Exterior time scaling with stiffness-free Lanczos time propagator: formulation and application to atoms interacting with strong mid-infrared lasers

Aiming at efficient numerical treatment of tunneling ionization of atoms and molecules by mid-infrared (IR) lasers, exterior time scaling (ETS) theory is formulated as a generalization of the time-scaled coordinate approach. The key idea of ETS is the division of the spatial volume into a small region around the nucleus and its outside; the radial coordinates are time scaled only in the outer region. The continuum components of photoelectron wave packets are prevented from reaching the edge of the spatial simulation volume, enabling the long-time evolution of wave packets with a relatively small number of basis functions without concerns of electron reflections. On the other hand, the bound-state components are free from shrinking toward the origin because of non-time scaling in the inner region. Hence, the equations of motion in ETS are less stiff than the ones in the original time-scaled coordinate approach in which the shrinking bound states make the equations of motion seriously stiff. For numerical implementation of ETS, the working equations are derived in terms of finite-element discrete-variable-representation functions. Furthermore, the stiffness-free Lanczos time propagator is introduced to remove any persistent stiffness in the treatment of mid-IR lasers due to the involvement of hundreds of angular-momentum states. The test calculations for atomic hydrogen interacting with linearly polarized mid-IR pulses demonstrate the accuracy and numerical efficiency of the new scheme, and exhibit its special capability if there is no recollision with the parent ion. Hence, ETS will show its true potential for the detailed analysis of photoelectron wave packet dynamics in circularly or near-circularly polarized mid-IR fields.