Dynamics of valence-shell electrons and nuclei probed by strong-field holography and rescattering

Probing and Steering Attosecond Electron Motion Using Tailored Ultrafast Laser Fields

An ultrafast intense laser field is one of the most important tools to observe and manipulate electronic and nuclear dynamics with subcycle precision in highly nonlinear light-matter interactions, which provides access to attosecond chemistry and physics. In this review, we briefly summarize the protocol of attosecond chronoscopy and its application in probing the attosecond photoemission dynamics from atoms and molecules. We also review the control schemes of attosecond electron motion in atoms and molecules as well as molecular bond formation and cleavage with the assistance of tailored femtosecond laser fields.

Unraveling the ultrafast dynamics of thermal-energy chemical reactions

In this perspective, we discuss how one can initiate, image, and disentangle the ultrafast elementary steps of thermal-energy chemical dynamics, building upon advances in technology and scientific insight. We propose that combinations of ultrashort mid-infrared laser pulses, controlled molecular species in the gas phase, and forefront imaging techniques allow to unravel the elementary steps of general-chemistry reaction processes in real time. We detail, for prototypical first reaction systems, experimental methods enabling these investigations, how to sufficiently prepare and promote gas-phase samples to thermal-energy reactive states with contemporary ultrashort mid-infrared laser systems, and how to image the initiated ultrafast chemical dynamics. The results of such experiments will clearly further our understanding of general-chemistry reaction dynamics.

Probing the electron motion in molecules using forward-scattering photoelectron holography

Charge migration initiated by the coherent superposition of several electronic states is a basic process in intense laser-matter interactions. Observing this process on its intrinsic timescale is one of the central goals of attosecond science. Here, using forward-scattering photoelectron holography we theoretically demonstrate a scheme to probe the charge migration in molecules. In our scheme, by solving the time-dependent Schrödinger equation, the photoelectron momentum distributions (PEMDs) for strong-field tunneling ionization of the molecule are obtained. For a superposition state, it is shown that an intriguing shift of the holographic interference appears in the PEMDs, when the molecule is aligned perpendicularly to the linearly polarized laser field. With the quantum-orbit analysis, we demonstrate that this shift of the interference fringes is caused by the time evolution of the non-stationary superposition state. By analyzing the dependence of the shift on the final parallel momentum of the electrons, the relative phase and the expansion coefficient ratio of the two electronic states involved in the superposition state are determined accurately. Our study provides an efficient method for probing the charge migration in molecules. It will facilitate the application of the forward-scattering photoelectron holography to survey the electronic dynamics in more complex molecules.

Decoherence and Revival in Attosecond Charge Migration Driven by Non-adiabatic Dynamics

Attosecond charge migration is a periodic evolution of the charge density at specific sites of a molecule on a time scale defined by the energy intervals between the electronic states involved. Here, we report the observation of charge migration in neutral silane (SiH₄) in 690 as, its decoherence within 15 fs, and its revival after 40-50 fs, using X-ray attosecond transient absorption spectroscopy. We observe the migration of charge as pairs of quantum beats with a characteristic spectral phase in the transient spectrum, in agreement with theory. The decay and revival of the degree of electronic coherence is found to be a result of both adiabatic and non-adiabatic dynamics in the populated Rydberg and valence states. The experimental results are supported by fully quantum-mechanical ab-initio calculations that include both electronic and nuclear dynamics, which additionally support the experimental evidence that conical intersections can mediate the transfer of electronic coherence from an initial superposition state to another one involving a different lower-lying state.

Different timescales during ultrafast stilbene isomerization in the gas and liquid phases revealed using time-resolved photoelectron spectroscopy

Directly contrasting ultrafast excited-state dynamics in the gas and liquid phases is crucial to understanding the influence of complex environments. Previous studies have often relied on different spectroscopic observables, rendering direct comparisons challenging. Here, we apply extreme-ultraviolet time-resolved photoelectron spectroscopy to both gaseous and liquid cis-stilbene, revealing the coupled electronic and nuclear dynamics that underlie its isomerization. Our measurements track the excited-state wave packets from excitation along the complete reaction path to the final products. We observe coherent excited-state vibrational dynamics in both phases of matter that persist to the final products, enabling the characterization of the branching space of the S₁-S₀ conical intersection. We observe a systematic lengthening of the relaxation timescales in the liquid phase and a red shift of the measured excited-state frequencies that is most pronounced for the complex reaction coordinate. These results characterize in detail the influence of the liquid environment on both electronic and structural dynamics during a complete photochemical transformation.

