Imaging orbitals with attosecond and Ångström resolutions: toward attochemistry?

Modeling of High-Harmonic Generation in the C60 Fullerene Using Ab Initio, DFT-Based, and Semiempirical Methods

We report calculations of the high-harmonic generation spectra of the C60 fullerene molecule carried out by employing a diverse set of real-time time-dependent quantum chemical methods. All methodologies involve expanding the propagated electronic wave function in bases consisting of the ground and singly excited time-independent eigenstates obtained through the solution of the corresponding linear-response equations. We identify the correlation and exchange effect in the spectra by comparing the results from methods relying on the Hartree-Fock reference determinant with those obtained using approaches based on the density functional theory with different exchange-correlation functionals. The effect of the full random-phase approximation treatment of the excited electronic states is also analyzed and compared with the configuration interaction singles and the Tamm-Dancoff approximation. We also showcase the fact that the real-time extension of the semiempirical method INDO/S can be effectively applied for an approximate description of laser-driven dynamics in large systems.

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.

Probing the molecular frame of uracil and thymine with high-harmonic generation spectroscopy

In this work, we present the computed high-harmonic generation (HHG) spectra of uracil and thymine molecules, by means of the real-time time-dependent formulation of Gaussian-based configuration interaction with single excitations (RT-TD-CIS). According to the experimental work [Hutchison et al., Comparison of high-order harmonic generation in uracil and thymine ablation plumes, Phys. Chem. Chem. Phys., 2013, 15, 12308], a pulse wavelength of 780 nm has been used, together with an intensity of 1014 W cm-2 and a pulse duration of 23 optical cycles. In order to examine the effect of pulse polarisation, rotationally averaged (to mimic the gas-phase sample) and single-polarisations have been computed for both molecules. Our results show that the HHG signal for both molecules possibly originates from different ionisation channels, involving HOMO, HOMO-1, HOMO-2 and HOMO-3 orbitals, which lie within 4 eV. We characterize the HHG spectrum of thymine, supporting the idea that the absence of the thymine signal in the original work does not depend on the single-molecule behaviour. Present results for uracil are consistent with the experimental data. Moreover, we have observed that states below and above the chosen ionisation threshold provide different contributions to the HHG spectrum in averaged and single-polarisation calculations.

Application of the Phase-Space Path Integral to Strong-Laser-Field-Assisted Electron-Ion Radiative Recombination: A Gauge-Covariant Formulation

We consider the problem of the choice of gauge in nonrelativistic strong-laser-field physics. For this purpose, we use the phase-space path-integral formalism to obtain the momentum-space matrix element of the exact time-evolution operator. With the assumption that the physical transition amplitude corresponds to transitions between eigenstates of the physical energy operator rather than the unperturbed Hamiltonian H0=(−i∂/∂r)2/2+V(r), we prove that the aforementioned momentum-space matrix elements obtained in velocity gauge and length gauge are equal. These results are applied to laser-assisted electron-ion radiative recombination (LAR). The transition amplitude comes out identical in length gauge and velocity gauge, and the expression agrees with the one conventionally obtained in length gauge. In addition to the strong-field approximation (SFA), which is the zeroth-order term of our expansion, we present explicit results for the first-order and the second-order terms, which correspond to LAR preceded by single and double scattering, respectively. Our general conclusion is that in applications to atomic processes in strong-field physics the length-gauge version of the SFA (and its higher-order corrections) should be used. Using the energy operator as the basis-defining Hamiltonian, we have shown that the resulting transition amplitude is gauge invariant and agrees with the form commonly derived in length gauge.

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.

Scaling Law of High Harmonic Generation in the Framework of Photon Channels

A photon channel perspective on high harmonic generation (HHG) is proposed by quantizing both the driving laser and high harmonics. It is shown that the HHG yield can be expressed as a sum of the contribution of all the photon channels. From this perspective, the contribution of a specific photon channel follows a simple scaling law and the competition between the channels is well interpreted. Our prediction is shown to be in good agreement with the simulations by solving the time-dependent Schrödinger equation. It also can explain well the experimental results of the HHG in the noncollinear two-color field and bicicular laser field.

Helicity asymmetry in strong-field ionization of atoms by a bicircular laser field

Ionization of atoms by an intense bicircular laser field is considered, which consists of two coplanar corotating or counterrotating circularly polarized field components with frequencies that are integer multiples of a fundamental frequency. Emphasis is on the effect of a reversal of the helicities of the two field components on the photoelectron spectra. The velocity maps of the liberated electrons are calculated using the direct strong-field approximation (SFA) and its improved version (ISFA), which takes into account rescattering off the parent ion. Under the SFA all symmetries of the driving field are preserved in the velocity map while the ISFA violates certain reflection symmetries. This allows one to assess the significance of rescattering in actual data obtained from an experiment or a numerical simulation.

