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Offset angle of various atoms and wavelengths. (a) Offset angles of various atoms at the wavelength of 800 nm as a function of intensity. Black circle, red square, green diamond, blue triangle-up and magenta triangle-down represent Xe, H, Ar, Ne and He, respectively. (b) Details of (a) in the low laser intensity range. (c) Offset angles of hydrogen at various wavelengths as a function of laser intensity. Black circle, red square, green diamond and blue triangle represent the wavelengths 600 nm, 800 nm, 1000 nm and 1200 nm, respectively.
Source publication
Photocurrents generated in few-cycle circularly polarized laser fields are investigated by solving the time-dependent Schrodinger equation and by means of classical-trajectory Monte Carlo simulations. After confirming the offset angles of the total final currents and electron spectra to be the same, we illustrate the effects of the Coulomb potentia...
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Synopsis
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Citations
... We calculate the PMD using Eqs. (26) and (27) (for the asymptotic time t → ∞) as a function of the time delay t d . The chosen parameters are E 0 = 0.25 a.u., ω = 0.075 a.u., and ξ = 0.05. ...
Tunneling ionization is characterized by a negative time delay, observed asymptotically as a specific shift of the photoelectron momentum distribution, which is caused by the interference of the sub-barrier recolliding and direct ionization paths. In contrast, a Gedankenexperiment following the peak of the wave function shows a positive tunneling time delay at the tunnel exit, considering only the direct ionization path. In this paper we investigate the effects of sub-barrier recollisions on the time delay pattern at the tunnel exit. We conclude that the interference of the direct and recolliding trajectories decreases the tunneling time delay at the exit by the value equal to the asymptotic time delay, maintaining, however, its sizable positive value. Finally, we discuss the recent experiment [Yu et al., Light Sci. Appl. 11, 215 (2022)] addressing the tunneling time in a modified two-color attoclock setup. The analysis of the experimental findings with our theoretical model indicates the physical necessity to introduce an additional time characteristic for tunneling ionization—the time delay describing the initiation of the tunneling wave packet.
... Thus, the total amplitude (in the p x plane) in three dimensions can be calculated from the amplitude in one dimension, using Eqs. (24), (26): ...
Tunneling ionization is characterized by a time delay, observed asymptotically as a specific shift of the photoelectron momentum distribution. This shift corresponds to a negative time delay which is caused by the interference of the sub-barrier recolliding and direct ionization paths, as laid out in [Phys. Rev. Lett. 120, 013201 (2018)]. While a \textit{Gedankenexperiment} following the peak of the wavefunction considering only the direct ionization path shows a positive tunneling time delay at the tunnel exit, in this paper we investigate the effects of sub-barrier recollisions on the time delay pattern at the tunnel exit. We conclude that the interference of the direct and recolliding trajectories slightly decreases the tunneling time delay at the exit, the latter nevertheless maintaining its sizeable positive value. The relation of the variation of the exit time delay due to the sub-barrier recollisions to the asymptotic momentum shift, and its dependence on the laser field are discussed.
... 1(b) and 2(e). In the previous studies, some classical or semiclassical models, such as classical-trajectory Monte Carlo [10,15,47] and quantumtrajectory Monte Carlo [48], are used to study the PMD, in which it is usually assumed that the electron has an initial transverse velocity distribution at the exit of the tunnel. The initial velocity distribution at the tunnel exit is directly related to the radial distribution of the PMD. ...
We theoretically study the attoclock using the Bohmian trajectory. The Bohmian trajectory includes a nonlocal quantum potential and the Coulomb potential, allowing us to separately study their effects on the attoclock. We show that the quantum potential has a significant effect on the photoelectron momentum spread while it only slightly affects the offset angle in the attoclock. We further show how the Coulomb potential affects the offset angle, revealing that the Coulomb force before the electron reaches the position of the classical tunnel exit has a significant contribution to the offset angle. Thus taking the full Coulomb effect into account during the tunneling ionization is a prerequisite for our understanding of the origin of the offset angle in the attoclock experiment. Our study also implies that there might be an inertia time for the initial wave function in response to the change of the laser electric field.
... tunnel ionization and afterwards accelerated by the laser electric field as well as the Coulomb field of the parent ion such that the initial conditions (ionization time, velocity and position) are mapped to final electron momenta. Recent theoretical investigations using various techniques such as the analytical R-matrix theory [28], the classical backpropagation method [31][32][33], classical trajectory Monte-Carlo simulations [34], trajectory-free ionization times from Dyson integrals [35] and a classical Rutherford scattering model [36] have shown that deviations from the simple man's model are mostly related to the Coulomb field of the residual ion. Since this interaction depends strongly on the initial position (tunnel exit) of the electron, the attoclock can be used as a fine 'nano-ruler' that measures the width of the tunnel barrier [37]. ...
Strong-field ionization of atoms can be investigated on the attosecond time scale by using the attoclock method, i.e. by observing the peak of the photoelectron momentum distribution (PMD) after applying a laser pulse with a two-dimensional polarization form. Examples for such laser fields are close-to-circular or bicircular fields. Here, we report numerical solutions of the time-dependent Schrödinger equation for bicircular fields and a comparison with a compact classical model to demonstrate that the tunnel-exit position, i.e. the position where the electron emerges after tunnel ionization, is encoded in the PMD. We find that the tunnel-exit position depends on the transverse velocity of the tunneling electron. This gives rise to a momentum-dependent attoclock shift, meaning that the momentum shift due to the Coulomb force on the outgoing electron depends on which slice of the momentum distribution is analysed. Our finding is supported by a momentum-space-based implementation of the classical backpropagation method.
