Transitions in Electromagnetic Fields of Arbitrary Intensity

A nonperturbative method presented earlier for the interaction of electromagnetic radiation with bound quantum systems is refined and extended. A general $T$ matrix is derived which is valid for arbitrarily high field intensity, subject to certain conditions on the frequency of the electromagnetic field. An example is done in detail for a $1s-2s$ transition in a hydrogen atom. It is shown that after a perturbative low-intensity region, the transition amplitude experiences oscillations which pass through zero as the field intensity increases. Qualitative arguments are also given to show why this is to be expected, and why it is inexplicable in perturbation theory. It is found that high-order processes are more important than low-order processes when the intensity is high, and in the hydrogen-atom example it is shown that the transition amplitude has peaks as a function of the order of the process. However, for sufficiently high orders, it is shown that there is an eventual exponential decline in the transition amplitude as the order of the process gets very large. Simple results are obtained for the low-intensity limit with any number of photons and for the high-intensity, large-photon-number limit. The last result should be useful, for example, in calculating optical transitions caused by intense microwave radiation.