$q$-Deformed Barut–Girardello $su(1,1)$ coherent states and Schrödinger cat states

We define Schrödinger cat states as superpositions of $q$-deformed Barut–Girardello $su(1,1)$ coherent states with an adjustable angle $\varphi$ in a $q$-deformed Fock space. We study the statistical properties of the $q$-deformed Barut–Girardello $su(1,1)$ coherent states and Schrödinger cat states. The statistical properties of photons are always sub-Poissonian for $q$-deformed Barut–Girardello $su(1,1)$ coherent states. For Schrödinger cat states in the cases $\varphi=0,\pi/2,\pi$, the statistical properties of photons are always sub-Poissonian if $\varphi=\pi/2$, and the other cases are hard to determine because they depend on the parameters $q$ and $k$. Moreover, we find some interesting properties of Schrödinger cat states in the limit $|z|\to0$, where $z$ is the parameter of those states. We also derive that the statistical properties of photons are sub-Poissonian in the undeformed case where $\pi/2\le\varphi\le3\pi/2$.