The frequency doubling of laser light was one of the first new phenomena observed following the invention of the laser over 50 years ago. Since then, the quest to extend nonlinear optical upconversion to ever-shorter wavelengths has been a grand challenge in laser science. Two decades of research into high-order harmonic generation has recently uncovered several feasible routes for generating bright coherent X-ray beams using small-scale femtosecond lasers. The physics of this technique combines the microscopic attosecond science of atoms driven by intense laser fields with the macroscopic extreme nonlinear optics of phase matching, thus essentially realizing a coherent, tabletop version of the Roentgen X-ray tube.

The attosecond nonlinear optics of bright coherent X-ray generation

A strong laser field may tunnel ionize a molecule from several orbitals simultaneously, forming an attosecond electron–hole wavepacket. Both temporal and spatial information on this wavepacket can be obtained through the coherent soft X-ray emission resulting from the laser-driven recollision of the liberated electron with the core. By characterizing the emission from aligned N 2 molecules, we demonstrate the attosecond contributions of the two highest occupied molecular orbitals. We determine conditions where they are disentangled in the real and imaginary parts of the emission dipole moment. This allows us to carry out a tomographic reconstruction of both orbitals with angstrom spatial resolution. Their coherent superposition provides experimental images of the attosecond wavepacket created in the ionization process. Our results open the prospect of imaging ultrafast intramolecular dynamics combining attosecond and angstrom resolutions.

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Attosecond imaging of molecular electronic wavepackets

The ultrafast motion of electrons and holes after light-matter interaction is fundamental to a broad range of chemical and biophysical processes. We advanced high-harmonic spectroscopy to resolve spatially and temporally the migration of an electron hole immediately after ionization of iodoacetylene while simultaneously demonstrating extensive control over the process. A multidimensional approach, based on the measurement and accurate theoretical description of both even and odd harmonic orders, enabled us to reconstruct both quantum amplitudes and phases of the electronic states with a resolution of ~100 attoseconds. We separately reconstructed quasi-field-free and laser-controlled charge migration as a function of the spatial orientation of the molecule and determined the shape of the hole created by ionization. Our technique opens the prospect of laser control over electronic primary processes.

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Measurement and laser control of attosecond charge migration in ionized iodoacetylene

Electron–molecule collision calculations using the R-matrix method

The emerging application of attosecond techniques to molecular systems allows the role of electronic coherence in the control of chemical reactions to be investigated. Prompt ionization of molecules by an attosecond pulse may induce charge migration across a molecular structure on attosecond to few-femtosecond timescales, thereby possibly determining the subsequent relaxation pathways that a molecule may take. We discuss how proposals for this 'charge-directed reactivity' fit within the current understanding of quantum control and review the current state of the art of attosecond molecular science. Specifically, we review the role of electronic coherence and coupling of the electronic and nuclear degrees of freedom in high-harmonic spectroscopy and in the first attosecond pump–probe experiments on molecular systems. Attosecond science allows the role of electronic coherence in the control of chemical reactions in molecular systems to be investigated. This article reviews recent activities in attosecond molecular science and identifies some promising directions for further development.

Attosecond molecular dynamics: fact or fiction?

The quantitative rescattering theory (QRS) for high-order harmonic generation (HHG) by intense laser pulses is presented. According to the QRS, HHG spectra can be expressed as a product of a returning electron wave packet and the photorecombination differential cross section of the laser-free continuum electron back to the initial bound state. We show that the shape of the returning electron wave packet is determined mostly by the laser. The returning electron wave packets can be obtained from the strong-field approximation or from the solution of the time-dependent Schr\"odinger equation (TDSE) for a reference atom. The validity of the QRS is carefully examined by checking against accurate results for both harmonic magnitude and phase from the solution of the TDSE for atomic targets within the single active electron approximation. Combining with accurate transition dipoles obtained from state-of-the-art molecular photoionization calculations, we further show that available experimental measurements for HHG from partially aligned molecules can be explained by the QRS. Our results show that quantitative description of the HHG from aligned molecules has become possible. Since infrared lasers of pulse durations of a few femtoseconds are easily available in the laboratory, they may be used for dynamic imaging of a transient molecule with femtosecond temporal resolutions.

https://www.phys.ksu.edu/personal/cdlin/articles/cdl/j333-atle-QRS-hhg.pdf

Quantitative rescattering theory for high-order harmonic generation from molecules

Elliptic coordinates within the hyperspherical formalism for three-body problems were proposed some time ago [V. Aquilanti, S. Cavalli, and G. Grossi, J. Chem. Phys. 85, 1362 (1986)] and recently have also found application, for example, in chemical reaction theory [see O. I. Tolstikhin and H. Nakamura, J. Chem. Phys. 108, 8899 (1998)]. Here we consider their role in providing a smooth transition between the known “symmetric” and “asymmetric” parametrizations, and focus on the corresponding hyperspherical harmonics. These harmonics, which will be called hyperspherical elliptic, involve products of two associated Lame polynomials. We will provide an expansion of these new sets in a finite series of standard hyperspherical harmonics, producing a powerful tool for future applications in the field of scattering and bound-state quantum-mechanical three-body problems.

Three-body problem in quantum mechanics: hyperspherical elliptic coordinates and harmonic basis sets.

A theoretical description of the dissociative recombination process for the ${\mathrm{HCO}}^{+}$ ion suggests that the nonadiabatic Renner-Teller coupling between electronic and vibrational degrees of freedom plays an important role. This finding is consistent with a recent study of this process for another closed-shell molecule, the ${{\mathrm{H}}_{3}}^{+}$ ion, where Jahn-Teller coupling was shown to generate a relatively high rate. The cross section obtained here for the dissociative recombination of ${\mathrm{HCO}}^{+}$ exhibits encouraging agreement with a merged-beam experiment.

Renner-Teller effects in HCO+ dissociative recombination

A theoretical description of the dissociative recombination process for the HCO+ ion suggests that the nonadiabatic Renner-Teller coupling between electronic and vibrational degrees of freedom plays an important role. This finding is consistent with a recent study of this process for another closed-shell molecule, the H-3(+) ion, where Jahn-Teller coupling was shown to generate a relatively high rate. The cross section obtained here for the dissociative recombination of HCO+ exhibits encouraging agreement with a merged-beam experiment.

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Renner-Teller effects in HCO$^{+}$ dissociative recombination

We report on ab initio calculations relevant for dissociative recombination of HCO+. From accurate quantum chemistry calculations, it is found that the electron collision is driven by capture into Rydberg states. The Renner-Teller effect is not important for higher Rydberg states. From calculated potentials, the effective quantum numbers are fitted in three dimensions. The model of the electron collision is simplified using an adiabatic approximation, where one of the Jacobi coordinates is treated adiabatically. Then the electron collision is described using two different theoretical methods. Preliminary results on autoionization widths for Rydberg states and dissociative recombination cross section are given.

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Stefano Tonzani

Papers

Quantitative rescattering theory for high-order harmonic generation from molecules

Three-body problem in quantum mechanics: hyperspherical elliptic coordinates and harmonic basis sets.

Renner-Teller effects in HCO+ dissociative recombination

Renner-Teller effects in HCO$^{+}$ dissociative recombination

Dissociative recombination of HCO