I have a question on paper https://doi.org/10.1038/nature02054. In this paper, the CNOT gate is realized using a beam displacer for higher stability. The figure below shows a 1:1 mapping from (a) to (b). In (a), the interference where $C_1$ meets with $T_0$ is needed for a CNOT gate. This interference happens when the two photons meet at "1/3" BS. However, in (b), the interference occurs from the indistinguishability between {($C_1\rightarrow T_{out}$ mode) + ($T_+\rightarrow C_{out}$)} and {($C_1\rightarrow C_{out}$ mode) + ($T_+\rightarrow T_{out}$)}. This is a problem because both cases have a different time delay due to the beam displacer [see drawing below]. Therefore interference cannot happen, and thus, the CNOT gate fails.
Can someone help me understand where I'm wrong?
