Single-electron wavefunctions, or orbitals, are the mathematical constructs used to describe the multi-electron wavefunction of molecules. Because the highest-lying orbitals are responsible for chemical properties, they are of particular interest. To observe these orbitals change as bonds are formed and broken is to observe the essence of chemistry. Yet single orbitals are difficult to observe experimentally, and until now, this has been impossible on the timescale of chemical reactions. Here we demonstrate that the full three-dimensional structure of a single orbital can be imaged by a seemingly unlikely technique, using high harmonics generated from intense femtosecond laser pulses focused on aligned molecules. Applying this approach to a series of molecular alignments, we accomplish a tomographic reconstruction of the highest occupied molecular orbital of N2. The method also allows us to follow the attosecond dynamics of an electron wave packet.

Tomographic imaging of molecular orbitals

Summary form only given. Fundamental processes in atoms, molecules, as well as condensed matter are triggered or mediated by the motion of electrons inside or between atoms. Electronic dynamics on atomic length scales tends to unfold within tens to thousands of attoseconds (1 attosecond [as] = 10-18 s). Recent breakthroughs in laser science are now opening the door to watching and controlling these hitherto inaccessible microscopic dynamics. The key to accessing the attosecond time domain is the control of the electric field of (visible) light, which varies its strength and direction within less than a femtosecond (1 femtosecond = 1000 attoseconds). Atoms exposed to a few oscillations cycles of intense laser light are able to emit a single extreme ultraviolet (XUV) burst lasting less than one femtosecond. Full control of the evolution of the electromagnetic field in laser pulses comprising a few wave cycles have recently allowed the reproducible generation and measurement of isolated sub-femtosecond XUV pulses, demonstrating the control of microscopic processes (electron motion and photon emission) on an attosecond time scale. These tools have enabled us to visualize the oscillating electric field of visible light with an attosecond "oscilloscope", to control single-electron and probe multi-electron dynamics in atoms, molecules and solids. Recent experiments hold promise for the development of an attosecond X-ray source, which may pave the way towards 4D electron imaging with sub-atomic resolution in space and time.

Attosecond Physics

Modern laser sources nowadays deliver ultrashort light pulses reaching few cycles in duration and peak powers exceeding several terawatt (TW). When such pulses propagate through optically transparent media, they first self-focus in space and grow in intensity, until they generate a tenuous plasma by photo-ionization. For free electron densities and beam intensities below their breakdown limits, these pulses evolve as self-guided objects, resulting from successive equilibria between the Kerr focusing process, the chromatic dispersion of the medium and the defocusing action of the electron plasma. Discovered one decade ago, this self-channeling mechanism reveals a new physics, widely extending the frontiers of nonlinear optics. Implications include long-distance propagation of TW beams in the atmosphere, supercontinuum emission, pulse shortening as well as high-order harmonic generation. This review presents the landmarks of the 10-odd-year progress in this field. Particular emphasis is laid on the theoretical modeling of the propagation equations, whose physical ingredients are discussed from numerical simulations. The dynamics of single filaments created over laboratory scales in various materials such as noble gases, liquids and dielectrics reveal new perspectives in pulse shortening techniques. Far-field spectra provide promising diagnostics. Attention is also paid to the multifilamentation instability of broad beams, breaking up the energy distribution into small-scale cells along the optical path. The robustness of the resulting filaments in adverse weathers, their large conical emission exploited for multipollutant remote sensing, nonlinear spectroscopy and the possibility of guiding electric discharges in air are finally addressed on the basis of experimental results.

