Phase and time delays of atomic above-threshold ionization were recently experimentally explored in an $ω-2ω$ setting [Zipp et al, Optica 1, 361 (2014)]. The phases of wavepackets ejected from argon by a strong $2ω$ pulse were probed as a function of the relative phase of a weaker $ω$ probe pulse. Numerical simulations solving the time-dependent Schrödinger equation (TDSE) displayed a sensitive dependence of the doubly differential momentum distribution on the relative phase between the $ω$ and $2ω$ fields. Moreover, a surprisingly strong variation of the extracted phase delays on the intensity of the probe pulse was found. We present a semiclassical strong-field description of the phase delays in the emission of electrons in an $ω-2ω$ setting and apply it to atomic hydrogen. Non-perturbative effects in both the $2ω$ pump and the $ω$ probe field are included. The semiclassical description allows tracing phase delays to path interferences between emission during different points in time of emission within the temporal unit cell of the two-color laser field. We find good agreement between the semiclassical saddle-point approximation, the full strong field approximation (SFA), and previous results applicable in the perturbative limit of probe fields. We show that the RABBIT-like perturbative description of phase delays breaks down for stronger fields and higher-energy electron emission. In this regime, characterization of the ionization signal requires an entire ensemble of phase delays {$δ_i(E)$} with $i=1,2,\ldots$ the difference in photon numbers of the strong $2ω$ field involved in the interfering paths. Comparison between SFA and TDSE calculations reveals the influence of the Coulomb field even in this strong-field scenario.
We review the semiclassical two-step model for strong-field ionization. The semiclassical two-step model describes quantum interference and accounts for the ionic potential beyond the semiclassical perturbation theory. We discuss formulation and implementation of this model, its further developments, as well as some of the applications. The reviewed applications of the model include strong-field holography with photoelectrons, multielectron polarization effects in ionization by an intense laser pulse, and strong-field ionization of the hydrogen molecule.
We present a mixed quantum-classical approach to strong-field ionization - a semiclassical two-step model with quantum input. In this model the initial conditions for classical trajectories that simulate electron wave packet after ionization are determined by the exact quantum dynamics. As a result, the model allows to overcome deficiencies of standard semiclassical approaches in describing the ionization step. The comparison with the exact numerical solution of the time-dependent Schrödinger equation shows that for ionization of a one-dimensional atom the model yields quantitative agreement w…
Fully differential cross sections are calculated for the ionization of H$_2$ by fast charged projectiles using a semiclassical model developed previously for the ionization of atoms. The method is tested in case of 4 keV electron and 6 MeV proton projectiles. The obtained results show good agreement with the available experimental data. Interference effects due to the two-center character of the target are also observed and analyzed.
We extend the semiclassical two-step model for strong-field ionization that describes quantum interference and accounts for the Coulomb potential beyond the semiclassical perturbation theory to the hydrogen molecule. In the simplest case of the molecule oriented along the polarization direction of a linearly polarized laser field, we predict significant deviations of the two-dimensional photoelectron momentum distributions and the energy spectra from the case of atomic hydrogen. Specifically, for the hydrogen molecule the electron energy spectrum falls off slower with increasing energy, and th…
A theoretical study of the interference pattern imprinted on the doubly differential momentum distribution of the photoelectron due to atomic ionization induced by a short laser pulse is developed from a semiclassical standpoint. We use the semiclassical two-step model of Shvetsov-Shilovski et al. (Phys. Rev. A 94, 013415) to elucidate the nature of the holographic structure. Three different types of trajectories are characterized during the ionization process by a single cycle pulse with three different types of interferences. We show that the holographic interference arises from the ionizati…
We consider the time delay of electron detachment from a Coulomb center and two-center systems in the process of ionization. It is shown that the attosecond streaking, most usual method of time delay measure, can be formally described by placing a virtual detector of the arrival time delay at a certain distance from the center of the system. This approach allows derivation of a simple formula for Coulomb-laser coupling that perfectly agrees with the results of numerical solution of the time-dependent Schrodinger equation. The dependence of the time delay upon the energy, the angular momentum p…
Quasi-static models of barrier suppression have played a major role in our understanding of the ionization of atoms and molecules in strong laser fields. Despite their success, in the case of diatomic molecules these studies have so far been restricted to fields aligned with the molecular axis. In this paper we investigate the locations and heights of the potential barriers in the hydrogen molecular ion in an electric field of arbitrary orientation. We find that the barriers undergo bifurcations as the external field strength and direction are varied. This phenomenon represents an unexpected l…
