Nonselective two-level Zeno dynamics ($P_b(T)=\frac{1}{2}[1-{\cos }^n(\Omega T/n)]$)

Start in $a$, drive resonantly with Hamiltonian operator $\hslash \Omega X/2$, and perform $n$ ideal basis measurements separated by $T/n$. An interval changes either basis state with probability $p={\sin }^2(\Omega T/(2n))$. Ignoring measurement records leaves a two-state Markov chain whose population difference is multiplied by $1-2p=\cos (\Omega T/n)$ each step. This proves the displayed final-population formula.

For a pi pulse, $\Omega T=\pi$, it is ${\pi }^2/(4n)+O(n^{-2})$. The repeated projective-measurement survival probability instead counts records in which every result is $a$, giving $P_{\mathrm{all}\mathrm{a}}={\cos }^{2n}(\pi /(2n))$. Final return after an earlier departure separates the two finite-$n$ events.