https://iopscience.iop.org/article/10.1088/0034-4885/70/10/R03/pdf

Ultrashort filaments of light in weakly ionized, optically transparent media

High harmonic emission occurs when an electron, liberated from a molecule by an incident intense laser field, gains energy from the field and recombines with the parent molecular ion. The emission provides a snapshot of the structure and dynamics of the recombining system, encoded in the amplitudes, phases and polarization of the harmonic light. Here we show with CO2 molecules that high harmonic interferometry can retrieve this structural and dynamic information: by measuring the phases and amplitudes of the harmonic emission, we reveal ‘fingerprints’ of multiple molecular orbitals participating in the process and decode the underlying attosecond multi-electron dynamics, including the dynamics of electron rearrangement upon ionization. These findings establish high harmonic interferometry as an effective approach to resolving multi-electron dynamics with sub-Angstrom spatial resolution arising from the de Broglie wavelength of the recombining electron, and attosecond temporal resolution arising from the timescale of the recombination event. The high harmonic emission that accompanies the recombination of an electron with its parent molecular ion in an intense laser field provides a snapshot of the structure and dynamics of the recombining system. Experiments with CO2 molecules now show that high harmonic interferometry can retrieve this structural and dynamic information by measuring the phases and amplitudes of the harmonic emission. The resulting 'fingerprints' of the multiple molecular orbitals participating in the process can be used to decode the underlying attosecond multi-electron dynamics, including the dynamics of electron rearrangement upon ionization. The light emitted from the system contains images of moving electrons that can be processed into a movie. These findings establish high harmonic interferometry as an effective approach to resolving multi-electron dynamics with sub-Angstrom spatial resolution arising from the de-Broglie wavelength of the recombining electron, and attosecond temporal resolution arising from the timescale of the recombination event. The high harmonic emission that accompanies the recombination of an electron with its parent molecular ion in an intense laser field provides a snapshot of the structure and dynamics of the recombining system. Experiments on CO2 molecules now show how to extract information from the properties of the emitted light about the underlying multi-electron dynamics with sub-Angstrom spatial resolution and attosecond temporal resolution

High harmonic interferometry of multi-electron dynamics in molecules

Science and technologies based on terahertz frequency electromagnetic radiation (100 GHz–30 THz) have developed rapidly over the last 30 years. For most of the 20th Century, terahertz radiation, then referred to as sub-millimeter wave or far-infrared radiation, was mainly utilized by astronomers and some spectroscopists. Following the development of laser based terahertz time-domain spectroscopy in the 1980s and 1990s the field of THz science and technology expanded rapidly, to the extent that it now touches many areas from fundamental science to 'real world' applications. For example THz radiation is being used to optimize materials for new solar cells, and may also be a key technology for the next generation of airport security scanners. While the field was emerging it was possible to keep track of all new developments, however now the field has grown so much that it is increasingly difficult to follow the diverse range of new discoveries and applications that are appearing. At this point in time, when the field of THz science and technology is moving from an emerging to a more established and interdisciplinary field, it is apt to present a roadmap to help identify the breadth and future directions of the field. The aim of this roadmap is to present a snapshot of the present state of THz science and technology in 2017, and provide an opinion on the challenges and opportunities that the future holds. To be able to achieve this aim, we have invited a group of international experts to write 18 sections that cover most of the key areas of THz science and technology. We hope that The 2017 Roadmap on THz science and technology will prove to be a useful resource by providing a wide ranging introduction to the capabilities of THz radiation for those outside or just entering the field as well as providing perspective and breadth for those who are well established. We also feel that this review should serve as a useful guide for government and funding agencies.

The 2016 terahertz science and technology roadmap

We have extended the tunneling ionization model of Ammosov-Delone-Krainov (ADK) for atoms to diatomic molecules by considering the symmetry property and the asymptotic behavior of the molecular electronic wave function. The structure parameters of several molecules needed for calculating the ionization rates using this molecular ADK model have been obtained. The theory is applied to calculate the ratios of ionization signals for diatomic molecules with their companion atoms that have nearly identical binding energies. The origin of ionization suppression for some molecules has been identified. The predicted ratios for pairs with suppression $({\mathrm{D}}_{2}:\mathrm{Ar},$ ${\mathrm{O}}_{2}:\mathrm{Xe})$ and pairs without suppression $({\mathrm{N}}_{2}:\mathrm{Ar},$ CO:Kr) are in good agreement with the measurements. However, the theory predicts suppression for ${\mathrm{F}}_{2}:\mathrm{Ar},$ which is in disagreement with the experiment. The ionization signals of NO, ${\mathrm{S}}_{2},$ and of SO have also been derived from the experimental data, and the results are also shown to be in agreement with the prediction of the present molecular ADK theory.

