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

Quantum dynamics of H 2 + excited by two-cycle laser pulses with laser carrier frequencies corresponding to the wavelengths λ l = 800 and 200 nm (corresponding to the periods τ l = 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 H 2 + after the ends of the laser pulses, ⟨ E ⟩ ≈ −0.515 au, were close to the dissociation threshold of H 2 + . It is found that there exists a certain characteristic oscillation frequency ω osc = 0.2278 au (corresponding to the period τ osc = 0.667 fs and the wavelength λ osc = 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 λ l = 800 nm and at λ l = 200 nm. Moreover, at λ l = 200 nm, the expectation value ⟨ z ⟩ also demonstrates temporally shaped oscillations after the end of the laser pulse. In contrast, at λ l = 800 nm, the characteristic oscillation frequency ω osc = 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 τ shp ≈ 30 fs, correlates with the nuclear motion. It is also shown that vibrational excitation of H 2 + 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 H 2 + 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.

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