High-resolution Interferometric Diagnostics for Ultrashort Pulses

6. HIGH-HARMONIC GENERATIONstationary-phase approximation (SPA), and shall be described in a later section.6.2.2 Time-dependent Schrödinger equationThe TDSE in the length gauge and with the dipole approximation applied isi ∂ |ψ〉∂ t ˆp2=2 + V (x)+ (t ) · x |ψ〉 (6.3)where ˆp = −i ∇ x is the momentum operator, V (x) is the potential, (t ) the external field and |ψ〉is the electron wavefunction. Several simplifications lie behind (6.3). A single active electronis considered [369]. This gives accurate results for atoms but generally needs modification formolecules. The potential here is static, a reasonable approximation for atoms. In molecules thetime-evolution of the potential, caused, for example, by nuclear motion, can play a significant role.The electric field is homogeneous; this follows from the dipole approximation, valid as long as theelectron excursion is signifcantly smaller than the wavelength. As mentioned in section 6.2.1, themagnetic field is ignored since the electron motion is nonrelativistic.The result of a TDSE simulation of HHG is shown in Fig. 6.4. The parameters of this simulationare used in several examples throughout this chapter. At peaks of the electric field ionization isclearly visible: a significant component of the wavefunction detaches from the bound state whichis correspondingly depleted. Recombination events are rather subtle, manifesting in a slight ‘ripple’of the ground state where the returning electron trajectories overlap. This ripple leads to ahigh frequency oscillation of the dipole moment, which is the source of the harmonics. This illustratesthe weakness of the high harmonic generation process, with typical conversion efficienciesbeing 10 −6 .The radiation emitted by a single atom is proportional to the acceleration of the dipole moment(t )=〈ψ(t )|x|ψ(t )〉 of the electron. Although (t ) can be obtained directly from the calculationresults for |ψ(t )〉 and twice differentiated, it is instructive to use Ehrenfest’s theorem torelate the dipole acceleration to the expectation value of the force∂ 2 (t )∂ t 2 = 〈ψ(t )| −∇ x V (x)|ψ(t )〉 + (t ). (6.4)134