/pdf/theory-of-molecular-tunneling-ionization-1yoxqhwye5.pdf

Theory of molecular tunneling ionization

It was recently shown that the highest molecular orbital of ${\mathrm{N}}_{2}$ could be reconstructed from a series of high harmonic measurements. Existing theories of high harmonic generation and orbital tomographic imaging are based on the single active electron approximation that ignores essential quantum mechanical properties such as the indistinguishability of identical particles and the Pauli exclusion principle. We show that, when fully antisymmetrized multielectron wave functions and electronic relaxation in the cation are considered, molecular orbital tomography records the image of the Dyson orbital plus exchange contributions from inner shells. The mixing of contributions from more than one molecular orbital gives access to additional wave function information. By utilizing the exchange term, harmonic emission from a closed-shell 4-electron system can be interpreted as a complete Hartree-Fock wave function.

/pdf/high-harmonic-generation-and-molecular-orbital-tomography-in-3uc505csqf.pdf

High harmonic generation and molecular orbital tomography in multielectron systems : Beyond the single active electron approximation

The orientation-dependent strong-field ionization of CO molecules is investigated using the fully propagated three-dimensional time-dependent Hartree-Fock theory. The full ionization results are in good agreement with recent experiments. The comparisons between the full method and the single active orbital method show that although the core electrons are generally more tightly bound and contribute little to the total ionization yields, their dynamics cannot be ignored, which effectively modifies the behavior of electrons in the highest occupied molecular orbital. By incorporating it into the single active orbital method, we identify that the dynamic core polarization plays an important role in the strong-field tunneling ionization of CO molecules, which is helpful for the future development of the tunneling ionization theory of molecules beyond the single active electron approximation.

Dynamic core polarization in strong-field ionization of CO molecules.

It is shown that the alignment dependence of the yields of high-order harmonic generation (HHG) from molecules by intense lasers is governed by the orbital symmetry of the outermost electron(s) For ${\mathrm{N}}_{2}$, with its outermost ${\ensuremath{\sigma}}_{g}$ electron, the HHG yield is maximal when the molecules are aligned with the laser polarization direction, in agreement with the recent experiment of Itatani et al [Nature 432, 867 (2004)] For ${\mathrm{O}}_{2}$, with its outermost ${\ensuremath{\pi}}_{g}$ electron, the HHG yield peaks when the molecules are aligned at about 45\ifmmode^\circ\else\textdegree\fi{} from the polarization axis We emphasize that the alignment dependence is determined mostly by the orbital symmetry and weakly on the laser parameters or the species

https://www.phys.ksu.edu/personal/cdlin/articles/cdl/j283.pdf

Role of molecular orbital symmetry on the alignment dependence of high-order harmonic generation with molecules

The strong field approximation is used to investigate the harmonic generation of aligned ${\mathrm{N}}_{2}$ and ${\mathrm{O}}_{2}$ molecules. It is shown that the angular dependence of the yields of high-order harmonic generation from ${\mathrm{N}}_{2}$ and ${\mathrm{O}}_{2}$ molecules are different, which can be attributed to the orbital symmetry of the outmost electrons. The theory is applied to study the angular dependence of harmonic generation from recent pump-probe experiments where the pump pulse was used to produce the time-dependent partial alignment of molecules and the probe pulse was used to generate high-order harmonics. Good agreement with the experiments was found.

Zengxiu Zhao

Papers

Theory of molecular tunneling ionization

High harmonic generation and molecular orbital tomography in multielectron systems : Beyond the single active electron approximation

Dynamic core polarization in strong-field ionization of CO molecules.

Role of molecular orbital symmetry on the alignment dependence of high-order harmonic generation with molecules

Alignment dependence of high-order harmonic generation from N 2 and O 2 molecules in intense laser fields