Classicalization times of parametrically amplified “Schrödinger cat” states coupled to phase-sensitive reservoirs

► We study parametric excitation of a quantum oscillator in phase-sensitive baths. ► Exact time-dependent Wigner function for initial even/odd coherent states is found. ► The evolution of negativity of Wigner function is compared with the squeezing dynamics. ► The difference between initial and final “classicalization times” is emphasized. ► Both these times can be arbitrarily long for rigged reservoirs at infinite temperature.