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The original Figure 1 has the allowed chain $1\leftrightarrow 2\leftrightarrow 3\leftrightarrow 4$. Its positive resonance frequencies are ${\omega }_1=E_2-E_1$, ${\omega }_2=E_2-E_3$ and ${\omega }_3=E_4-E_3$ in units $\hslash =1$; level 2 lies above level 3. Use a pulse at ${\omega }_1$ to transfer level population from 1 to 2, then one at ${\omega }_2$ to transfer it from 2 to 3, and finally one at ${\omega }_3$ to transfer it from 3 to 4.

For a selected pair, choose the optical phase so the effective Quantum Hamiltonian has a fixed off-diagonal rotation axis. If its coupling coefficient is $g_{jk}(t)=A_{jk}(t)|d_{jk}|/2$ in this normalization, the active two-dimensional propagator iswhere $K_{jk}^2=I_{jk}$. A full level population transfer has $\beta =\pi /2$. Equivalently, in the Rabi-frequency convention ${\Omega }_R=2g_{jk}$, the required area is $\int {\Omega }_{R}dt=\pi$. On spectator levels the embedded two-level quantum rotation is the identity.

Thus three selective $\pi$ pulses at ${\omega }_1,{\omega }_2,{\omega }_3$, in that order, transfer $|1⟩$ to $|4⟩$ up to global phase. A strongly regular quantum Hamiltonian separates the resonances; envelopes long enough that their bandwidth and coupling strength are small relative to unwanted frequency gaps make off-resonant excitation arbitrarily small within the selective-pulse limit. The phases accumulated along this path do not affect the final level population.