Cat-States in the Framework of Wigner-Heisenberg Algebra

A one-parameter generalized Wigner-Heisenberg algebra( WHA) is reviewed in detail. It is shown that WHA verifies the deformed commutation rule $[\hat{x}, \hat{p}_{\lambda}] = i(1 + 2\lambda \hat{R})$ and also highlights the dynamical symmetries of the pseudo-harmonic oscillator( PHO). \textbf{The present article is devoted to the study of new cat-states} built from $\lambda$-deformed Schr\"{o}dinger coherent states, which according to the Barut-Girardello scheme are defined as the eigenstates of the generalized annihilation operator. Particular attention is devoted to the limiting case where the Schr\"{o}dinger cat states are obtained. Nonclassical features and quantum statistical properties of these states are studied by evaluation of Mandel's parameter and quadrature squeezing with respect to the $\lambda-$deformed canonical pairs $( \hat{x}, \hat{p}_{\lambda})$. It is shown that these states minimize the uncertainty relations of each pair of the $su(1,1)$ components.