Generation and complete polarimetry of ultrashort circularly polarized extreme-ultraviolet pulses

The generation of ultrashort circularly polarized pulses in the extreme-ultraviolet spectral range has recently attracted considerable interest for applications in time-resolved circular-dichroism experiments. Here, we demonstrate a simple approach to generate near-circularly polarized femtosecond pulses in the vacuum-ultraviolet. The ellipticity of the generated light can be continuously tuned from linear to near-circular, as demonstrated by detailed polarimetry measurements. Combining optical polarimetry with photoelectron circular-dichroism (PECD) measurements, we demonstrate a novel approach to characterizing the polarization state of light in terms of all four Stokes parameters. For photon energies of 9.3 eV, we obtained S₃ = 0.96 ± 0.02 and a degree of polarization of 97±2%, i.e. the highest values reported from any harmonic-generation source so far. This source is directly applicable to circular-dichroism experiments, also enabling time-resolved PECD in the extreme-ultraviolet, a general approach to probing time-dependent chirality during chemical processes on (sub)-femtosecond time scales.

Picosecond pulse-shaping for strong three-dimensional field-free alignment of generic asymmetric-top molecules

Fixing molecules in space is a crucial step for the imaging of molecular structure and dynamics. Here, we demonstrate three-dimensional (3D) field-free alignment of the prototypical asymmetric top molecule indole using elliptically polarized, shaped, off-resonant laser pulses. A truncated laser pulse is produced using a combination of extreme linear chirping and controlled phase and amplitude shaping using a spatial-light-modulator (SLM) based pulse shaper of a broadband laser pulse. The angular confinement is detected through velocity-map imaging of H+ and C2+ fragments resulting from strong-field ionization and Coulomb explosion of the aligned molecules by intense femtosecond laser pulses. The achieved three-dimensional alignment is characterized by comparing the result of ion-velocity-map measurements for different alignment directions and for different times during and after the alignment laser pulse to accurate computational results. The achieved strong three-dimensional field-free alignment of [Formula: see text] demonstrates the feasibility of both, strong three-dimensional alignment of generic complex molecules and its quantitative characterization.

Photoelectron interference of He atoms in the attosecond ionization gating

We theoretically investigate the photoelectron momentum distribution of He atoms by numerically solving the time-dependent Schro¨dinger equation (TDSE) in few-cycle ionization gating, which is synthesized by two linearly polarized laser pulses. When applying the TDSE, we can clearly see the spider-like structures in the photoelectron momentum spectra. We also find that the spider-like structures can be isolated by changing the relative phase. The directionality of the spider-like structure is changed from right-side to left-side and the ring-like interference structure gradually appears in the photoelectron momentum spectra when increasing the relative phase. The interference patterns observed in TDSE are recaptured well by the quantum-trajectory Monte Carlo (QTMC) model. We separate the ionization time window of the tunneling electron by analyzing the ionization rate. With the help of QTMC simulation, we illustrate the change of the interference structure and its directionality in the photoelectron momentum spectra. By changing the relative phase, the forward-backward asymmetry of the momentum distribution of the emitted electrons can also be controlled. Moreover, we find that the relative contribution of the nonrescattering and the rescattering trajectories can be controlled. These properties are beneficial for the application of photoelecron holography in probing atomic and molecular structures and dynamics.

Trends in angle-resolved molecular photoelectron spectroscopy

In this perspective article, main trends of angle-resolved molecular photoelectron spectroscopy in the laboratory up to the molecular frame, in different regimes of light-matter interactions, are highlighted with emphasis on foundations and most recent applications.

Picometer-Resolved Photoemission Position within the Molecule by Strong-Field Photoelectron Holography

Laser-induced tunneling ionization is one of the fundamental light-matter interaction processes. An accurate description of the tunnel-ionized electron wave packet is central to understanding and controlling subsequent electron dynamics. Because of the anisotropic molecular structure, tunneling ionization of molecules involves considerable challenges in accurately describing the tunneling electron wave packet. Up to now, some basic properties of the tunneling electron from molecules still remain unexplored. Here, we demonstrate that the tunneling electron from a molecule is not always emitted from the geometric center of the molecule along the tunnel direction. Rather, the photoemission position depends on the molecular orientation. Using a photoelectron holographic technique, we determine the photoemission position for a nitrogen molecule relative to the molecular geometric center to be 95±21  pm when the molecular axis is oriented along the tunnel direction. Our Letter poses, and answers experimentally, a fundamental question as to where the molecular photoionization actually begins, which has significant implications for time-resolved probing of valence electron dynamics in molecules.

