First, only one of the two kets in your initial state is the vacuum state, the other is the coherent state.
Second, you are missing the point. You have an initial state ##\ket{0} \ket{\alpha}##. This is a product of two kets. That means you have two physical degrees of freedom in the quantum system that this state describes. What are those two physical degrees of freedom, in, for example, the beam splitter scenario?
You then time evolve that initial state using a Hamiltonian to get a final state. That final state will also describe a quantum system with two physical degrees of freedom (but in general it won't be a product state, it will be entangled). What are those two physical degrees of freedom in the final state, in, for example, the beam splitter scenario? (Hint: they're not the same as the two physical degrees of freedom in the initial state, because of the beam splitter.)