Past exam of the laser science and quantum information processing course of the University of Oxford / 2021 / 6 / c / Solution

An ideal projective measurement of the field's photon number selects the coefficient of $|1⟩$. ThereforeWhenever this probability is nonzero, normalization of the projected joint quantum state gives the atomic Bell stateup to a common global phase. The maximum heralding probability is $1/3$, attained at $\sqrt{6}gt=\pi /2+n\pi$. At a time with $P_1=0$, that conditioning event is impossible, so a conditional state is not defined there. The interpretation is a photon number measurement at the selected time; continuous monitoring of escaping photons would instead change the dynamics and require a different model.