Recent attoclock experiments using the attsecond angular streaking technique enabled the measurement of the tunneling time delay during laser induced strong field ionization. Theoretically the tunneling time delay is commonly modelled by the Wigner time delay concept which is derived from the derivative of the electron wave function phase with respect to energy. Here, we present an alternative method for the calculation of the Wigner time delay by using the fixed energy propagator. The developed formalism is applied to the nonrelativistic as well as to the relativistic regime of the tunnel-ion…
We present a semiclassical two-step model for strong-field ionization that accounts for path interferences of tunnel-ionized electrons in the ionic potential beyond perturbation theory. Within the framework of a classical trajectory Monte-Carlo representation of the phase-space dynamics, the model employs the semiclassical approximation to the phase of the full quantum propagator in the exit channel. By comparison with the exact numerical solution of the time-dependent Schrödinger equation for strong-field ionization of hydrogen, we show that for suitable choices of the momentum distribution a…
In the semiclassical picture of photoionization process in intense laser fields, the ionization rate solely depends on the amplitude of the electric field and the final photoelectron momentum corresponds to the instant of ionization of the photoelectron, however, this picture has never been checked rigorously. Recently an attosecond angular streaking technique based on this semiclassical perspective has been widely applied to temporal measurement of the atomic and molecular dynamics in intense laser fields. We use a Wigner-distribution-like function to calculate the time-emission angle distrib…
The electron dynamics in the classically forbidden region during relativistic tunnel-ionization is investigated. The classical forbidden region in the relativistic regime is identified by defining a gauge invariant total energy operator. Introducing position dependent energy levels inside the tunneling barrier, we demonstrate that the relativistic tunnel-ionization can be well described by a one-dimensional intuitive picture. This picture predicts that, in contrast to the well-known nonrelativistic regime, the ionized electron wave packet arises with a momentum shift along the laser's propagat…
We present both full quantum mechanical and semiclassical calculations of above threshold ionization (ATI) of a hydrogen atom in the tunneling regime by a few-cycle linearly polarized infrared laser pulse. As a quantum treatment, we applied the direct integration of the time dependent Schrödinger equation (TDSE). In the semiclassical approximation (SCA), it is assumed that wavepacket propagation in the post-tunneling process can be well described within the classical framework. With these two methods, we analyze the similarities and deviations for ionization of the hydrogen atom. We found that…
Nonlinear relativistic ionization phenomena induced by a strong laser radiation with elliptically polarization are considered. The starting point is the classical relativistic action for a free electron moving in the electromagnetic field created by a strong laser beam. The application of the relativistic action to the classical barrier-suppression ionization is briefly discussed. Further the relativistic version of the Landau-Dykhne formula is employed to consider the semiclassical sub-barrier ionization. Simple analytical expressions have been found for: (i) the rates of the strong-field non…
This tutorial presents an introduction to the interaction of light and matter on the attosecond timescale. Our aim is to detail the theoretical description of ultra-short time-delays, and to relate these to the phase of extreme ultraviolet (XUV) light pulses and to the asymptotic phase-shifts of photoelectron wave packets. Special emphasis is laid on time-delay experiments, where attosecond XUV pulses are used to photoionize target atoms at well-defined times, followed by a probing process in real time by a phase-locked, infrared laser field. In this way, the laser field serves as a "clock" to…
We report full-dimensionality quantum and classical calculations for double ionization of laser-driven helium at 390 nm. Good qualitative agreement is observed. We show that the classical double ionization trajectories can be divided into two distinct pathways: direct and delayed. The direct pathway, with an almost simultaneous ejection of both electrons, emerges from small laser intensities. With increasing intensity its relative importance, compared to the delayed ionization pathway, increases until it becomes the predominant pathway for total electron escape energy below around 5.25 $U_{p}$…
We perform theoretical investigations of laser-induced nonsequential double ionization with few cycle pulses, with particular emphasis on the dependence of the electron-momentum distributions on the carrier-envelope phase. We focus on the recollision-excitation with subsequent tunneling ionization (RESI) pathway, in which a released electron, upon return to its parent ion, gives part of its kinetic energy to promote a second electron to an excited state. At a subsequent time, the second electron is freed through tunneling ionization. We show that the RESI electron-momentum distributions vary d…
When intense laser pulses release correlated electrons, the time delay between the ionizations may last more than one laser cycle. We show that this "Recollision-Excitation with Subsequent Ionization" pathway originates from the inner electron being promoted to a sticky region by a recollision where it is trapped for a long time before ionizing. We identify the mechanism which regulates this region, and predict oscillations in the double ionization yield with laser intensity.
