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Time evolution of cascade decay

We study non-perturbatively the time evolution of cascade decay for generic fields $π\rightarrow ϕ_1ϕ_2\rightarrow ϕ_2χ_1χ_2$ and obtain the time dependence of amplitudes and populations for the resonant and final states. We analyze in detail the different time scales and the manifestation of unitary time evolution in the dynamics of production and decay of resonant intermediate and final states. The probability of occupation (population) "flows" as a function of time from the initial to the final states. When the decay width of the parent particle $Γ_π$ is much larger than that of the intermediate resonant state $Γ_{ϕ_1}$ there is a "bottleneck" in the flow, the population of resonant states builds up to a maximum at $t^* = \ln[Γ_π/Γ_{ϕ_1}]/(Γ_π-Γ_{ϕ_1})$ nearly saturating unitarity and decays to the final state on the longer time scale $1/Γ_{ϕ_1}$. As a consequence of the wide separation of time scales in this case the cascade decay can be interpreted as evolving sequentially $π\rightarrow ϕ_1ϕ_2; ~ ϕ_1ϕ_2\rightarrow ϕ_2χ_1χ_2$. In the opposite limit the population of resonances ($ϕ_1$) does not build up substantially and the cascade decay proceeds almost directly from the initial parent to the final state without resulting in a large amplitude of the resonant state. An alternative but equivalent non-perturbative method useful in cosmology is presented. Possible phenomenological implications for heavy sterile neutrinos as resonant states and consequences of quantum entanglement and correlations in the final state are discussed.