Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 4 / 33B (Principles of Quantum Mechanics)

First-order time-dependent perturbation theory givesThe lowest nontrivial transition probability is $|c_1(t){|}^2$, with errors of higher order in $V$.

Move to a rotating frame. With detuning $\Delta ={\omega }_1-{\omega }_0+\omega$ (changing sign with the phase convention), the constant two-level Hamiltonian has generalized Rabi frequency $\Omega =\sqrt{{\Delta }^2+4v^2}$. Starting in state zero,The transition envelope is maximized on resonance, $\Delta =0$, namely $\omega ={\omega }_0-{\omega }_1$ for the matrix phases printed in the question.