Keldysh approximation,A2, and strong-field ionization

The Keldysh approximation as applied to above-threshold ionization (ATI) is compared with conventional (weak-field) perturbation theory and the Kroll-Watson theory of electron scattering in intense fields. Unlike the latter ‘‘nonperturbative’’ theory, the Keldysh approximation treats the effects of the field on the initial and final states differently, and this leads to a spurious contribution from the ${\mathrm{A}}^2$ part of the Hamiltonian in the A⋅p gauge. We attempt to clarify the role of the ponderomotive potential in ATI and the influence of the ${\mathrm{A}}^2$ term on the polarization dependence of the transition amplitude. The fundamental defect of the Keldysh approximation with regard to ${\mathrm{A}}^2$ is shown to be of little practical consequence for the experiments carried out thus far. However, it is shown that the Keldysh amplitude is not ‘‘exact’’ in the applied field, as sometimes claimed, and is best regarded as an ansatz rather than a leading term in a gauge-invariant perturbation expansion. A simple modification of the Keldysh theory is proposed.