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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 $\{|a⟩,|b⟩\}$. 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 $t_j=j{T}_{\pi }/n$, 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 isAfter each ideal projective measurement, the populations therefore follow the symmetric two-state Markov chainThe population difference is multiplied by $1-2p=\cos (\pi /n)$ at each step. Starting in $a$, the nonselective two-level Zeno dynamics givesFor $n=64$, the exact final population is $P_b\simeq 0.03712$, while the requested large-$n$ estimate is ${\pi }^2/256\simeq 0.03855$, about 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 $a$ isIt 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.

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.

Continue with the initial quantum state ${\psi }_1$ from part (ii), and let $\Delta t=T/N$. Conditional on every preceding outcome being $+\hslash /2$, the post-measurement state after each ideal projective measurement is again ${\psi }_1$, up to global phase. Part (ii) therefore gives the same conditional probability ${\cos }^2(\omega \Delta t)$ for each successive outcome. Multiplying these conditional probabilities yields the repeated projective-measurement survival probabilityThis multiplication uses the quantum state reset at each successful projective measurement; the outcomes need not be assumed statistically independent.

For fixed $T$, put $z=\omega T$. The Taylor expansion ${\cos }^{2}u=1-u^2+O(u^4)$ gives the quadratic short-time quantum survival probability. For sufficiently large $N$, taking logarithms givesHere the remainder uses Big O notation. Exponentiating provesThis is the quantum Zeno effect: increasingly frequent ideal projective measurements inhibit the transition away from the initial quantum state. The departure probability in a short interval is quadratic in its duration, so $N$ intervals of duration $T/N$ produce a total leading departure probability proportional to $T^2/N$. Without intervening measurements, the return probability at time $T$ would be ${\cos }^2(\omega T)$. The result concerns obtaining $+\hslash /2$ at every measurement in the ideal limit of instantaneous measurements with unitary time evolution between them.

Inhibition of a transition by increasingly frequent ideal projective measurements that check whether a quantum system remains in its initial quantum state. For a normalized vector $\psi$, a fixed Hamiltonian operator $H$ and the orthogonal projection $P=|\psi ⟩⟨\psi |$, the probability of passing every check at $N$ equally spaced times up to fixed time $T$ isFor a finite-dimensional Hilbert space, the quadratic short-time quantum survival probability implies ${\mathbb{P}}_N(T)\to 1$. This is an ideal limit with instantaneous measurements and unitary time evolution between them.