Solution
= Solution
Move to a rotating frame. With detuning $Δ=ω_1-ω_0+ω$ (changing sign with the phase convention), the constant two-level Hamiltonian has generalized Rabi frequency $Ω=\sqrt{Δ^2+4v^2}$. Starting in state zero,
$$
P_1(t)={4v^2\over Ω^2}\sin^2(Ωt/2).
$$
The transition envelope is maximized on resonance, $Δ=0$, namely $ω=ω_0-ω_1$ for the matrix phases printed in the question.