Relative phases of atomic above-threshold ionization wavepackets have been investigated in a recent experiment [L. J. Zipp, A. Natan, and P. H. Bucksbaum, Optica \textbf{1}, 361-364 (2014)] exploiting interferences between different pathways in a weak probe field at half the frequency of the strong ionization pulse. In this work we theoretically explore the extraction of phase delays and time delays of attosecond wavepackets formed in strong-field ionization. We perform simulations solving the time-dependent Schrödinger equation and compare these results with the strong-field and Coulomb-Volko…
More than 100 years after its discovery and its explanation in the energy domain, the duration of the photoelectric effect is still heavily studied. The emission time of a photoelectron can be quantified by the Wigner time delay. Experiments addressing this time delay for single-photon ionization became feasible during the last 10 years. A missing piece, which has not been studied, so far, is the Wigner time delay for strong-field ionization of molecules. Here we show experimental data on the Wigner time delay for tunnel ionization of $H_{2}$ molecules and demonstrate its dependence on the emi…
The problem of time delay in tunneling ionization is revisited. The origin of time delay at the tunnel exit is analysed, underlining the two faces of the concept of the tunnelling time delay: the time delay around the tunnel exit and the asymptotic time delay at a detector. We show that the former time delay, in the sense of a delay in the peak of the wavefunction, exists as a matter of principle and arises due to the sub-barrier interference of the reflected and transmitted components of the tunneling electronic wavepacket. We exemplify this by describing the tunnelling ionization of an elect…
We theoretically study the photoelectron angular distributions (PADs) from two-color two-photon near-threshold ionization of hydrogen and noble gas (He, Ne, and Ar) atoms by a combined action of femtosecond extreme ultraviolet (EUV) and near-infrared (IR) laser pulses. Using the second-order time-dependent perturbation theory, we clarify how the two-photon ionization process depends on EUV-IR pulse delay and how it is connected to the interplay between resonant and nonresonant ionization paths. Furthermore, by solving the time-dependent Schrödinger equation, we calculate the anisotropy paramet…
We study the formation of highly excited neutral atoms during the break-up of strongly-driven molecules. Past work on this significant phenomenon has shown that during the formation of highly excited neutral atoms ($\mathrm{H^{*}}$) during the break-up of H$_{2}$ in a linear laser field the electron that escapes does so either very quickly or after remaining bound for a few periods of the laser field. Here, we address the electron-nuclear dynamics in $\mathrm{H^{*}}$ formation in elliptical laser fields, through Coulomb explosion. We show that with increasing ellipticity two-electron effects a…
Several recent attoclock experiments have investigated the fundamental question of a quantum mechanically induced time delay in tunneling ionization via extremely precise photoelectron momentum spectroscopy. The interpretations of those attoclock experimental results were controversially discussed, because the entanglement of the laser and Coulomb field did not allow for theoretical treatments without undisputed approximations. The method of semiclassical propagation matched with the tunneled wavefunction, the quasistatic Wigner theory, the analytical R-matrix theory, the backpropagation metho…
It is shown that it is feasible to use ultrashort time delay between two XUV femtosecond pulses in order to control two photon resonant ionization. The proposal is demonstrated on the spectrum of Helium, in terms of nonperturbative solutions of the time dependent Schroedinger equation. Comparison with results from second order time dependent perturbation theory provides additional insight.
