= Solution
There is at most one field <photon> in <3/b>. Projection onto its one-photon sector gives
$$
\boxed{P(n=1;t)=\sin^2(gt),\qquad
\rho_{\rm atom\mid n=1}=|g\rangle\langle g|.}
$$
Before detector absorption, the normalized conditional joint state is $|g,1\rangle$ up to an overall phase. A destructive <photon> count changes the field to its <vacuum state>[vacuum] but leaves the <atom> in $|g\rangle$. This conditional state is meaningful whenever the one-photon outcome has nonzero <probability>.
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