= 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 is
$$
|\widetilde\chi_5\rangle=c_5e^{-i11\omega t/2}\cos(g\sqrt6\,t)|e\rangle-i c_4e^{-i9\omega t/2}\sin(g\sqrt5\,t)|g\rangle.
$$
Thus the five-count <probability> and normalized post-measurement state are
$$
\boxed{P_5=|c_5|^2\cos^2(g\sqrt6\,t)+|c_4|^2\sin^2(g\sqrt5\,t),\qquad
|\chi_5\rangle=\frac{|\widetilde\chi_5\rangle}{\sqrt{P_5}}.}
$$
The 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 $z=0$.
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