Generation of mixtures of Schrödinger-cat states from a competitive two-photon process

We study the formation of mixtures of superposition states of the coherent states {}±{}{α}〉, also known as Schr\"odinger-cat states, from a competitive process involving a two-photon parametric process and two-photon absorption. Using the fact that photons are created or destroyed in pairs, we deduce the long-time steady-state form of the density operator. We numerically study the evolution at all times and show that generally the field is in a mixed state. We also examine nonclassical properties of the field such as squeezing and sub-Poissonian statistics, and also the evolution of the Q function.