 Solution

Conservation of $\mathcal{N}$ confines evolution from $|e,0⟩$ to $|e,0⟩,|g,1⟩$. On this subspace,The interaction matrix squares to the identity, so its exponential separates into cosine and sine terms. The state isThe overall phase can be discarded or removed by the interaction picture. The relative phase $-i$ follows from Schrödinger evolution with the positive coupling in a. The atom and field are generally entangled; complete excitation transfer occurs at $gt=\pi /2$.