Quantum path analysis of high-order above-threshold ionization

High-order above-threshold ionization spectra are calculated via an improved Keldysh approximation that takes rescattering into account. An approximate method of evaluating the crucial multidimensional integral proceeds via the saddle point method. The saddle points define complex orbits in position space that depart from the ion and return to it to rescatter. The real parts of these orbits are very closely related to the trajectories of the simple-man model. The spectra are analyzed in terms of these quantum orbits whose constructive and destructive interferences generate the spectrum's intricate structures. In most spectral regions, the six trajectories having the shortest travel times between start and return already provide an excellent approximation to the exact calculation. In exceptional cases, more orbits are required. The quantum orbits provide an illuminating illustration of the quantum mechanical path integral.