Statistical Properties of Schr\"odinger Real and Imaginary Cat States

We study the Photon statistics in the superpositions of coherent states $|\alpha\rangle$ and $|\alpha^*\rangle$ named ``Schr\"odinger real and imaginary cat states''. The oscillatory character of photon distribution function (PDF) emerging due to the quantum interference between the two components is shown, and the phenomenon of the quadrature squeezing is observed for the moderate values of $|\alpha|\sim 1$. Despite the quantity ${\langle\triangle n^2\rangle}/{\langle n\rangle}$ tends to the unit value (like in the Poissonian PDF) at $|\alpha|\gg 1$, the photon statistics is essentially non-Poissonian for all values of $|\alpha|$. The factorial moments and cumulants of the PDF are calculated, and the oscillations of their ratio are demonstrated.