= Solution
The original Figure 1 has the allowed chain $1\leftrightarrow2\leftrightarrow3\leftrightarrow4$. 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 $\hbar=1$; 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 is
$$
U_{jk}=\cos\beta\,I_{jk}-i\sin\beta\,K_{jk},\qquad \beta=\int g_{jk}(t)\,dt,
$$
where $K_{jk}^2=I_{jk}$. A full <level population> transfer has $\beta=\pi/2$. Equivalently, in the Rabi-frequency convention $\Omega_R=2g_{jk}$, the required area is $\int\Omega_Rdt=\pi$. On spectator levels the <embedded two-level quantum rotation> is the identity.
Thus \b[three selective $\pi$ pulses at $\omega_1,\omega_2,\omega_3$, in that order, transfer $|1\rangle$ to $|4\rangle$ 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>.
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