 Solution

ID: past-exam-of-the-laser-science-and-quantum-information-processing-course-of-the-university-of-oxford/2017/6/a/solution

Use

a ∣ n ⟩ = n ∣ n - 1 ⟩

a^{†} ∣ n ⟩ = n + 1 ∣ n + 1 ⟩

σ_{z} ∣ e ⟩ = ∣ e ⟩

and

σ_{z} ∣ g ⟩ = - ∣ g ⟩

. Only one interaction term acts on each of the two quantum states. Hence

H ∣ e, n ⟩ = ℏ (nω + \frac{ω _{0}}{2}) ∣ e, n ⟩ + ℏ g n + 1 ∣ g, n + 1 ⟩,

H ∣ g, n + 1 ⟩ = ℏ ((n + 1) ω - \frac{ω _{0}}{2}) ∣ g, n + 1 ⟩ + ℏ g n + 1 ∣ e, n ⟩ .

The interaction exchanges one atomic excitation with one field quantum. The square-root factor is the photon-number enhancement of the transition matrix element.

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