 Solution

Project the field onto the five-photon number state. The excited-atom contribution comes from the initial $n=5$ block, while the ground-atom contribution comes from the initial $n=4$ block. The unnormalized conditional atomic state isThus the five-count probability and normalized post-measurement state areThe relative free-evolution phase is physically retained; it is absent only in an explicitly rotating atomic frame. The conditional state is defined when $P_5>0$. In particular, an initial vacuum cannot produce five photons in this one-atom model, so that conditioning event has probability zero when $z=0$.