,\,n+1\rangle\big{]}, \tag{18}\]  
Figure 2: The parameter \(M_{p}\) of the SCSs in the context of su\({}_{q}\)(2) algebra, defined in Equation (13), in terms of \(|z|\) for specific values of the deformed parameter \(q\) with \(j=2\). (**a**) is for \(\Phi=0\), (**b**) is for \(\Phi=\pi/2\), (**c**) is for \(\Phi=\pi/4\), and (**d**) is for \(\Phi=\pi\). Black (solid line) corresponds to \(q=1\) which displays the case of ordinary SCSs, blue (dashed line) corresponds to \(q=\) 0.3, and red (dotted-dashed line) corresponds to \(q=\) 0.8. The parameter \(M_{p}\) is negative or positive depending of values of the physical parameters, exhibiting sub-Poissonian and super-Poissonian distribution of photons. Figure 4: The dynamics of the different quantum quantifiers for specific values of the deformed parameter \(q\) with \(|z|=1\), \(j=2\), and \(\Phi=0\). (**a**) is for the atomic inversion, (**b**) and Figure 5d are for the Mandel’s parameter and (**c**) is for the von Neumann entropy. Blue (solid) line corresponds to \(q=1\), black (dotted-dashed) line corresponds to \(q=0.3\) and red (dashed) line corresponds to \(q=0.8\). Figure 3: The dynamics of the different quantum quantifiers for specific values of the deformed parameter \(q\) with \(|z|=1\), \(j=1\), and \(\Phi=0\). (**a**) is for the atomic inversion, (**b**) is for the Mandel’s parameter and (**c**) is for the von Neumann entropy. Blue (solid) line corresponds to \(q=1\), black (dotted-dashed) line corresponds to \(q=0.3\) and red (dashed) line corresponds to \(q=0.8\). The temporal evolution of the Mandel's parameter is displayed in Figures 3b and 4b against the dimensionless time \(gt\) for \(j=1\) and \(j=2\), respectively, considering different values of the deformed parameter \(q\) with \(|z|=1\) and \(\Phi=0\). In general, the parameter \(M_{p}\) exhibits oscillations that depend on the values of the spin number \(j\) and the values of the \(M_{p}\) are very sensitive to the deformed parameter \(q\) during the dynamics. For small values of spin, the Mandel's parameter takes negative values as the deformed parameter gets farther from one, exhibiting sub-Poissonian distribution. For the undeformed case, \(q\to 1\), we can see that the photon statistics seem to be fluctuated around super-Poissonian and sub-Poissonian distributions during the time evolution. On the other hand, when the spin number increases, the Mandel's parameter tends to get positive values for different times, exhibiting super-Poissonian distribution of photons (see Figure 5b,d).