High-order above-threshold ionization spectra are calculated via an improve
d Keldysh approximation that takes rescattering into account. An approximat
e method of evaluating the crucial multidimensional integral proceeds via t
he 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 par
ts of these orbits are very closely related to the trajectories of the simp
le-man model. The spectra are analyzed in terms of these quantum orbits who
se constructive and destructive interferences generate the spectrum's intri
cate 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 th
e quantum mechanical path integral. (C) 2000 Elsevier Science B.V. All righ
ts reserved.