Influence of intense incident electromagnetic fields on weakly bound electrons: An exact Keldysh approximation

In this paper, the classical Keldysh approximation for evaluating the tunnel ionization rate of a bound electron under the action of an intense radiation field is revisited. We transform the original Keldysh expression for the transition amplitude to a simplified one used by Gribakin and Kuchiev [Phys. Rev. A 55, 3760 (1997)], and show that the two expressions are equivalent up to a phase factor which does not contribute to ionization. Using the residue theorem to perform the contour integral, rather than the saddle point method employed by Keldysh and others, a revised closed form analytical formula for the photoionization rate is obtained. The formula also avoids using the redundant assumption of small kinetic momentum employed by Keldysh. This paper thus realizes a long-sought goal to calculate the transition rate within only the Keldysh approximation and provides an "exact" formula to compare with other approaches using different gauges at various levels of approximation.