We have seen in Section 1.2 that a transition between two discrete levels |i〉 and |k〉 is possible when the system is submitted to a perturbation which has a non-zero Fourier component close to the Bohr frequency ω0 = (Ek − Ei)/ħ of the transition. This is the case when the perturbation Ĥ1(t) is sinusoidal, of the form Ĥ1(t) = Ŵ cos(ωt + φ), and oscillates at a frequency ω close to ω0. But in many real physical situations, the perturbation is not perfectly mastered: it is a pure sine function only for times shorter than a limit, called the coherence time. At longer times, the oscillating perturbation has a phase and an amplitude that vary randomly. This is, for example, the case when the perturbation is due to an incident electromagnetic wave produced by a thermal lamp, in which the thermal fluctuations induce random variations of the amplitude and phase of the wave. As the perturbation is no longer a pure sinusoidal wave, its Fourier spectrum is no longer a Dirac delta function. It has some finite width around a mean frequency. For this reason a random perturbation is also called a ‘broadband’ perturbation.

We shall determine in this complement the temporal evolution of the quantum system submitted to such a random perturbation. We shall see that the transition probability between two discrete levels is proportional to T for small interaction time T, and has an exponential behaviour at long times, a result which is very similar to that obtained for a transition between a discrete level and a continuum.