Time-dependent ab initio approaches for high-harmonic generation spectroscopy

High-harmonic generation (HHG) is a nonlinear physical process used for the production of ultrashort pulses in XUV region, which are then used for investigating ultrafast phenomena in time-resolved spectroscopies. Moreover, HHG signal itself encodes information on electronic structure and dynamics of the target, possibly coupled to nuclear degrees of freedom. Investigating HHG signal leads to HHG spectroscopy, which is applied to atoms, molecules, solids and recently also to liquids. Analysing the number of generated harmonics, their intensity and shape gives a detailed insight of, e.g., ionisation and recombination channels occurring in the strong-field dynamics. A number of valuable theoretical models has been developed over the years to explain and interpret HHG features, with the three-step model being the most known one. Originally, these models neglect the complexity of the propagating electronic , by only using an approximated formulation of ground and continuum states. Many effects unravelled by HHG spectroscopy are instead due to electron correlation effects, quantum interference, and Rydberg-state contributions, which are all properly captured by an ab initio electronic-structure approach. In this Review we have collected recent advances in modelling HHG by means of ab initio time-dependent approaches relying on the propagation of the time-dependent Schr\"odinger equation (or derived equations) in presence of a very intense electromagnetic field. We limit ourselves to gas-phase atomic and molecular targets, and to solids. We focus on the various levels of theory for describing the electronic structure of the target, coupled with strong-field dynamics and ionisation approaches, and on the basis used to represent electronic states. Selected applications and perspectives for future developments are also given.