Control of Nuclear Dynamics through Conical Intersections and Electronic Coherences

The effect of nuclear dynamics and conical intersections on electronic coherences is investigated employing a two-state, two-mode linear vibronic coupling model. Exact quantum dynamical calculations are performed using the multiconfiguration time-dependent Hartree method. It is found that the presence of a nonadiabatic coupling close to the Franck-Condon point can preserve electronic coherence to some extent. Additionally, the possibility of steering the nuclear wave packets by imprinting a relative phase between the electronic states during the photoionization process is discussed. It is found that the steering of nuclear wave packets is possible given that a coherent electronic wave packet embodying the phase difference passes through a conical intersection. A conical intersection close to the Franck-Condon point is thus a necessary prerequisite for control, providing a clear path towards attochemistry.

Tomography of asymmetric molecular orbitals with a one-color inhomogeneous field

We demonstrate image asymmetric molecular orbitals via high-order harmonic generation in a one-color inhomogeneous field. Due to the broken inversion symmetry of the inhomogeneous field in space, the returning electrons with energy in a broad range can be forced to recollide from only one direction for all the orientation angles of molecules, which therefore can be used to reconstruct asymmetric molecular orbitals. Following the procedure of molecular orbital tomography, the highest occupied molecular orbital of carbon monoxide (CO) is satisfactorily reconstructed with high-order harmonic spectra driven by the inhomogeneous field. This scheme is helpful to relax the requirement of laser conditions and is also applicable to other asymmetric molecules.

Single-shot molecular orbital tomography with orthogonal two-color fields

Molecular orbital tomography (MOT) based on high-order-harmonic generation opens a way to track the molecular electron dynamics or even follow a chemical reaction. However, the real-time imaging of the evolution of electron orbitals is hampered by the multi-shot measurement of high-order harmonics. Here, we report a single-shot MOT scheme with orthogonal two-color (OTC) fields. This scheme enables the tomographic imaging of molecular orbital with single-shot measurement in experiment, owing to the two-dimensional manipulation of the electron motion in OTC fields. Our work paves the way towards tracking the molecular electron dynamics with combined attosecond temporal and sub-Ångström spatial resolutions.

Laser-induced blurring of molecular structure information in high harmonic spectroscopy

High harmonic spectroscopy gives access to molecular structure with Angström resolution. Such information is encoded in the destructive interferences occurring between the harmonic emissions from the different parts of the molecule. By solving the time-dependent Schrödinger equation, either numerically or with the molecular strong-field approximation, we show that the electron dynamics in the emission process generally results in a strong spectral smoothing of the interferences, blurring the structural information. However we identify specific generation conditions where they are unaffected. These findings have important consequences for molecular imaging and orbital tomography using high harmonic spectroscopy.

Molecular attochemistry in non-polar liquid environments: ultrafast charge migration dynamics through gold-thiolate and gold-selenolate linkages

Molecular attosecond science has already started contributing to our fundamental understanding of ultrafast purely electron dynamics in isolated molecules under vacuum. Extending attosecond science to the liquid phase is expected to offer new insight into the influence of a surrounding solvent environment on the attosecond electron dynamics in solvated molecules. A systematic theoretical investigation of the attochemistry of solvated molecules would help one design attosecond experiments under ambient conditions to explore the attochemistry in a liquid environment. With this goal in mind, for the first time, we have explored the attochemistry of molecules surrounded by different non-polar solvent environments. For this work, we have focused on the attosecond charge conduction through gold-thiolate and gold-selenolate linkages following the vertical ionization of the S/Se(CH₃)-CH₂-phenyl-X unit anchored to a gold dimeric cluster (Au₂), where X represents either a strong electron donating N(CH₃)₂ group or a strong electron withdrawing NO₂ group. To model solvation effects on the attochemistry of molecules containing gold-chalcogen linkages, we have used an implicit solvent model (Polarizable Continuum Model) under the density functional theory (DFT) formalism for non-polar solvents. We have found that the charge migration time scale in molecules becomes faster in the presence of the solvent environment as compared to that under vacuum. Charge oscillation does not damp quickly in molecules surrounded by the solvent environment as compared to that under vacuum. Furthermore, the direction of the charge migration may change in molecules when they are surrounded by the solvent environment as compared to that under vacuum. Thus, the present work has laid the foundation, for the first time, for thinking of the attochemistry into the realm of liquids.