Tracking the Ionization Site in Neutral Molecules

When a diatomic molecule is exposed to intense light, the valence electron may tunnel from a higher potential (corresponding to an upfield atom) due to the suppressed internuclear barrier. This process is known as ionization enhancement and is a key mechanism in strong field ionization of molecules. Alternatively, the bound electron wave function can evolve adiabatically in the laser field, resulting in ionization from the downfield atom. Here, we introduce a method to quantify the relative contribution of these two processes. Applying this method to experimentally measured electron momenta distributions following strong field ionization of N_{2} with infrared laser light, we find approximately a 2∶1 ratio of electrons ionized from a downfield atom, relative to upfield. This suggests that the bound state wave function largely adapts adiabatically to the changing laser field, although the nonadiabatic process of ionization enhancement still contributes even in neutral molecules. Our method can be applied to any diatomic neutral molecule to better understand the evolution of the initially bound electron wave packet and hence the nature of the molecular ionization process.

Dynamic Core Polarization in High Harmonic Generation from Solids: The Example of MgO Crystals

Previously, the strong field processes in solids have always been explained by the single-active-electron (SAE) model with a frozen core excluding the fluctuation of background electrons. In this work, we demonstrate the strong field induced dynamic core polarization effect and propose a model for revealing its role in high harmonic generation (HHG) from solids. We show that the polarized core induces an additional polarization current beyond the SAE model based on the frozen cores. It gives a new mechanism for HHG and leads to new anisotropic structures, which are experimentally observed with MgO. Our experiments indicate that the influences of dynamic core polarization on HHG are obvious for both linearly and elliptically polarized laser fields. Our work establishes the bridge between the HHG and the dynamic changes of the effective many-electron interaction in solids, which paves the way to probe the ultrafast electron dynamics.

Imaging of electron transition and bond breaking in the photodissociation of H2+ via ultrafast X-ray photoelectron diffraction

We theoretically investigate the photodissociation dynamics of H2+ using the methodology of ultrafast X-ray photoelectron diffraction (UXPD). We use a femtosecond infrared pulse to prompt a coherent excitation from the molecular vibrational state (v = 9) of the electronic ground state (1sσ_g) and then adopt another time-delayed attosecond X-ray pulse to probe the dynamical properties. We have calculated photoionization momentum distributions by solving the non-Born-Oppenheimer time-dependent Schrödinger equation (TDSE). We unambiguously identify the phenomena associated with the g - u symmetry breakdown in the time-resolved photoelectron diffraction spectra. Using the two-center interference model, we can determine the variation in nuclear spacing with high accuracy. In addition, we use a strong field approximation (SFA) model to interpret the UXPD profile, and the SFA simulations can reproduce the TDSE results in a quantitative way.

High-Order Phase-Dependent Asymmetry in the Above-Threshold Ionization Plateau

Above-threshold ionization spectra from cesium are measured as a function of the carrier-envelope phase (CEP) using laser pulses centered at 3.1  μm wavelength. The directional asymmetry in the energy spectra of backscattered electrons oscillates three times, rather than once, as the CEP is changed from 0 to 2π. Using the improved strong-field approximation, we show that the unusual behavior arises from the interference of few quantum orbits. We discuss the conditions for observing the high-order CEP dependence, and draw an analogy with time-domain holography with electron wave packets.

Setting the photoelectron clock through molecular alignment

The interaction of strong laser fields with matter intrinsically provides a powerful tool for imaging transient dynamics with an extremely high spatiotemporal resolution. Here, we study strong-field ionisation of laser-aligned molecules, and show a full real-time picture of the photoelectron dynamics in the combined action of the laser field and the molecular interaction. We demonstrate that the molecule has a dramatic impact on the overall strong-field dynamics: it sets the clock for the emission of electrons with a given rescattering kinetic energy. This result represents a benchmark for the seminal statements of molecular-frame strong-field physics and has strong impact on the interpretation of self-diffraction experiments. Furthermore, the resulting encoding of the time-energy relation in molecular-frame photoelectron momentum distributions shows the way of probing the molecular potential in real-time, and accessing a deeper understanding of electron transport during strong-field interactions.

