Proof

Proof. The mutually orthogonal spaces $\mathrm{span}\{|a,0⟩,|0,a⟩\}$ are invariant. On each, $S_j$ is the exchange Pauli matrix; it vanishes on their orthogonal complement. Thus ${S_j}^2=P_j$ and ${S_j}^3=S_j$, which gives (eq:Uj). All Hamiltonians within an interval are scalar multiples of $S_j$, so time ordering reduces to the pulse area. The expression solves the finite-dimensional Schrödinger equation almost everywhere for integrable pulses.

Both members of each coupled pair have energy $\omega$. This proves the energy commutator and (eq:rotation). Tensoring all propagators and the preparation with $I_Q$ proves the reference statement.