Progress in study of low-energy photoelectron in ultra-fast strong fields-analytical R-matrix theory based semiclassical trajectory method

The semi-classical method based on the recently developed analytical R-matrix theory is reviewed in this work. The method is described with the application to ultra-fast strong-field direct ionization of atoms with one active electron in a linearly polarized field[Torlina L, Smirnova O 2012 Phys. Rev. A 86 043408]. The analytical R-matrix theory separates the space into inner and outer regions, naturally allowing the possibility of an analytical or semi-analytical description of wave function in the outer region, which can be approximated by Eikonal-Volkov solutions while the inner region provides well-defined boundary conditions. Applying the stationary phase method, the calculation of the ionization amplitude is cast into a superposition of components from trajectories and their associated phase factors. The shape of the tunneling wave packets associated with different instants of ionization is presented. It shows the exponential cost of deviating from the optimal tunneling trajectory renders the tunneling wave packet a Gaussian shape surrounding the semi-classical trajectory. The intrinsically non-adiabatic corrections to the sub-cycle ionization amplitude in the presence of both the Coulomb potential and the laser field is shown to have different influences on the probability of ionization. As a specific study case, soft recollisions of the released electron near the ionic core is investigated by using pure light-driven trajectories with Coulomb-corrected phase factor[Pisanty E, Ivanov M 2016 Phys. Rev. A 93 043408]. Incorporating the Coulomb potential, it is found problematic to use the conventional integration contour as chosen in other methods with trajectory-based Coulomb corrections, because the integration contour may run into the Coulomb-induced branch cuts and hence the analyticity of the integrand fails. In order to overcome the problem, the evolution time of the post-tunneling electron is extended into the complex domain which allows a trajectory to have an imaginary component. As the soft recollision occurs, the calculation of the ionization amplitude requires navigating the branch cuts cautiously. The navigating scheme is found based on closest-approach times which are the roots of closest-approach times equations. The appropriately selected closest-approach times that always present in the middle of branch-cut gate may serve to circumvent these branch cuts. The distribution of the closest-approach times presents rich geometrical structures in both the classical and quantum domains, and intriguing features of complex trajectories emerge as the electron returns near the core. Soft recollisions responsible for the low-energy structures are embedded in the geometry, and the underlying emergence of near-zero energy structures is discussed with the prediction of possible observations in experiments.