In this answer dealing with details of decay theory (incl. references) it is shown that
[Given] a system initialized at $t = 0$ in the state [...] $| \varphi \rangle$ and left to evolve under a time-independent hamiltonian $H$ [... its] probability of decay is at small times only quadratic, and the survival probability is slightly rounded near $t = 0$ before of going down [exponentially].
Is it correct that therefore it is also possible to prepare (initialize) an entire ensemble of $N \gg 1$ such states $| \varphi \rangle$, such that their survival probability is at small times only quadratic ?
Is it instead possible at all to prepare an ensemble of $N$ states (which would likewise "evolve under the Hamiltonian $H$") such that their survival probability is (at least to a good approximation) not quadratic but rather drops linearly as a function of the duration since completion of the preparation ?
In particular, if an ensemble of $2~N$ states $| \varphi \rangle$ had been given and (in the process of an extended preparation procedure) half of those (i.e. $N$ systems) had decayed, do the remaining/surviving $N$ systems together then constitute such an ensemble? What exactly is the survival probability of these given, momentarily remaining/surviving $N$ systems; as a function e.g. of $t_{\text{(extended prep.)}} := t - \tau_{1/2}$, where $\tau_{1/2} = \tau~\text{Ln}[2]$ is the specific overall "half-life" duration?
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