Dalgarno-Lewis equation and its application to ultrafast two-photon processes: Short- and long-range terms for intermediate states

One long-term goal of attophysics is to deepen our knowledge on the ultrafast dynamics of quantum systems with the prospect of stirring electrons in their natural time and space scales. Two-photon processes are the lowest-order methods able to provide such information. Despite the long history backing two-photon studies and the success of many different exact and approximate methods, there are still some details to be unraveled. We use the Dalgarno-Lewis method to obtain both numerically exact results and accurate analytic expressions for short- and long-range contributions to the wave packets condensing information from the initial state and its coupling with the infinitely many virtual intermediate states in these two-photon processes. We identify some specific conditions under which short- and long-range terms adding up into the intermediate state induce radial matrix elements to hold competing coherent contributions for both single- and two-color cases. In the first case, interference structures marking a regime change may appear. On the second, short-range contributions may mask the compliance with propensity rules.