Molecular frame photoemission by a comb of elliptical high-order harmonics: a sensitive probe of both photodynamics and harmonic complete polarization state

Due to the intimate anisotropic interaction between an XUV light field and a molecule resulting in photoionization (PI), molecular frame photoelectron angular distributions (MFPADs) are most sensitive probes of both electronic/nuclear dynamics and the polarization state of the ionizing light field. Consequently, they encode the complex dipole matrix elements describing the dynamics of the PI transition, as well as the three normalized Stokes parameters s₁, s₂, s₃ characterizing the complete polarization state of the light, operating as molecular polarimetry. The remarkable development of advanced light sources delivering attosecond XUV pulses opens the perspective to visualize the primary steps of photochemical dynamics in time-resolved studies, at the natural attosecond to few femtosecond time-scales of electron dynamics and fast nuclear motion. It is thus timely to investigate the feasibility of measurement of MFPADs when PI is induced e.g., by an attosecond pulse train (APT) corresponding to a comb of discrete high-order harmonics. In the work presented here, we report MFPAD studies based on coincident electron-ion 3D momentum imaging in the context of ultrafast molecular dynamics investigated at the PLFA facility (CEA-SLIC), with two perspectives: (i) using APTs generated in atoms/molecules as a source for MFPAD-resolved PI studies, and (ii) taking advantage of molecular polarimetry to perform a complete polarization analysis of the harmonic emission of molecules, a major challenge of high harmonic spectroscopy. Recent results illustrating both aspects are reported for APTs generated in unaligned SF₆ molecules by an elliptically polarized infrared driving field. The observed fingerprints of the elliptically polarized harmonics include the first direct determination of the complete s₁, s₂, s₃ Stokes vector, equivalent to (ψ, ε, P), the orientation and the signed ellipticity of the polarization ellipse, and the degree of polarization P. They are compared to so far incomplete results of XUV optical polarimetry. We finally discuss the comparison between the outcomes of photoionization and high harmonic spectroscopy for the description of molecular photodynamics.

Nonperturbative Twist in the Generation of Extreme-Ultraviolet Vortex Beams

High-order harmonic generation (HHG) has been recently proven to produce extreme-ultraviolet (XUV) vortices from the nonlinear conversion of infrared twisted beams. Previous works have demonstrated a linear scaling law of the vortex charge with the harmonic order. We demonstrate that this simple law hides an unexpectedly rich scenario for the buildup of orbital angular momentum (OAM) due to the nonperturbative behavior of HHG. The complexity of these twisted XUV beams appears only when HHG is driven by nonpure vortex modes, where the XUV OAM content is dramatically increased. We explore the underlying mechanisms for this diversity and derive a general conservation rule for the nonperturbative OAM buildup. The simple scaling found in previous works corresponds to the collapse of this scenario for the particular case of pure (single-mode) OAM driving fields.

Coulomb-corrected molecular orbital tomography of nitrogen

High-order harmonic generation (HHG) from aligned molecules has provided a promising way to probe the molecular orbital with an Ångström resolution. This method, usually called molecular orbital tomography (MOT) replies on a simple assumption of the plane-wave approximation (PW), which has long been questioned due to that PW approximation is known to be valid in the keV energy region. However, the photon energy is usually no more than 100 eV in HHG. In this work, we experimentally reconstruct the highest occupied molecular orbital (HOMO) of nitrogen (N₂) by using a Coulomb-corrected MOT (CCMOT) method. In our scheme, the molecular continuum states are described by a Coulomb wave function instead of the PW approximation. With CCMOT, the reconstructed orbital is demonstrated to agree well with the theoretical prediction and retain the main features of the HOMO of N₂. Compared to the PW approximation method, the CCMOT shows a significant improvement in eliminating the artificial structures caused by PW approximation.

Multidimensional high harmonic spectroscopy of polyatomic molecules: detecting sub-cycle laser-driven hole dynamics upon ionization in strong mid-IR laser fields

High harmonic generation (HHG) spectroscopy has opened up a new frontier in ultrafast science, where electronic dynamics can be measured on an attosecond time scale. The strong laser field that triggers the high harmonic response also opens multiple quantum pathways for multielectron dynamics in molecules, resulting in a complex process of multielectron rearrangement during ionization. Using combined experimental and theoretical approaches, we show how multi-dimensional HHG spectroscopy can be used to detect and follow electronic dynamics of core rearrangement on sub-laser cycle time scales. We detect the signatures of laser-driven hole dynamics upon ionization and reconstruct the relative phases and amplitudes for relevant ionization channels in a CO₂ molecule on a sub-cycle time scale. Reconstruction of channel-resolved complex ionization amplitudes on attosecond time scales has been a long-standing goal of high harmonic spectroscopy. Our study brings us one step closer to fulfilling this initial promise and developing robust schemes for sub-femtosecond imaging of multielectron rearrangement in complex molecular systems.