Interaction of strong laser fields with matter provides powerful tools to image transient dynamics with high spatiotemporal resolution. The authors investigate strong-field ionisation of laser-aligned molecules showing the effect of molecular alignment on the photoelectron dynamics and the resulting influence of the molecular frame in imaging experiments.

Gouy's Phase Anomaly in Electron Waves Produced by Strong-Field Ionization

Ionization of atoms by strong laser fields produces photoelectron momentum distributions that exhibit modulations due to the interference of outgoing electron trajectories. For a faithful modeling, it is essential to include previously overlooked phase shifts occurring when trajectories pass through focal points. Such phase shifts are known as Gouy's phase anomaly in optics or as Maslov phases in semiclassical theory. Because of Coulomb focusing in three dimensions, one out of two trajectories in photoelectron holography goes through a focal point as it crosses the symmetry axis in momentum space. In addition, there exist observable Maslov phases already in two dimensions. Clustering algorithms enable us to implement a semiclassical model with the correct preexponential factor that affects both the weight and the phase of each trajectory. We also derive a simple rule to relate two-dimensional and three-dimensional models for linear polarization. It explains the shifted interference fringes and weaker high-energy yield in three dimensions. The results are in excellent agreement with solutions of the time-dependent Schrödinger equation.

Deep Learning for Feynman's Path Integral in Strong-Field Time-Dependent Dynamics

Feynman's path integral approach is to sum over all possible spatiotemporal paths to reproduce the quantum wave function and the corresponding time evolution, which has enormous potential to reveal quantum processes in the classical view. However, the complete characterization of the quantum wave function with infinite paths is a formidable challenge, which greatly limits the application potential, especially in the strong-field physics and attosecond science. Instead of brute-force tracking every path one by one, here we propose a deep-learning-performed strong-field Feynman's formulation with a preclassification scheme that can predict directly the final results only with data of initial conditions, so as to attack unsurmountable tasks by existing strong-field methods and explore new physics. Our results build a bridge between deep learning and strong-field physics through Feynman's path integral, which would boost applications of deep learning to study the ultrafast time-dependent dynamics in strong-field physics and attosecond science and shed new light on the quantum-classical correspondence.

It is all about phases: ultrafast holographic photoelectron imaging

Photoelectron holography constitutes a powerful tool for the ultrafast imaging of matter, as it combines high electron currents with subfemtosecond resolution, and gives information about transition amplitudes and phase shifts. Similarly to light holography, it uses the phase difference between the probe and the reference waves associated with qualitatively different ionization events for the reconstruction of the target and for ascertaining any changes that may occur. These are major advantages over other attosecond imaging techniques, which require elaborate interferometric schemes in order to extract phase differences. For that reason, ultrafast photoelectron holography has experienced a huge growth in activity, which has led to a vast, but fragmented landscape. The present review is an organizational effort towards unifying this landscape. This includes a historic account in which a connection with laser-induced electron diffraction is established, a summary of the main holographic structures encountered and their underlying physical mechanisms, a broad discussion of the theoretical methods employed, and of the key challenges and future possibilities. We delve deeper in our own work, and place a strong emphasis on quantum interference, and on the residual Coulomb potential.

Two-dimensional photoelectron holography in strong-field tunneling ionization by counter rotating two-color circularly polarized laser pulses

Strong-field photoelectron holography (SFPH), originating from the interference of the direct electron and the rescattering electron in tunneling ionization, is a significant tool for probing structure and electronic dynamics in molecules. We theoretically study SFPH by counter rotating two-color circularly (CRTC) polarized laser pulses. Different from the case of the linearly polarized laser field, where the holographic structure in the photoelectron momentum distribution (PEMD) is clustered around the laser polarization direction, in the CRTC laser fields, the tunneling ionized electrons could recollide with the parent ion from different angles and thus the photoelectron hologram appears in the whole plane of laser polarization. This property enables structural information delivered by the electrons scattering the molecule from different angles to be recorded in the two-dimensional photoelectron hologram. Moreover, the electrons tunneling at different laser cycles are streaked to different angles in the two-dimensional polarization plane. This property enables us to probe the sub-cycle electronic dynamics in molecules over a long time window with the multiple-cycle CRTC laser pulses.

Atomic-resolution imaging of carbonyl sulfide by laser-induced electron diffraction

Measurements on the strong-field ionization of carbonyl sulfide molecules by short, intense, 2 µm wavelength laser pulses are presented from experiments where angle-resolved photoelectron distributions were recorded with a high-energy velocity map imaging spectrometer, designed to reach a maximum kinetic energy of 500 eV. The laser-field-free elastic-scattering cross section of carbonyl sulfide was extracted from the measurements and is found in good agreement with previous experiments, performed using conventional electron diffraction. By comparing our measurements to the results of calculations, based on the quantitative rescattering theory, the bond lengths and molecular geometry were extracted from the experimental differential cross sections to a precision better than ±5 pm and in agreement with the known values.

