 Solution

An ideal zero-photon number result projects onto the first term in part (b). ConsequentlyThis is a product state, not an entangled success. The remaining ideal outcome hasand leaves $|gg⟩$. 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 $|ee;0⟩$, 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 $|ee⟩$ 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.