Above-threshold ionization (ATI) is an important process in the interaction between atoms and an intense laser field . The photoelectron energy spectrum shows that ATI occurs either in the form of multiphoton ionization or tunneling ionization. Calculations based on numerical solution of the 3D time-dependent Schro¨dinger equation (TDSE) provide quantitative agreement but little physical insight.

Researchers at Shanghai Institute of Optics and Fines Mechanics (SIOM/China)——Key Laboratory for Quantum Optics and Center for Cold Atom Physics present a heuristic analysis showing that the LES structure, which cannot be accounted for by the lowest-order S-matrix theory (i.e.,the SFA), is reproduced by the second-order amplitude.The good agreement between our heuristic theory and the experimental data supports the conclusion that the LES can be attributed to the dominance of the long-range electron–parent-ion Coulomb interaction in the tunneling regime.

A calculation of the lowest-order rescattering term (analogous to the lowest-order Born approximation) in a quantum mechanical S-matrix expansion of the ionization amplitude was presented and compared with extensive experimental data for various atoms, wavelengths, and intensities, which display the recently discovered low-energy structure. [Physical Review Letters 110, 013001 (2013)]

The calculation was carried out for very low momenta of the liberated electron and for an unscreened Coulomb potential. It reproduces many features of the data, including their wavelength and intensity dependence and the corresponding scaling. Technically,divergences due to the long-range Coulomb potential,which are normally encountered in high-order terms of the S-matrix expansion, were removed by taking into account the actual value of the depletion of the atomic ground state due to ionization. It is proposed that to some extent the lowest-order rescattering term provides an ‘‘effective’’description of forward rescattering and that the inclusion of higher-order terms may neither be necessary nor even desirable, in the same way that the first-order Born approximation yields the exact Coulomb scattering cross section.

FIG. 2 (color online). The dependence of the LES high-energy boundary EH on the Keldysh parameter _ and the ponderomotive energy Up for various atoms and wavelengths. Left-hand column: Theoretical results; right-hand column: experimental results.