Nonadiabatic tunnel ionization in strong circularly polarized laser fields: counterintuitive angular shifts in the photoelectron momentum distribution

We perform time-dependent calculation of strong-field ionization of neon, initially prepared in 2p(-1) and 2p(+1) states, with intense near-circularly polarized laser pulses. By solving the three-dimensional time-dependent Schrödinger equation, we find clear different offset angles of the maximum in the photoelectron momentum distribution in the polarization plane of the laser pulses for the two states. We provide clear interpretation that this different angular offset is linked to the sign of the magnetic quantum number, thus it can be used to map out the orbital angular momentum of the initial state. Our results provide a potential tool for studying orbital symmetry in atomic and molecular systems.

Dynamic Exchange in the Strong Field Ionization of Molecules

We show that dynamic exchange is a dominant effect in strong field ionization of molecules. In CO(2) it fixes the peak ionization yield at the experimentally observed angle of 45° between polarization direction and the molecular axis. For O(2) it changes the angle of peak emission and for N(2) the alignment dependence of yields is modified by up to a factor of 2. The effect appears on the Hartree-Fock level as well as in full ab initio solutions of the Schrödinger equation.

Ultrafast molecular orbital imaging based on attosecond photoelectron diffraction

We present ab initio numerical study of ultrafast ionization dynamics of H(2)(+) as well as CO(2) and N(2) exposed to linearly polarized attosecond extreme ultraviolet pulses. When the molecules are aligned perpendicular to laser polarization direction, photonionization of these molecules show clear and distinguishing diffraction patterns in molecular attosecond photoelectron momentum distributions. The internuclear distances of the molecules are related to the position of the associated diffraction patterns, which can be determined with high accuracy. Moreover, the relative heights of the diffraction fringes contain fruitful information of the molecular orbital structures. We show that the diffraction spectra can be well produced using the two-center interference model. By adopting a simple inversion algorithm which takes into account the symmetry of the initial molecular orbital, we can retrieve the molecular orbital from which the electron is ionized. Our results offer possibility for imaging of molecular structure and orbitals by performing molecular attosecond photoelectron diffraction.

Cavity-enhanced field-free molecular alignment at a high repetition rate

Extreme ultraviolet frequency combs are a versatile tool with applications including precision measurement, strong-field physics, and solid-state physics. Here we report on an application of extreme ultraviolet frequency combs and their driving lasers for studying strong-field effects in molecular systems. We perform field-free molecular alignment and high-order harmonic generation with aligned molecules in a gas jet at a repetition rate of 154 MHz using a high-powered optical frequency comb inside a femtosecond enhancement cavity. The cavity-enhanced system provides a means to reach suitable intensities to study field-free molecular alignment and enhance the observable effects of the molecule-field interaction. We observe modulations of the driving field, arising from the nature of impulsive stimulated Raman scattering responsible for coherent molecular rotations. We foresee the impact of this work on the study of molecule-based strong-field physics, with improved precision and a more fundamental understanding of the interaction effects on both the field and molecules.

Signatures of molecular orbital structure in lateral electron momentum distributions from strong-field ionization

Strong-field ionization of aligned diatomic and polyatomic molecules such as O2, N2, C2H4, and others in circularly polarized laser fields is investigated theoretically. By calculating the emission-angle-resolved lateral width of the momentum distribution perpendicular to the polarization plane, we show that nodal planes in molecular orbitals are directly imprinted on the angular dependence of the width. We demonstrate that orbital symmetries can be distinguished with the information obtained by observing the lateral width in addition to the angular distributions.

Flexible attosecond beamline for high harmonic spectroscopy and XUV/near-IR pump probe experiments requiring long acquisition times

We describe the versatile features of the attosecond beamline recently installed at CEA-Saclay on the PLFA kHz laser. It combines a fine and very complete set of diagnostics enabling high harmonic spectroscopy (HHS) through the advanced characterization of the amplitude, phase, and polarization of the harmonic emission. It also allows a variety of photo-ionization experiments using magnetic bottle and COLTRIMS (COLd Target Recoil Ion Momentum Microscopy) electron spectrometers that may be used simultaneously, thanks to a two-foci configuration. Using both passive and active stabilization, special care was paid to the long term stability of the system to allow, using both experimental approaches, time resolved studies with attosecond precision, typically over several hours of acquisition times. As an illustration, applications to multi-orbital HHS and electron-ion coincidence time resolved spectroscopy are presented.

