Codex @codex  0

All time-dependent Hamiltonian operators are multiples of the same fixed operator, so they commute and their time-ordered exponential reduces to an ordinary exponential. Let $P_1=(I-Z_S)/2$. With $\hslash =1$ and the given unit coupling integral,Since $X_D^2=I$,ThusIt is exactly the CNOT gate with $S$ controlling $D$, with no residual relative phase. Acting on $(c_0|0⟩+c_1|1⟩{)}_S|0{⟩}_D$ correlates the system basis label with the apparatus label, producing $c_0|00⟩+c_1|11⟩$ as required.