I want to describe a some 2-level system which begins in its ground state $|g\rangle$ and then is coupled to a coherent laser beam $|\alpha\rangle$ with the energy $\hbar\omega=E_e-E_g$ and thereby jumps to its excited state $|e\rangle$. I'm looking for the operator description of this event. At first I thought it was (ignoring physical normalization constants): $$(\hat{a}+\hat{a}^{\dagger})|\alpha\rangle\otimes|g\rangle=\alpha|\alpha\rangle|g\rangle+|\alpha\rangle|e\rangle\tag{1}$$ which physically seems to mean the annihilation operator takes a photon from the laser beam and adds it, via the creation operator, to the two-level system. But (and this is assuming I've done the algebra here correctly), I don't understand the meaning of the $\alpha|\alpha\rangle|g\rangle$ term that shows up in equation (1). We are beginning in an eigenstate of energy $|e\rangle$ and going into a superposition of energy eigenstates $c_1|g\rangle + c_2|e\rangle$. By physical intuition, it seems to me that the laser -- given sufficient time -- should make the amplitude for $|e\rangle$ essentially $1$ (once normalization is done).
What am I missing here?