Nuclear reaction induced by carrier-envelope-phase controlled proton recollision in a laser-driven molecule

Nuclear reactions induced by proton recollision with a nearby nucleus are studied in a setup where a neutral molecule is exposed to an extremely intense, few-cycle laser pulse. At the rising edge of the laser pulse, all electrons in the molecule are first ejected by field ionization, resulting in a molecule consisting of the bare nuclei only. A proton in the bare molecule is subsequently accelerated by the laser field in such a way that it recollides with a nearby, heavier nucleus, with a kinetic energy high enough to induce a nuclear reaction. As a specific example, the probability of triggering the (15)N(p,α)(12)C reaction by exposing either a (15)NH molecule or a (15)NH3 molecule to an intense laser pulse is calculated using the classical trajectory Monte Carlo method. We show that the proton recollision process can be controlled both by varying the carrier-envelope phase of the laser field and by the degree of molecular orientation. We also find that the magnetic field of the laser pulse plays a crucial role in the recollision dynamics.

Attosecond extreme ultraviolet vortices from high-order harmonic generation

We present a theoretical study of high-order harmonic generation (HHG) and propagation driven by an infrared field carrying orbital angular momentum (OAM). Our calculations unveil the following relevant phenomena: extreme-ultraviolet harmonic vortices are generated and survive to the propagation effects, vortices transport high-OAM multiples of the corresponding OAM of the driving field and, finally, the different harmonic vortices are emitted with similar divergence. We also show the possibility of combining OAM and HHG phase locking to produce attosecond pulses with helical pulse structure.

Asymmetric molecular imaging through decoding odd-even high-order harmonics

The exquisite procedure for imaging a molecular orbital with high-order harmonics proposed by Itatani et al. [Nature (London) 432, 867 (2004)] encounters difficulty when extended to an asymmetric molecule because the wave function there usually does not have a definite parity. With the observation that the wave function can be decomposed into a sum of odd and even functions and that the ionization process in harmonic generation is usually not sensitive to the asymmetry of the molecular potential, we predict that asymmetric molecular orbital imaging can be implemented through decoding odd-even high-order harmonics. A generalized tomography procedure is proposed, which has been certified by analytic deduction and numerical simulation. The above finding greatly extends the molecular tomography procedure and will further stimulate related experiments.

Theoretical methods for ultrafast spectroscopy

Time-resolved spectroscopy in the femtosecond and attosecond time domain is a tool to unravel the dynamics of nuclear and electronic motion in molecular systems. Theoretical insight into the underlying physical processes is ideally gained by solving the time-dependent Schrödinger equation. In this work, methods currently used to solve this equation are reviewed in a compact presentation. These methods involve numerical representations of wavefunctions and operators, the calculation of time evolution operators, the setting up of the Hamiltonian operators and the types of coordinates to be used hereto. The advantages and disadvantages of some methods are discussed.

Influence of large permanent dipoles on molecular orbital tomography

The influence of large permanent dipoles on molecular orbital tomography via high-order harmonic generation (HHG) is investigated in this work. It is found that, owing to the modification of the angle-dependent ionization rate resulting from the Stark shift, the one-side-recollision condition for the tomographic imaging can not be satisfied even with the few-cycle driving pulses. To overcome this problem, we employ a tailored driving pulse by adding a weak low-frequency pulse to the few-cycle laser pulse to control the HHG process and the recollision of the continuum electrons are effectively restricted to only one side of the core. Then we carried out the orbital reconstruction in both the length and velocity forms. The results show that, the orbital structure can only be successfully reproduced by using the dipole matrix elements projected perpendicular to the permanent dipole in both forms.

Tomographic imaging of asymmetric molecular orbitals with a two-color multicycle laser field

We theoretically demonstrate a scheme for tomographic reconstruction of asymmetric molecular orbitals based on high-order harmonic generation with a two-color multicycle laser field. It is shown that by adjusting the relative phase of the two fields, the returning electrons can be forced to recollide from one direction for all the orientations of molecules. Thus, the reconstruction of the asymmetric orbitals can be carried out with multicycle laser field. This releases the stringent requirement of a single-cycle pulse with a stabilized and controllable carrier-envelope phase for the tomographic imaging of asymmetric molecular orbitals.