Tomographic Extraction of the Internuclear Separation Based on Two-Center Interference with Aligned Diatomic Molecules

We experimentally investigate the two-dimensional photoelectron momentum spectra of aligned diatomic molecules in an intense laser field. Our results reveal a novel prominent valley structure in the molecular alignment dependence of the high-energy photoelectron spectra along the laser polarization. Resorting to the molecular strong-field approximation and a simple semiclassical analysis, we show that this valley structure stems from the destructive two-center interference of the laser-driven rescattered electrons in diatomic molecules. Based on this two-center interference with aligned diatomic molecules, we demonstrate for the first time a tomographic method to extract the molecular internuclear separation, providing a more straightforward approach of molecular imaging, in comparison with, e.g., laser-induced electron diffraction and fixed-angle broadband laser-driven electron scattering.

Photoelectron Holographic Interferometry to Probe the Longitudinal Momentum Offset at the Tunnel Exit

Laser-induced electron tunneling underlies numerous emerging spectroscopic techniques to probe attosecond electron dynamics in atoms and molecules. The improvement of those techniques requires an accurate knowledge of the exit momentum for the tunneling wave packet. Here we demonstrate a photoelectron interferometric scheme to probe the electron momentum longitudinal to the tunnel direction at the tunnel exit by measuring the photoelectron holographic pattern in an orthogonally polarized two-color laser pulse. In this scheme, we use a perturbative 400-nm laser field to modulate the photoelectron holographic fringes generated by a strong 800-nm pulse. The fringe shift offers direct experimental access to the intermediate canonical momentum of the rescattering electron, allowing us to reconstruct the momentum offset at the tunnel exit with high accuracy. Our result unambiguously proves the existence of nonzero initial longitudinal momentum at the tunnel exit and provides fundamental insights into the nonquasistatic nature of the strong-field tunneling.

Accurate measurement of laser intensity using photoelectron interference in strong-field tunneling ionization

Accurate determination of laser intensity is of fundamental importance to study various phenomena in intense laser-atom/molecule interactions. We theoretically demonstrate a scheme to measure laser intensity by examining the holographic structure originating from the interference between the direct and near-forward rescattering electrons in strong-field tunneling ionization. By adding a weak second-harmonic field with polarization orthogonal to the strong fundamental driving field, the interference pattern oscillates with the changing relative phases of the two-color fields. Interestingly, the amplitude of this oscillation in the photoelectron momentum spectrum depends on the parallel momentum. With the quantum-orbit analysis, we show that the amplitude of the oscillation minimizes when the time difference between the recollision and ionization of near-forward rescattering electron is half cycle of the fundamental driving field. This enables us to measure accurately the laser intensity by seeking the minimum of the oscillation amplitude. Moreover, we show that this minimum can be determined without scanning the relative phases, instead, by just monitoring the interference patterns for two relative phases. This facilitates the application of our scheme in experiment.

Photoelectron holography and forward scattering in atomic ionization by elliptically polarized laser pulses

We report on the scaling of the photoelectron holography with the laser ellipticity in strong-field atomic ionization. We find that the spacing of the holographic fringe gradually decreases with increasing of the ellipticity. In terms of the strong-field approximation, the scaling of the fringe spacing with the laser ellipticity is explained by the effect of the initial transverse momenta at the tunnel exit. With increasing of the laser ellipticity, a ridge structure arising from forward scattering electrons is observed in the low-energy region of the electron momentum distribution. An analytic formula is obtained that demarcates the phase diagram for the observation of the holographic pattern and ridge structure in elliptically polarized laser fields.

Molecular orbital imprint in laser-driven electron recollision

Electrons released by strong-field ionization from atoms and molecules or in solids can be accelerated in the oscillating laser field and driven back to their ion core. The ensuing interaction, phase-locked to the optical cycle, initiates the central processes underlying attosecond science. A common assumption assigns a single, well-defined return direction to the recolliding electron. We study laser-induced electron rescattering associated with two different ionization continua in the same, spatially aligned, polyatomic molecule. We show by experiment and theory that the electron return probability is molecular frame-dependent and carries structural information on the ionized orbital. The returning wave packet structure has to be accounted for in analyzing strong-field spectroscopy experiments that critically depend on the interaction of the laser-driven continuum electron, such as laser-induced electron diffraction.

