Past exam of the laser science and quantum information processing course of the University of Oxford / 2017 / 6 / a

Use $a|n⟩=\sqrt{n}|n-1⟩$, $a^{†}|n⟩=\sqrt{n+1}|n+1⟩$, ${\sigma }_z|e⟩=|e⟩$ and ${\sigma }_z|g⟩=-|g⟩$. Only one interaction term acts on each of the two quantum states. HenceThe interaction exchanges one atomic excitation with one field quantum. The square-root factor is the photon-number enhancement of the transition matrix element.