= Solution
Write $\cos\omega_1t=(e^{i\omega_1t}+e^{-i\omega_1t})/2$ and $\Delta\omega=\omega_2-\omega_1$. Multiplication of the matrix from part (e) gives
$$
\begin{aligned}
fH_I=\frac{A_1(t)}2\{&d_1(1+e^{-2i\omega_1t})E_{12}+d_1(1+e^{2i\omega_1t})E_{21}\\
&+d_2[e^{i\Delta\omega t}+e^{i(2\omega_1+\Delta\omega)t}]E_{23}\\
&+d_2[e^{-i\Delta\omega t}+e^{-i(2\omega_1+\Delta\omega)t}]E_{32}\}.
\end{aligned}
$$
These are exactly the constant, difference-frequency, and sum-frequency terms in the PDF, with each Hermitian pair visible. For a weak, slowly varying envelope, the terms at $2\omega_1$ and $\omega_1+\omega_2$ average out by the <rotating-wave approximation>. To discard the other transition as well, its detuning must be nonzero and large compared with the drive coupling and envelope bandwidth. Sufficient scale conditions are
$$
|A_1d_j|,\ B\ll\omega_1,\omega_2,\qquad
|A_1d_2|,\ B\ll|\Delta\omega|,
$$
where $B$ is a characteristic envelope bandwidth; turn-on transients must satisfy the same spectral restrictions. Then
$$
\boxed{fH_I\simeq\frac{A_1(t)d_1}{2}(E_{12}+E_{21}).}
$$
Small coupling compared with the optical frequencies alone is not enough for this selective approximation: if $|\Delta\omega|$ is comparable with the pulse bandwidth or Rabi scale, the spectator transition remains driven. A long selective gate also accumulates a small off-resonant Stark phase of order $(A_1d_2)^2/\Delta\omega$, so that error must be acceptable or compensated. These assumptions distinguish <resonant quantum control> from an exact decoupling.
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