q-Cat states revisited: two families in a Fock representation space of q-oscillator algebra with different nonclassical behaviors

This paper has been motivated by our previous paper (Eur. Phys J Plus 135:253, 2020) on q-coherent states of the Arik–Coon q-oscillator. The quadratic powers of unbounded and bounded annihilation operators associated with the Arik–Coon q-oscillator are used to obtain two different families of the even and odd q-cat states for 0<q<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<q<1$$\end{document} in even and odd subspaces of the Fock representation space of the system. The resolutions of the identity condition for q-cat states of the unbounded and bounded annihilation operators are realized by two appropriate positive definite q-measures in the q-integral on the whole complex plane and a disk in radius 1/1-q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1/\sqrt{1-q}$$\end{document}, respectively. It is shown that the antibunching effect and sub-Poissonian statistics as well as the bunching effect and super-Poissonian statistics are simultaneously exhibited by the first and second families of q-cat states, respectively. Otherwise, a fixed observation cannot be concluded. The strong and weak squeezing conditions for two different types of q-generalizations of the position and momentum quadratures, corresponding to the unbounded and bounded annihilation operators, on their associated even and odd q-cat states are considered. Using an example, we show that contrary to the odd q-cat states of the first family that exhibit both strong and weak squeezing effects by both quadratures, the ones of the second family demonstrate only the weak squeezing.