Proof

Proof. For one interval, (eq:Wunnormalized) is (eq:rotation). After emission at an earlier interval the memory is in $|0⟩$, so a later pulse with a fresh vacant cell acts trivially on that branch. The surviving branch acquires $\cos {\theta }_j$, or emits $-\mathrm{i}\sin {\theta }_j{\prod }_{k<j}\cos {\theta }_k$. Induction proves the complete formula.

Let $q_0=1$ and $q_j=q_{j-1}{\cos }^2{\theta }_j$. Then

Orthogonality of the time and label basis vectors proves ${L_N}^{†}{L}_N=(1-q_N)I_C$. The memory ground and excited sectors are orthogonal, as are radiation vacuum and excitation sectors, so ${W_N}^{†}{W}_N=I_C$.

Tracing memory kills the cross terms because $⟨0|J_C=0$. Tracing radiation kills them because $⟨\mathrm{vac}|L_N=0$. The respective remaining traces give (eq:Nall) and (eq:C).