We study laser-assisted photoionization by attosecond pulses using a time-independent formalism based on diagrammatic many-body perturbation theory. Our aim is to provide an ab inito route to the "delays" for this above-threshold ionization process, which is essential for a quantitative understanding of attosecond metrology. We present correction curves for characterization schemes of attosecond pulses, such as "streaking", that account for the delayed atomic response in ionization from neon and argon. We also verify that photoelectron delays from many-electron atoms can be measured using simi…
High harmonic spectroscopy has the potential to combine attosecond temporal with sub-Angstrom spatial resolution of the early nuclear and multielectron dynamics in molecules. It involves strong field ionization of the molecule by the IR laser field followed by time-delayed recombination of the removed electron with the molecular ion. The time-delay is controlled on the attosecond time scale by the oscillation of the IR field and is mapped into the harmonic number, providing a movie of molecular dynamics between ionization and recombination. One of the challenges in the analysis of high harmoni…
Recent angle-resolved RABBITT experiments have shown that the photoionization time delay depends on the emission angle of the photoelectron. In this work we demonstrate that for photoemission from helium accompanied by shake-up (correlation satellites), the angular variation of the time delay is dramatically enhanced by the dipolar coupling between the photoelectron and the highly polarizable bound electron in the IR field. We show that the additivity rule for the time delays due to the atomic potential, the continuum-continuum (cc) coupling by the IR field, and due to this two-electron proces…
A three-dimensional semiclassical model is used to study double ionization of Ar when driven by a near-infrared and near-single-cycle laser pulse for intensities ranging from 0.85$\times$10$^{14}$ W/cm$^{2}$ to 5$\times$10$^{14}$ W/cm$^{2}$. Asymmetry parameters, distributions of the sum of the two electron momentum components along the direction of the polarization of the laser field and correlated momenta are computed as a function of intensity and of the carrier envelope phase. A very good agreement is found with recently obtained results in kinematically complete experiments employing near…
We present experimental measurements and theoretical calculations of photoionization time delays from the $3s$ and $3p$ shells in Ar in the photon energy range of 32-42 eV. The experimental measurements are performed by interferometry using attosecond pulse trains and the infrared laser used for their generation. The theoretical approach includes intershell correlation effects between the 3s and 3p shells within the framework of the random phase approximation with exchange (RPAE). The connection between single-photon ionization and the two-color two-photon ionization process used in the measur…
The intermediate domain of strong-field ionization between the tunneling and the multiphoton regimes is investigated using the strong field approximation and the imaginary-time method. An intuitive model for the dynamics is developed which describes the ionization process within a nonadiabatic tunneling picture with a coordinate dependent electron energy during the under-the-barrier motion. The nonadiabatic effects in the elliptically polarized laser field induce a transversal momentum shift of the tunneled electron wave packet at the tunnel exit, a delayed appearance in the continuum as well …
During the ionization of atoms irradiated by linearly polarized intense laser fields, we find for the first time that the transverse momentum distribution of photoelectrons can be well fitted by a squared zeroth-order Bessel function because of the quantum interference effect of Glory rescattering. The characteristic of the Bessel function is determined by the common angular momentum of a bunch of semiclassical paths termed as Glory trajectories, which are launched with different nonzero initial transverse momenta distributed on a specific circle in the momentum plane and finally deflected to …
We apply a fundamental definition of time delay, as the difference between the time a particle spends within a finite region of a potential and the time a free particle spends in the same region, to determine results for photoionization of an electron by an extreme ultraviolet (XUV) laser field using numerical simulations on a grid. Our numerical results are in good agreement with those of the Wigner-Smith time delay, obtained as the derivative of the phase shift of the scattering wave packet with respect to its energy, for the short-range Yukawa potential. In case of the Coulomb potential we …
