Properties of the momentum-translation approximation

It is shown that the momentum-translation approximation (MTA) for the treatment of the interaction of bound states with a plane-wave field can be developed in at least five different ways. This multiplicity of approaches is used to exhibit some of the basic features of the approximation, as well as to contrast it with a gauge transformation, with which it is sometimes confused. Validity conditions are $\frac{\mathrm{ea}{a}_0\omega }{E}\ll 1$, $\omega a_0\ll 1$ (where $a$ is the amplitude of the vector potential in the Coulomb gauge for a plane wave of circular frequency $\omega$, $E$ is a characteristic bound-state energy, and $a_0$ is a characteristic size of the bound system), and an absence of intermediate near resonances in a transition. The momentum-translation technique predicts replica states very directly, but level shifts are found only by a method equivalent to perturbation theory. A general formalism is developed for transitions caused by linearly or circularly polarized applied plane-wave fields, in which a second field may or may not be involved. In first order, the MTA always reduces exactly to first-order perturbation theory, but when treated in their entireties, it is shown that a perturbation series and a momentum-translation series are rearrangements of each other. All the criticisms which have been directed at the MTA are reviewed and evaluated. The principal limitations of the MTA are found to be the difficulty of treating problems with intermediate near resonances, and the absence of a systematic way to improve on the basic approximation. The strengths of the MTA are its simple analytical form, its good accuracy when there are no intermediate near resonances, and the fact that high-order multiphoton processes are treated as easily as first-order processes.