# Quantum Features of Atom-Field Systems in the Framework of Deformed Fields
## 3 Deformed Schrodinger Cat Spin Coherent States  
A SCS is obtained through acting a displacement operator on the slightest weight state of the space basis [7; 9]. Nevertheless, for a deformed algebra, it is not insignificant to create such an operator as the framework of the algebra is not conserved. The DSCSs are introduced as  
\[|z,\,j\rangle_{q}=\mathbf{N}_{q}\Big{(}|z|^{2}\Big{)}\,E_{q}^{zJ_{+}^{q}}|j,- j\rangle\,,\,\,z\in\mathbf{C}\,, \tag{7}\]  
where, we considered the deformed exponential  
\[E_{q}^{x}=\sum_{m=0}^{\infty}\frac{x^{m}}{[m]_{q}!}\,,\,\,\,\,[m]_{q}!=[m]_{q} [m-1]_{q}\dots[1]_{q^{\prime}}\,\,[0]!=1. \tag{8}\]  
The normalization function is given by  
\[\mathbf{N}_{q}\Big{(}|z|^{2}\Big{)}=\frac{1}{\sqrt{(1+|z|^{2})_{q}^{2}}}, \tag{9}\]  
where, the [PERSON]'s deformed binomial formula is used  
\[(x+y)_{q}^{n}\,\,:=\sum_{m=0}^{n}[^{n}_{m}]_{q}x^{n-m}y^{m}. \tag{10}\]Here, the deformed binomial function is,  
\[[^{n}_{m}]_{q}=\frac{[n]_{q}!}{[n]_{q}![n-m]_{q}!}\text{ for }n\geq m. \tag{11}\]  
Using these formulations, the generalized spin coherent is written as:  
\[|z,\,j\rangle_{q}=\left((1+|z|^{2})_{q}^{2j}\right)^{-\frac{1}{2}}\sum_{m=-j}^{ j}\left({2j\atop[j+m]_{q}}\right)^{\frac{1}{2}}\!z^{(m+j)}|j,\,m\rangle. \tag{12}\]  
To examine the physical features of the deformed states, we display the basis states \(|j,\,m\rangle\) as a function of the Fock states \(|n\rangle\) (\(|j,\,m\rangle\sim|n\rangle\)). To do this, we use the deformation of HPR given in Equation (4). Applying this realization, we obtain the change of variable \(n=j+m\), which When used in Equation (11), gives the following deformed states  
\[|z,\,j\rangle_{q}=\left((1+|z|^{2})_{q}^{2j}\right)^{-\frac{1}{2}}\sum_{n=0}^ {2j}\left({2j\atop[n]_{q}}\right)^{\frac{1}{2}}\!z^{n}|n\rangle. \tag{13}\]  
The [PERSON]'s criteria for the DSCSs are widely studied in [55]. The realization of the su\({}_{q}(2)\) quantum algebra with respect to the deformed HO operators has originated loads of studies in different areas. The DSCSs have been employed to explain a sizeable group of quantum systems made from different potentials, for example, Morse, Poschl-Teller and infinite potentials. Let us now introduce the deformed Schrodinger cat spin states and study their statistical properties of photons. Here, we consider superposition of two DSCSs, which are often called \(\mathbf{su}(2)\) Schrodinger cat states, and discuss the physical properties of the photon statistics. We define \(\mathbf{su}(2)\) Schrodinger cat states as  
\[|z,\,j,\,\Phi\rangle_{q}=\mathbf{N}\Big{(}|z|^{2},\,j,\,\Phi\rangle\,\Big{)} \left[|z,\,j\rangle_{q}+e^{i\Phi}|-z,\,j\rangle_{q}\right], \tag{14}\]  
where \(0\leq\Phi\leq 2\pi\) is an adjustable angle and the normalization constant \(\mathbf{N}_{q}\big{(}|z|^{2},\,j,\,\Phi\rangle\big{)}\) is obtained from the normalization condition \({}_{q}\langle z,\,j,\,\Phi|z,\,j,\,\Phi\rangle_{q}=1\),  
\[\mathbf{N}_{q}\Big{(}|z|^{2},\,j,\,\Phi\rangle=\frac{1}{\sqrt{2}}\left[1+ \frac{\big{(}1-|z|^{2}\big{)}_{q}^{2j}}{\big{(}1+|z|^{2}\big{)}_{q}^{2j}}\cos \Phi\right]^{-\frac{1}{2}}. \tag{15}\]  
The deformed cat spin states exhibit a richer structure than the non-DSCSs. We examine the physical features of these states by evaluating the Mandel's parameter [56]. This parameter is given by  
\[M_{p}=\frac{\langle(\Delta N)^{2}\rangle-\langle N\rangle}{\langle N\rangle}, \tag{16}\]  
where \(\langle N\rangle\) is the mean photon number and \(\langle(\Delta N)^{2}\rangle\) is the photon number variance. This parameter depends on \(z,j,q\) and \(\Phi\). Let's start with the non-deformed case that corresponds to the CSs defined by Equation (13). Figure 1, displays the graph of [PERSON]'s \(M_{p}\) parameter in terms of \(|z|\) for specific values of the parameters \(q\) and \(j\). We can see from the first graph that \(M_{p}<0\) for various values of the physical parameters. Interestingly, the parameter \(M_{p}\) decreases with \(|z|\) and approaches the value \(-1\) as \(|z|\) becomes significantly large.