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Rotating-frame solution for a harmonically driven two-level system

For

H (t) = (0 c e^{- iω t /ℏ} c e^{iω t /ℏ} 0),

set

a = U (t) b

with

U (t) = diag (e^{iω t / (2ℏ)}, e^{- iω t / (2ℏ)}) .

Then

b

evolves under the constant Hamiltonian

H_{eff} = (ω /2 c c - ω /2), H_{eff}^{2} = (c^{2} + \frac{ω ^{2}}{4}) I .

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