Direct Visualization of Valence Electron Motion Using Strong-Field Photoelectron Holography

Watching the valence electron move in molecules on its intrinsic timescale has been one of the central goals of attosecond science and it requires measurements with subatomic spatial and attosecond temporal resolutions. The time-resolved photoelectron holography in strong-field tunneling ionization holds the promise to access this realm. However, it remains to be a challenging task hitherto. Here we reveal how the information of valence electron motion is encoded in the hologram of the photoelectron momentum distribution (PEMD) and develop a novel approach of retrieval. As a demonstration, applying it to the PEMDs obtained by solving the time-dependent Schrödinger equation for the prototypical molecule H_{2}^{+}, the attosecond charge migration is directly visualized with picometer spatial and attosecond temporal resolutions. Our method represents a general approach for monitoring attosecond charge migration in more complex polyatomic and biological molecules, which is one of the central tasks in the newly emerging attosecond chemistry.

Visualizing rotational wave functions of electronically excited nitric oxide molecules by using an ion imaging technique

Here we report the dissociative ionization imaging of electronically excited nitric oxide (NO) molecules to visualize rotational wave functions in the electronic excited state (A ²Σ⁺). The NO molecules were excited to a single rotational energy eigenstate in the first electronic excited state by a resonant nanosecond ultraviolet pulse. The molecules were then irradiated by a strong, circularly polarized femtosecond imaging pulse. Spatial distribution of the ejected N⁺ and O⁺ fragment ions from the dissociative NO²⁺ was recorded as a direct measure of the molecular axis distribution using a high-resolution slice ion imaging apparatus. The circularly polarized probe pulse realizes the isotropic ionization and thus undistorted shapes of the functions can be visualized. Due to the higher ionization efficiency of the excited molecules relative to the ground state ones, signals from the excited NO were enhanced. We can, therefore, extract shapes of the square of rotational wave functions in the electronic excited state although the unexcited ground state molecules are the majority in an ensemble. The observed images show s-function-like and p-function-like shapes depending on the excitation wavelengths. These shapes well reflect the rotational (angular momentum) character of the prepared states. The present approach directly leads to the evaluation method of the molecular axis alignment in photo-excited ensembles, and it could also lead to a visualization method for excited state molecular dynamics.

Signatures of Electronic Structure in Bicircular High-Harmonic Spectroscopy

High-harmonic spectroscopy driven by circularly polarized laser pulses and their counterrotating second harmonic is a new branch of attosecond science which currently lacks quantitative interpretations. We extend this technique to the midinfrared regime and record detailed high-harmonic spectra of several rare-gas atoms. These results are compared with the solution of the Schrödinger equation in three dimensions and calculations based on the strong-field approximation that incorporate accurate scattering-wave recombination matrix elements. A quantum-orbit analysis of these results provides a transparent interpretation of the measured intensity ratios of symmetry-allowed neighboring harmonics in terms of (i) a set of propensity rules related to the angular momentum of the atomic orbitals, (ii) atom-specific matrix elements related to their electronic structure, and (iii) the interference of the emissions associated with electrons in orbitals corotating or counterrotating with the laser fields. These results provide the foundation for a quantitative understanding of bicircular high-harmonic spectroscopy.

Charge migration and charge transfer in molecular systems

The transfer of charge at the molecular level plays a fundamental role in many areas of chemistry, physics, biology and materials science. Today, more than 60 years after the seminal work of R. A. Marcus, charge transfer is still a very active field of research. An important recent impetus comes from the ability to resolve ever faster temporal events, down to the attosecond time scale. Such a high temporal resolution now offers the possibility to unravel the most elementary quantum dynamics of both electrons and nuclei that participate in the complex process of charge transfer. This review covers recent research that addresses the following questions. Can we reconstruct the migration of charge across a molecule on the atomic length and electronic time scales? Can we use strong laser fields to control charge migration? Can we temporally resolve and understand intramolecular charge transfer in dissociative ionization of small molecules, in transition-metal complexes and in conjugated polymers? Can we tailor molecular systems towards specific charge-transfer processes? What are the time scales of the elementary steps of charge transfer in liquids and nanoparticles? Important new insights into each of these topics, obtained from state-of-the-art ultrafast spectroscopy and/or theoretical methods, are summarized in this review.