# Do all unitary operations manifest from time-evolution?

Let $$|\psi\rangle$$ be an element of a Hilbert space $$\mathcal{H}$$ and $$U$$ a unitary operator on $$\mathcal{H}$$. I am concerned with the actual physical manifestation of such a unitary operator in the sense portrayed in the following parable.

I am in an inertial reference frame, and $$|\psi\rangle$$ is a quantum system at rest relative to me. Thus, I make the natural choice to parametrize the time evolution of $$|\psi\rangle$$ with my proper time. Now, I am given an apparatus that takes as input a quantum state like $$|\psi\rangle$$ and outputs $$U|\psi\rangle$$. Being the good note-taking scientist that I am, I write in my notebook that at time $$t_0$$ (I glanced at my watch), the state of the quantum system before me is $$|\psi\rangle$$. I then, at time $$t_{\text{in}} > t_0$$, feed the state $$|\psi\rangle$$ into the apparatus. At time $$t_{\text{out}}$$, the output state, $$U|\psi\rangle$$, emerges. Finally, knowing that the apparatus performs a $$U$$ operation, I write that the final state is $$U|\psi\rangle$$ at time $$t_f > t_{\text{out}}$$. My question is if $$t_{\text{out}} > t_{\text{in}}$$ necessarily? In other words, do actual unitary operations on quantum systems ever happen instantaneously? Is this even possible to answer?

If indeed $$t_{\text{out}} > t_{\text{in}}$$ for all $$U$$, does this imply that all unitary operations that actually happen in nature manifest from a Hamiltonian (or set of Hamiltonians) that generates an appropriate time evolution operator and implements the desired unitary $$U$$?