Past exam of the laser science and quantum information processing course of the University of Oxford / 2019 / 3 / c / Solution

From b, the excited-state probability isFor a single short measurement interval $\Delta t$ beginning in $|e,0⟩$, requiring $P_e>0.9$ before the first departure from unity givesA zero-photon outcome resets the conditional state to $|e,0⟩$. After $n=T/\Delta t$ such tests, the probability of the all-zero-photon record isThus a 90 percent survival requirement over a specified total duration $T$ is stronger than the single-interval condition: approximately $\Delta t<-\ln (0.9)/(g^{2}T)$ in the short-interval limit. The limiting preservation as $\Delta t\to 0$ is the quantum Zeno effect.

The measurement instrument matters. If photons are absorbed and their first detection removes the excitation, this survival probability also equals the unconditional excited population. For ideal nondestructive photon-number measurements, excitation can transfer back after a one-photon result; the unconditional result after $n$ equal intervals is instead $P_e^{(n)}=[1+{\cos }^n(2g\Delta t)]/2$. It also tends to one for fixed $T$ as the intervals shrink. Neither interpretation guarantees indefinite 90 percent survival merely by imposing a 90 percent bound on each separate interval.