Shaped Post-Field Electronic Oscillations in H₂⁺ Excited by Two-Cycle Laser Pulses: Three-Dimensional Non-Born–Oppenheimer Simulations

Abstract

Quantum dynamics of H2 + excited by two-cycle laser pulses with laser carrier frequencies corresponding to the wavelengths λl = 800 and 200 nm (corresponding to the periods τl = 2.667 and 0.667 fs, respectively) and being linearly polarized along the molecular axis have been studied by the numerical solution of the non-Born–Oppenheimer time-dependent Schrödinger equation within a three-dimensional (3D) model, including the internuclear distance R and electron coordinates z and ρ. The amplitudes of the pulses have been chosen such that the energies of H2 + after the ends of the laser pulses, ⟨E⟩ ≈ −0.515 au, were close to the dissociation threshold of H2 +. It is found that there exists a certain characteristic oscillation frequency ωosc = 0.2278 au (corresponding to the period τosc = 0.667 fs and the wavelength λosc = 200 nm) that plays the role of a “carrier” frequency of temporally shaped oscillations of the expectation values ⟨−∂V/∂z⟩ emerging after the ends of the laser pulses, both at λl = 800 nm and at λl = 200 nm. Moreover, at λl = 200 nm, the expectation value ⟨z⟩ also demonstrates temporally shaped oscillations after the end of the laser pulse. In contrast, at λl = 800 nm, the characteristic oscillation frequency ωosc = 0.2278 au appears as the frequency of small-amplitude oscillations of the slowly varying expectation value ⟨z⟩ which makes, after the end of the pulse, an excursion with an amplitude of about 4.5 au along the z axis and returns back to ⟨z⟩ ≈ 0 afterward. It is found that the period of the temporally shaped post-field oscillations of ⟨−∂V/∂z⟩ and ⟨z⟩, estimated as τshp ≈ 30 fs, correlates with the nuclear motion. It is also shown that vibrational excitation of H2 + is accompanied by the formation of “hot” and “cold” vibrational ensembles along the R degree of freedom. Power spectra related to the electron motion in H2 + calculated for both the laser-driven z and optically passive ρ degrees of freedom in the acceleration form proved to be very interesting. In particular, both odd and even harmonics can be observed.

Excitation of H+ 2 with one-cycle laser pulses: shaped post-laser-field electronic oscillations, generation of higher- and lower-order harmonics

“…We can conclude therefore, that periodic elongation-contraction of the bond of H + 2 ( Figure 6) is controlled by compressing-expanding electron acceleration along the z axis ( Figure 7) which takes place with the period of τ shp ≈ 30 fs corresponding to the frequency ω shp = 2π/τ shp of shaped post-laser-field oscillations occurring with the carrier oscillation frequency ω osc ≃ 0.2278 au (corresponding to the wavelength of λ osc ≃ 200 nm). Similar electron-nuclei correlations were found in our recent work [3] where two-cycle laser pulses were used to excite H + 2 and a more detailed explanation is given for post-laser-field electron-nuclei correlations in terms of below-resonance vibrational frequency [9,15] and for the existence of characteristic oscillation frequency ω osc (corresponding to λ osc ≃ 200 nm) in terms of a continuum state Ψ C prepared by the laser pulses. Afterwards, at t > 7 − 9 fs, expectation values ρ demonstrate a rather smooth behavior, with the electron excursion along the transversal ρ coordinate being quite strongly dependent of the laser pulse amplitude used to excite H + 2 initially along the z axis.…”

“…For the sake of completeness, some results were obtained by making use of two-cycle laser pulses as well. The present work provides a detailed and in-depth extension of our previous study [3] where shaped two-cycle laser pulses at the laser wavelengths of λ l =800 and 200 nm were used to excite H + 2 close to its dissociation threshold. The basic results obtained in the present work are summarized as follows.…”

“…It can be expected therefore that power spectra of electron oscillations resulting from excitation of H + 2 to E ≈ −0.515 au at various laser wavelengths λ l = 200 nm should always contain harmonics corresponding to λ ≈ 200 nm. Indeed, it was shown in our recent work [3], where excitation of H + 2 with two-cycle laser pulses at λ l = 200 and 800 nm was studied, that the respective power spectra contain the strongest harmonics corresponding to λ = 200 nm, the fourth harmonic at λ l = 800 nm and the first (identical) harmonic at λ l = 200 nm. It will be interesting to check therefore whether the excitation of H + 2 with laser wavelengths λ l < 200 nm would result in the generation of lower-order harmonics.…”

“…The laser-driven dynamics of H + 2 excited by two-cycle laser pulses was studied in our recent work [3]. It was found, in particular, that at λ l = 800 nm, the expectation value z follows the field out-of-phase only approximately (during the first optical cycle), while at λ l = 200 nm, the out-of-phase electron-field following does not take place even approximately.…”

“…It is seen from Figure 13(a) that the strongest harmonic in the A z spectrum generated by the optically active z degree of freedom is the first, or identical harmonic corresponding to the wavelength of λ ≃ 200 nm, while a much weaker, the second-order harmonic corresponds to λ ≃ 97 nm. The appearance of the second-order harmonic in the power spectrum A z of the symmetric H + 2 molecule is the specific feature of onecycle and two-cycle [3] laser pulses, because if laser pulses with many optical cycles are used to excite a symmetric molecule, even harmonics should not appear in the power spectra generated by optically active degrees of freedom at all, as suggested by the concept of inversion symmetry [5]. In the power spectrum A ρ generated by the optically passive ρ degree of freedom, Figure 13(b), the strongest second-order harmonic at λ ≃ 112 nm corresponds to the doubled frequency of ρ-oscillations ω ρ 2 ≈ 2ω osc , while a weaker, the first-order harmonic at λ ≃ 194 nm, corresponds to ω ρ 1 ≈ ω osc .…”