= Solution
Use $t_0=0$ as in the displayed carrier-phase convention. The drift is diagonal, and conjugation gives
$$
\widetilde H_1(t)=\begin{pmatrix}0&e^{i\omega t}\\e^{-i\omega t}&0\end{pmatrix}.
$$
Multiplying by $A(t)\cos(\omega t+\phi)$ and expanding the cosine into exponentials yields
$$
f(t)\widetilde H_1(t)=\frac{A(t)}2
\begin{pmatrix}
0&e^{-i\phi}+e^{i(2\omega t+\phi)}\\
e^{i\phi}+e^{-i(2\omega t+\phi)}&0
\end{pmatrix}.
$$
The <rotating-wave approximation> averages away the components oscillating at $2\omega$, retaining
$$
\boxed{H_I^{\rm RWA}(t)=\frac{A(t)}2\begin{pmatrix}0&e^{-i\phi}\\e^{i\phi}&0\end{pmatrix}
=\frac{A(t)}2(\cos\phi\,\sigma_x+\sin\phi\,\sigma_y).}
$$
The envelope is slowly varying and the effective coupling is weak compared with the carrier frequency. The pulse phase selects the transverse rotation axis on the <Bloch sphere>; its <pulse area> determines the angle. Counter-rotating corrections are neglected under this approximation, not identically zero.
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