We perform a detailed analysis of the importance of causality within the strong-field approximation and the steepest descent framework for the recollision-excitation with subsequent tunneling ionization (RESI) pathway in laser-induced nonsequential double ionization (NSDI). In this time-delayed pathway, an electron returns to its parent ion, and, by recolliding with the core, gives part of its kinetic energy to excite a second electron at a time $t^{\prime}$. The second electron then reaches the continuum at a later time $t$ by tunneling ionization. We show that, if $t^{\prime}$ and $t$ are co…
We study the ionization of atomic hydrogen in the direction of polarization due to a linearly polarized XUV pulse in the presence a strong field IR. We describe the photoelectron spectra as an interference problem in the time domain. Electron trajectories steming from different optical laser cycles give rise to intercycle interference energy peaks known as sidebands. These sidebands are modulated by a grosser structure coming from the intracycle interference of the two electron trajectories born during the same optical cycle. We make use of a simple semiclassical model which offers the possibi…
We perform a systematic analysis of quantum interference in nonsequential double ionization focusing on the recollision-excitation with subsequent ionization (RESI) mechanism, employing the strong-field approximation (SFA). We find that interference has a major influence on the shape, localization and symmetry of the correlated electron momentum distributions. In particular, the fourfold symmetry with regard to the parallel momentum components observed in previous SFA studies is broken. Two types of interference are observed and thoroughly analyzed, namely that caused by electron indistinguish…
A theoretical study of ionization of the hydrogen atom due to an XUV pulse in the presence of an IR laser is presented. Well-established theories are usually used to describe the problem of laser assisted photoelectron effect. However, the well-known soft-photon approximation firstly posed by Maquet et al in Journal of Modern Optics 54, 1847 (2007) and Kazansky's theory in Phys. Rev. A 82, 033420 (2010) completely fails to predict the electron emission prependicularly to the polarization direction. Making use of a simple semiclassical model, we study the angle-resolved energy distribution of p…
We theoretically demonstrate a one-to-one mapping between the direction of electron ionization and the phase delay between a linearly polarized VUV and a circular IR laser pulse. To achieve this, we use an ultrashort VUV pulse that defines the moment in time and space when an above threshold electron is released in the IR pulse. The electron can then be accelerated to high velocities escaping in a direction completely determined by the phase delay between the two pulses. The dipole matrix element to transition from an initial bound state of the N$_2$ molecule, considered in this work, to the c…
Interferometrically Enhanced Asymmetry in Strong-field Ionization with Bright Squeezed Vacuum
Perimeter Institute for Theoretical Physics（Perimeter理论物理研究所）；University of Waterloo（滑铁卢大学）；University of Oxford（牛津大学）；University College London（伦敦大学学院）；ICFO – Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology（ICFO-光子科学研究所，巴塞罗那科技研究所）
Ionization of atoms by linearly polarized strong laser fields produces cylindrically symmetric 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 jumps occurring when trajectories pass through focal points. Such phase jumps are known as Gouy's phase anomaly in optics or as Maslov phases in semiclassical theory. Most importantly, because of Coulomb focusing in three dimensions, one out of two trajectories in photoelectron holography goes through a…
The use of semiclassical models based on the Strong Field Approximation (SFA) is ubiquitous in strong field multi-photon ionization and underlies many key developments in attosecond science, including the description of High Harmonic Generation (HHG). However, such models are notably lacking in streaking experiments, which use an attosecond pulse to initiate single-photon ionization and a lower frequency infrared pulse to provide timing information. Here, we introduce a classical Wigner propagation (CWP) method, which analogously to semiclassical models in strong field ionization, treats the i…
We introduce a method to investigate the longitudinal momentum spread resulting from strong-field tunnel ionization of Helium which, unlike other methods, is valid for all ellipticities of laser pulse. Semiclassical models consisting of tunnel ionization followed by classical propagation in the combined ion and laser field reproduce the experimental results if an initial longitudinal spread at the tunnel exit is included. The values for this spread are found to be of the order of twice the transverse momentum spread.
