Excitation of H${}_2^{+}$ with one-cycle laser pulses: Shaped post-laser-field electronic oscillations, generation of higher- and lower-order harmonics

Abstract

Non Born-Oppenheimer quantum dynamics of H${}_2^{+}$ excited by shaped one-cycle laser pulses linearly polarized along the molecular axis have been studied by the numerical solution of the time-dependent Schrödinger equation within a %three-body three-dimensional model, including the internuclear separation, $R$, and the electron coordinates $z$ and $\rho$. Laser carrier frequencies corresponding to the wavelengths ${\lambda }_l=25$~nm through ${\lambda }_l=400$~nm were used and the amplitudes of the pulses were chosen such that the energy of H${}_2^{+}$ was close to its dissociation threshold at the end of any laser pulse applied. It is shown that there exists a characteristic oscillation frequency ${\omega }_{\mathrm{osc}}\simeq 0.2265$~au (corresponding to the period of ${\tau }_{\mathrm{osc}}\simeq 0.671$~fs and the wavelength of ${\lambda }_{\mathrm{osc}}\simeq 200$~nm) that manifests itself as a "carrier" frequency of temporally shaped oscillations of the time-dependent expectation values $⟨z⟩$ and $⟨\partial V/\partial z⟩$ that emerge at the ends of the laser pulses and exist on a timescale of at least 50~fs. Time-dependent expectation values $⟨\rho ⟩$ and $⟨\partial V/\partial \rho ⟩$ of the optically-passive degree of freedom, $\rho$, demonstrate post-laser-field oscillations at two basic frequencies ${\omega }_1^{\rho }\approx {\omega }_{\mathrm{osc}}$ and ${\omega }_2^{\rho }\approx 2{\omega }_{\mathrm{osc}}$. Power spectra associated with the electronic motion show higher- and lower-order harmonics with respect to the driving field.