Given a fixed value of the coupling:
For a value of the energy splitting Δ = (E1 - E2)/2, the Rabi Frequency is
If the system is prepared in one state, and we ask "what is the time-dependent amplitude in the other?", the exact treatment of the probability is
The approximation to this solution from first-order perturbation theory is
Now let's compare the two forms of the solution for the time dependence of the pobability, using different values of the energy splitting Δ.
First the time-domain:
The exact is plotted as dashes, while the perturbation theory result is solid.
Now lets look at the dependence of P on energy splitting Δ: for different values of the time delay.
Set time points:
The exact is plotted as dashes, while the perturbation theory result is solid.