 Solution

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

Subtract the common sector energy and call the remaining eigenvalue

ℏ λ

. Its characteristic equation is

λ^{2} - \frac{Δ ^{2}}{4} - g^{2} (n + 1) = 0.

The dressed-state energies are consequently

E_{\pm n} = ℏ ω (n + \frac{1}{2}) \pm \frac{ℏ}{2} Ω_{n}, Ω_{n} = Δ^{2} + 4 g^{2} (n + 1) .

On resonance their splitting is

2ℏ∣ g ∣ n + 1

; far off resonance the eigenstates approach the uncoupled quantum states with small dispersive energy shifts. The eigenvalue splitting is an angular-frequency difference

Ω_{n}

, so an ordinary frequency would be

Ω_{n} / (2 π)

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