= Solution
In the ideal <state-selective fluorescence detection> model, driving the $a\leftrightarrow c$ optical transition produces a bright record only for $a$, while $b$ is dark. A sufficiently discriminating pulse therefore measures the basis $\{|a\rangle,|b\rangle\}$. The optical cycle is assumed to return population to the appropriate logical state without leakage. Ignoring the recorded outcome removes the off-diagonal <quantum coherence> between $a$ and $b$.
Take $n$ equally spaced measurements at $t_j=jT_\pi/n$, including the final time. Between checks the resonant drive rotates through angle $\Omega T_\pi/n=\pi/n$. The probability of changing basis state in one interval is
$$
p=\sin^2\frac\pi{2n}.
$$
After each ideal <projective measurement>, the populations therefore follow the symmetric two-state <Markov chain>
$$
\begin{pmatrix}P_a'\\P_b'\end{pmatrix}
=\begin{pmatrix}1-p&p\\p&1-p\end{pmatrix}
\begin{pmatrix}P_a\\P_b\end{pmatrix}.
$$
The population difference is multiplied by $1-2p=\cos(\pi/n)$ at each step. Starting in $a$, the <nonselective two-level Zeno dynamics> gives
$$
\boxed{P_b(T_\pi)=\frac{1-\cos^n(\pi/n)}2
=\frac{\pi^2}{4n}+O(n^{-2})\longrightarrow0.}
$$
For $n=64$, the exact final population is \b[$P_b\simeq0.03712$], while the requested large-$n$ estimate is \b[$\pi^2/256\simeq0.03855$], about \b[four percent]. Thus the transition that an uninterrupted pi pulse would complete is strongly inhibited.
If “leaving” instead means departing from $a$ at least once in the complete measurement record, use the <repeated projective-measurement survival probability>. The probability that every check returns $a$ is
$$
P_{\rm all\,a}=\cos^{2n}\frac\pi{2n},\qquad
1-P_{\rm all\,a}\simeq0.03782\quad(n=64).
$$
It has the same leading estimate $\pi^2/(4n)$, but is not exactly the final $b$ population: a trajectory can leave $a$ and subsequently return. Keeping these events separate also distinguishes postselected records from an experiment that measures only final populations.
\Image[/past-exam-of-the-laser-science-and-quantum-information-processing-course-of-the-university-of-oxford/2005/5-rabi-zeno-inhibition.png]
{title=Rabi transfer inhibited by repeated measurements, with final population and all-check departure distinguished}
{description=Original ideal two-level calculations, not historical data. Probe times include the end of the pi pulse. The first panel shows ensemble populations between nonselective probes; the second compares exact final population, departure at any recorded check, and their common large-n estimate.}
The mechanism is the <quadratic short-time quantum survival probability>: each short interval has transition probability of order $(T_\pi/n)^2$, so $n$ checks leave only an order-$1/n$ total transition probability. The <quantum Zeno effect> here requires effective, sufficiently fast measurements; a finite-duration optical probe is a physical interaction whose imperfections must be included when comparing with data.
The <beryllium-ion quantum Zeno experiment of 1990> used the same ground-state transition and a finite-duration optical probing sequence, followed by a calibrated ensemble <fluorescence> readout. Its measured final populations motivate the nonselective model; the instantaneous projections here idealize those physical pulses.
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