= Solution
{c}
An ideal zero-<photon number> result projects onto the first term in part (b). Consequently
$$
\boxed{|\psi_{\rm atoms}\rangle=|ee\rangle,\qquad
P_0(t)=\frac{[2+\cos(\sqrt6gt)]^2}{9}.}
$$
This is a <product state>, not an entangled success. The remaining ideal outcome has
$$
P_2(t)=\frac29[1-\cos(\sqrt6gt)]^2
$$
and leaves $|gg\rangle$. In particular, simply discarding the field outcome does not reproduce the pure heralded <Bell state>.
The practical requirements are strong coherent coupling of both atoms to the same optical <normal mode> with stable relative coupling <phases>, preparation of $|ee;0\rangle$, and interaction timing on the scale $g^{-1}$ before spontaneous emission or cavity loss destroys <quantum coherence>. The field must then be measured or extracted after halting the coherent exchange, with enough <photon number> resolution to distinguish one from two. Otherwise an inefficient detector can mistake a two-<photon> event for the desired herald. Even ideally the one-<photon> success probability is at most $1/3$, so trials must be repeated.
“No photons detected” implies $|ee\rangle$ only for the ideal zero-number projection specified above. With loss or finite detector efficiency, no click also includes missed one- and two-<photon> events and generally leaves a conditional <mixed state>. Detecting a leaking <photon> during the interaction is also a different measurement protocol from the end-time <projective measurement> analyzed here.
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