High-resolution Interferometric Diagnostics for Ultrashort Pulses

6.2 Theory of the single-atom responsewhereS ω (t r ,t b ,p)=− trt b p 2 + A(t )dt+ I p + ωt r (6.14)2is simply the action modified to include the phase ωt r of the emitted photon. Equation (6.13)expresses the radation as an integral over all birth times, momenta and return times.6.2.4 Quantum trajectoriesOne aspect of the semi-classical description which is not recovered by (6.13) is exactly which birthtimes, return times, and momenta contribute significantly to the harmonic emission. However, in(6.13), the action has a much greater dependence on the integration variables t b , t r , and p thanthe transition amplitudes. Since the cycle-averaged integral of a rapidly varying phase is approximatelyzero, significant contributions to the integral only occur for those values of t b , t r , and pfor which the action is stationary. The stationary-point approximation [372] provides a way of formalisingthis intuitive notion. Specifically, the dipole response integral (6.13) is approximated bya finite sum over each of the stationary points. Besides reducing the computational complexity,the stationary-point approximation provides physical insight and makes explicit the connectionbetween the quantum and classical pictures of high-harmonic generation.Requiring the birth time, return time, and momentum to be stationary leads to three conditions,each with a physical interpretation that justifies the assumptions made in the classicalthree-step picture.• Saddle points of the momentum integral occur when∇ p S ω (t r ,t b ,p)=x(t r ) − x(t b )=0 (6.15)which shows that strong contributions occur only when the electron returns to the positionat which it was born. One thus recovers one of the assumptions of the classical description.This equation can be solved directly to givep =−1 trt r − t bt bdt A(t ). (6.16)139