Tunnel ionization belongs to the fundamental processes of atomic physics. The so-called two-step model, which describes the ionization as instantaneous tunneling at the electric field maximum and classical motion afterwards with zero exit momentum, is commonly employed to describe tunnel ionization in adiabatic regimes. In this contribution, we show by solving numerically the time-dependent Schrödinger equation in one dimension and employing a virtual detector at the tunnel exit that there is a nonvanishing positive time delay between the electric field maximum and the instant of ionization. M…
Time-delays in the photoionization of molecules are investigated. As compared to atomic ionization, the time-delays expected from molecular ionization present a much richer phenomenon, with a strong spatial dependence due to the anisotropic nature of the molecular scattering potential. We investigate this from a scattering theory perspective, and make use of molecular photoionization calculations to examine this effect in representative homonuclear and hetronuclear diatomic molecules, nitrogen and carbon monoxide. We present energy and angle-resolved maps of the Wigner delay time for single-ph…
We measure the photoelectron energy spectra from strong-field ionization of Kr in a two-color laser pulse consisting of a strong 400-nm field and a weak 800-nm field. The intensities of the main above-threshold ionization (ATI) and sideband peaks in the photoelectron energy spectra oscillate roughly oppositely with respect to the relative phase between the two-color components. We study the photoelectron interferometry in strong-field ATI regime from the view of interference of different electron trajectories in order to extend RABBITT type analysis to the strong-field regime. Based on the str…
We analyze strong field atomic dynamics semiclassically, based on a full time-dependent description with the Hermann-Kluk propagator. From the properties of the exact classical trajectories, in particular the accumulation of action in time, the prominent features of above threshold ionization (ATI) and higher harmonic generation (HHG) are proven to be interference phenomena. They are reproduced quantitatively in the semiclassical approximation. Moreover, the behavior of the action of the classical trajectories supports the so called strong field approximation which has been devised and postula…
The measurement of the tunneling time in attosecond experiments, termed attoclock, triggered a hot debate about the tunneling time, the role of time in quantum mechanics, where the interaction with the laser pulse involves two regimes of a different character, the multiphoton and the tunneling (field-) ionization. In the adiabatic field calibration, one of us (O. K.) developed in earlier works a real tunneling time model and showed that the model fits well to the experimental data of Landsmann et al. (Optica {\bf 1}, 343 2014). In the present work, it is shown that the model explains the exper…
Strong-field photoelectron holography denotes the interference of various electron paths in laser-induced ionization of atoms, leading to interference patterns in the final momentum distribution. For a quantitative description of holography beyond the electric dipole approximation and in the presence of the Coulomb potential, we develop a semiclassical model in which the initial conditions of outgoing electrons are set according to the beyond-dipole strong-field approximation for the ionization step. The phases associated with trajectories are evaluated following the prescription for semiclass…
We analyze interference processes in atomic ionization induced by a two-color laser with fundamental frequency $ω$ and its second harmonic $2ω$. The interplay between inter- and intracycle interference processes give rise to multiphoton peaks which can be named as main or ATI peaks and sidebands, in analogy to the well-known RABBIT (reconstruction of attosecond harmonic beating by interference of two-photon transitions). We use the saddle point approximation (SPA) to extract the complex ionization times of the interfering electron trajectories. Changing the relative phase between the two color…
We describe an approach defining instantaneous ionization rate (IIR) as a functional derivative of the total ionization probability. The definition is based on physical quantities which are directly measurable, such as the total ionization probability and the waveform of the pulse. The definition is, therefore, unambiguous and does not suffer from gauge non-invariance. We compute IIR by solving numerically the time-dependent Schrodinger equation for the hydrogen atom in a strong laser field. We find that the IIR lags behind the electric field, but this lag is entirely due to the long tail effe…
