Quantum simulation of high-order harmonic spectra of the hydrogen atom

Three alternative forms of harmonic spectra, based on the dipole moment, dipole velocity, and dipole acceleration, are compared by a numerical solution of the Schrödinger equation for a hydrogen atom interacting with a linearly polarized laser pulse, whose electric field is given by E(t)=E₀f(t)cos(ω₀t+η) with Gaussian carrier envelope f(t)=exp(-t²/δ²) . The carrier frequency ω₀ is fixed to correspond to a wavelength of 800nm . Spectra for a selection of pulses, for which the intensity I₀=cɛ₀E₀² , duration T∝δ , and carrier-envelope phase η are systematically varied, show that, depending on η , all three forms are in good agreement for “weak” pulses with I₀<I_b , the over-barrier ionization threshold, but that marked differences among the three appear as the pulse becomes shorter and stronger (I₀>I_b) . Except for scalings by powers of the harmonic frequency, the three forms differ from one another only by “limit contributions” proportional to the expectation values of the dipole moment ⟨z(t_f)⟩ or dipole velocity ⟨ż(t_f)⟩ at the end (t_f) of the pulse. For long, weak pulses the limit contributions are negligible, whereas for short, strong ones they are not. In the short, strong limit, where ⟨ż(t_f)⟩≠0 and therefore ⟨z(t)⟩ may increase without bound (i.e., the atom may ionize), depending on η , an “infinite-time” spectrum based on the acceleration form provides a convenient computational pathway to the corresponding infinite-time dipole-velocity spectrum, which is related directly to the experimentally measured “harmonic photon number spectrum” (HPNS). For short, intense pulses the HPNS is quite sensitive to η and exhibits not only the usual odd harmonics but also even ones. The analysis also reveals that most of the harmonic photons are emitted during the passage of the pulse. Because of the divergence of ⟨z(t)⟩ the dipole-moment form does not provide a numerically reliable route to the harmonic spectrum for very short (few-cycle), very intense laser pulses.