= Solution
Conservation of $\mathcal N$ confines evolution from $|e,0\rangle$ to $|e,0\rangle,|g,1\rangle$. On this subspace,
$$
H=\hbar\omega I+\hbar g\begin{pmatrix}0&1\\1&0\end{pmatrix}.
$$
The interaction <matrix> squares to the identity, so its exponential separates into cosine and sine terms. The state is
$$
\boxed{|\Psi(t)\rangle=e^{-i\omega t}
\left[\cos(gt)|e,0\rangle-i\sin(gt)|g,1\rangle\right].}
$$
The overall phase can be discarded or removed by the <interaction picture>. The <relative phase> $-i$ follows from <Schrödinger equation>[Schrödinger evolution] with the positive coupling in <3/a>. The <atom> and field are generally <quantum entanglement>[entangled]; complete excitation transfer occurs at $gt=\pi/2$.
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