where the 3rd step can be proved by using a Taylor series expansion, and using the orthogonality of the states | g ⟩ ⟩ and | e ⟩ ⟩ . Note that a multiplication by an overall phase of e i ω ω 0 t / 2 on a unitary operator does not affect the underlying physics, so in the further usages of U we will neglect it. Applying U gives: