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 electron bound by a short-range potential within the strong field approximation in a "deep tunnelling" regime. If sub-barrier reflections are extracted from this wavefunction, then the time delay of the peak is shown to vanish. Thus, we assert that the disturbance of the tunnelling wavepacket by the reflection from the surface of the barrier causes a time delay in the neighbourhood of the tunnel exit.
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 $U_{p}$…
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…
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…
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…
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…
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 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…
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…
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…
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.
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 theoretical study of the dissociative tunneling ionization process. Analytic expressions for the nuclear kinetic energy distribution of the ionization rates are derived. A particularly simple expression for the spectrum is found by using the Born-Oppenheimer (BO) approximation in conjunction with the reflection principle. These spectra are compared to exact non-BO ab initio spectra obtained through model calculations with a quantum mechanical treatment of both the electronic and nuclear degrees freedom. In the regime where the BO approximation is applicable imaging of the BO nucle…
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 measurements of single-photon ionization time delays between valence electrons of argon and neon using a coincidence detection technique that allows for the simultaneous measurement of both species under identical conditions. Taking into account the chirp of the ionizing single attosecond pulse (attochirp) ensures that the clock of our measurement technique is started at the same time for both types of electrons, revealing with high accuracy and resolution energy-dependent time delays of a few tens of attoseconds. By comparing our results with theoretical predictions, we confirm tha…
We propose a method to study the tunneling process by analyzing the time-dependent ionization yield in circularly polarized laser. A numerical calculation shows that for an atom exposed to a long laser pulse, if its initial electronic state wave function is non-spherical symmetric, the delayed phase shift of the ionization rate vs. the laser cycle period in real time in the region close to the peak intensity of the laser pulse can be used to probe the tunneling time. In this region, an obvious delay phase shift is observed, showing the tunneling time is in order of tens of attoseconds. Further…
A new pathway of strong laser field induced ionization of an atom is identified which is based on recollisions under the tunneling barrier. With an amended strong field approximation, the interference of the direct and the under-the-barrier recolliding quantum orbits are shown to induce a measurable shift of the peak of the photoelectron momentum distribution. The scaling of the momentum shift is derived relating the momentum shift to the tunneling delay time according to the Wigner concept. This allows to extend the Wigner concept for the quasistatic tunneling time delay into the nonadiabatic…
We investigate laser induced nonsequential multiple ionization using a simple statistical model in which an electron recollides inelastically with its parent ion. In this collision, it thermalizes with the remaining $N-1$ bound electrons within a time interval $Δt$. Subsequently, all the $N$ electrons leave. We address the question of how the above time delays influence the individual contributions from the orbits in which the first electron, upon return, is accelerated or decelerated by the field, respectively, to the ion momentum distributions. In both cases, the time delays modify the drift…
We show that time ordering underlying time-dependent quantum dynamics is a physical observable accessible by attosecond streaking. We demonstrate the extraction of time ordering for the prototypical case of time-resolved two-photon double ionization (TPDI) of helium by an attosecond XUV pulse. The Eisenbud-Wigner-Smith time delay for the emission of a two-electron wavepacket and the time interval between subsequent emission events can be unambiguously determined by attosecond streaking. The delay between the two emission events sensitively depends on the energy, pulse duration, and angular dis…
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…
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…
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 intensit…
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…
We present a theoretical analysis of a novel scheme for optical cooling of particles that does not in principle require a closed optical transition. A tightly confined laser beam interacting with a trapped particle experiences a phase shift, which upon reflection from a mirror or resonant microstructure produces a time-delayed optical potential for the particle. This leads to a nonconservative force and friction. A quantum model of the system is presented and analyzed in the semiclassical limit.
We investigate laser-induced nonsequential double ionization by a polarization-gated laser pulse, constructed employing two counter-rotating circularly polarized few cycle pulses with a time delay $T_{d}$. We address the problem within a classical framework, and mimic the behavior of the quantum-mechanical electronic wave packet by means of an ensemble of classical electron trajectories. These trajectories are initially weighted with the quasi-static tunneling rate, and with suitably chosen distributions for the momentum components parallel and perpendicular to the laser-field polarization, in…
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…
In a recent article by P. Hockett \textit{et al.}, time delays arising in the context of molecular single-photon ionization are investigated from a theoretical point of view. We argue that one of the central equations derived in the paper is incorrect and present a reformulation that is consistent with the established treatment of angle-dependent scattering delays.
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…
We study photoionization of argon atoms excited by attosecond pulses using an interferometric measurement technique. We measure the difference in time delays between electrons emitted from the $3s^2$ and from the $3p^6$ shell, at different excitation energies ranging from 32 to 42 eV. The determination of single photoemission time delays requires to take into account the measurement process, involving the interaction with a probing infrared field. This contribution can be estimated using an universal formula and is found to account for a substantial fraction of the measured delay.
We perform a detailed analysis of how high-order harmonic generation (HHG) and above-threshold ionization (ATI) can be controlled by a time-delayed attosecond-pulse train superposed to a strong, near-infrared laser field. In particular we show that the high-harmonic and photoelectron intensities, the high-harmonic plateau structure and cutoff energies, and the ATI angular distributions can be manipulated by changing this delay. This is a direct consequence of the fact that the attosecond pulse train can be employed as a tool for constraining the instant an electronic wave packet is ejected in …
Elliptically polarized laser pulses (EPLPs) are widely applied in many fields of ultrafast sciences, but the ellipticity~($\varepsilon$) has never been in situ measured in the interaction zone of the laser focus. In this work, we propose and realize a robust scheme to retrieve the $\varepsilon$ by temporally overlapping two identical counter-rotating EPLPs. The combined linearly electric field is coherently controlled to ionize Xe atoms by varying the phase delay between the two EPLPs. The electron spectra of the above-threshold ionization and the ion yield are sensitively modulated by the pha…
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…
Double ionization in intense laser fields can comprise electron correlations, which manifest in the non-independent emission of two electrons from an atom or molecule. However, experimental methods that directly access the electron emission times have been scarce. Here, we explore the application of an all-optical streaking technique to strong-field double ionization both theoretically and experimentally. We show that both sequential and non-sequential double ionization processes lead to streaking delays that are distinct from each other and single ionization. Moreover, coincidence detection o…
We develop a novel technique for modeling of atomic and molecular ionization in superposition of XUV and IR fields with characteristics typical for attosecond streaking and RABBITT experiments. The method is based on solving the time-dependent Schrödinger equation in the coordinate frame expanding along with the photoelectron wave packet. The efficiency of the method is demonstrated by calculating angular anisotropy of photoemission time delay of the H$_2^+$ ion in a field configuration of recent RABBITT experiments.
The effects of confinement resonances on photoelectron group delay (Wigner time delay) following ionization of an atom encapsulated inside a C$_{60}$ cage have been studied theoretically using both relativistic and non-relativistic random phase approximations. The results indicate clearly the resonant character of the confinement oscillations in time delay of the $4d$ shell of Xe@C$_{60}$ and present a most direct manifestation of Wigner time delay. These oscillations were missed in a previous theoretical investigation of Ar@C$_{60}$ [PRL 111, 203003 (2013)]
We propose a new scenario to apply IR-pump-XUV-probe schemes to resolving strong field ionization induced and attosecond pulse driven electron-hole dynamics and coherence in real time. The coherent driving of both the infrared laser and the attoscond pulse correlates the dynamics of the core-hole and the valence-hole which leads to the otherwise forbidden absorption and emission of XUV photon. An analytical model is developed based on the strong-field approximation by taking into account of the essential multielectron configurations. The emission spectra from the core-valence transition and th…
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…
The magnetic-field effect of the laser pulse is investigated in laser-assisted XUV ionization. By numerically solving the three-dimensional time-dependent Schrödinger equation, we find that the photoelectron momentum distribution is distorted by the magnetic-field effect of the IR streaking field. It results in a transverse-momentum- and time-delay-dependent longitudinal photoelectron momentum shift, which is further analytically confirmed by employing the nondipole corrected strong field approximation. We also demonstrate that the magnetic field does not affect the time shift retrieved from t…
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 …
Attosecond pulses can be used to initiate and control electron dynamics on a sub-femtosecond time scale. The first step in this process occurs when an atom absorbs an ultraviolet photon leading to the formation of an attosecond electron wave packet (EWP). Until now, attosecond pulses have been used to create free EWPs in the continuum, where they quickly disperse. In this paper we use a train of attosecond pulses, synchronized to an infrared (IR) laser field, to create a series of EWPs that are below the ionization threshold in helium. We show that the ionization probability then becomes a fun…