... The electrons from direct tunneling ionization of molecules have a near-Gaussian distribution peaking around 20 eV, as shown by the red line in Fig. 3(b). It agrees well with the electron spectrum coincident with CO þ obtained from single ionization (gray line), and the distribution is also well reproduced by classical trajectory Monte Carlo (CTMC) calculation (blue line) [42,43]. In contrast, the electrons from SFAI peaks around ∼1 eV and extends to ∼30 eV, as shown by the black line in Fig. 3(b). ...
... In contrast, the influence of polarization dominates when the high energy SFAI electron is released in a strong field (close to the peak of the laser pulse), and a double-peak structure (around AE90°) of angular distribution appears. In addition, for the energy below ∼3 eV, the angle of the single peak is distorted strongly by the molecular potential to ∼40°which is approximately equal to the offset angle at low laser intensity in attoclock measurement [42,57]. The above analyses prove the SFAI occurs in the falling edge of the laser pulse and demonstrate that sub-cycle ac-Stark effect steers the ejection of electrons by modifying the effective resonance width and the lifetime of the autoionizing state, which depends closely on the intensity of the laser field and the angle between the laser field with the molecular axis. ...
The novel strong field autoionization (SFAI) dynamics is identified and investigated by channel-resolved angular streaking measurements of two electrons and two ions for the double-ionized CO. Comparing with the laser-assisted autoionization calculations, we demonstrate the electrons from SFAI are generated from the field-induced decay of the autoionizing state with a following acceleration in the laser fields. The energy-dependent photoelectron angular distributions further reveal that the subcycle ac-Stark effect modulates the lifetime of the autoionizing state and controls the emission of SFAI electrons in molecular frame. Our results pave the way to control the emission of resonant high-harmonic generation and trace the electron-electron correlation and electron-nuclear coupling by strong laser fields. The lifetime modulation of quantum systems in the strong laser field has great potential for quantum manipulation of chemical reactions and beyond.
... The photoelectron momentum distribution (PMD) in such a field configuration is particularly simple. It can be simulated by various simplified, but more physically transparent, techniques such as an analytic R-matrix theory (Torlina et al 2015), a classical back-propagation analysis (Ni et al 2016), classical-trajectory Monte Carlo simulations (Liu et al 2017), a classical Rutherford scattering model (Bray et al 2018) and the strong field approximation (SFA) implemented within the saddle point method (SPM) . By making comparisons with these models, the numerical attoclock firmly points to a vanishing tunneling time (Torlina et al 2015, Ni et al 2016, Bray et al 2018. ...
... So the basic premise of the KR model remains valid. To test this model for other atoms, an extended set of numerical attoclock simulations by Liu et al (2017) was chosen. These simulations are conducted using the CTMC method. ...
Attosecond angular streaking, also known as the ‘attoclock’, employs a short elliptically polarized laser pulse to tunnel ionize an electron from an atom or a molecule and to put a time stamp on this process by deflecting the photoelectron in the angular spatial direction. This deflection can be used to evaluate the time the tunneling electron spends under the classically inaccessible barrier and to determine whether this time is finite. In this review, we examine the latest experimental and theoretical findings and present a comprehensive set of evidence supporting the zero tunneling time scenario.
... The photoelectron momentum distribution (PMD) in such a field configuration is particularly simple. It can be simulated by various simplified, but more physically transparent, techniques such as an analytic R-matrix theory (Torlina et al 2015), a classical back-propagation analysis , classical-trajectory Monte Carlo simulations (Liu et al 2017), a classical Rutherford scattering model (Bray et al 2018) and the strong field approximation (SFA) implemented within the saddle point method (SPM) (Serov et al 2019). By making comparisons with these models, the numerical attoclock firmly points to a vanishing tunnelling time (Torlina et al 2015, Ni et al 2016, Bray et al 2018. ...
... So the basic premise of the KR model remains valid. To test this model for other atoms, an extended set of numerical attoclock simulations (Liu et al 2017) was chosen. These simulations are conducted using the CTMC method. ...
Attosecond angular streaking, also known as the "attoclock", employs a short elliptically polarized laser pulse to tunnel ionize an electron from an atom or a molecule and to put a time stamp on this process by deflecting the photoelectron in the angular spatial direction. This deflection can be used to evaluate the time the tunneling electron spends under the classically inaccessible barrier and to determine whether this time is finite. In this review, we examine the latest experimental and theoretical findings and present a comprehensive set of evidence supporting the zero tunneling time scenario.
... First, we consider a circularly polarized (CP) field (ξ = 1), used for attoclock measurements [165,106]. For ellipticity close to circular, the initial drift momentum is large and the electron moves away from the ionic core quickly. ...
... where P = P x e x + P y e y + P z e z is the final momentum of the T-trajectory. In order to see the Coulomb asymmetry in a PMD from a CP field, a short laser pulse has to be used [165,106] (see Fig. 3.2 for the PMD in CP fields for short laser pulses). Otherwise, the PMD would resemble to a ring around the origin. ...
Subjecting atoms or molecules to intense laser pulses gives rise to a variety of highly nonlinear phenomena, such as for instance the ionization of electrons and the radiation of high-frequency photons. The distributions of the velocity of the ionized electrons or the frequency of the radiated photons measured at the detector encode relevant informations on the target atoms and molecules at the natural time scale of the electrons, the attosecond–that is, million, million, millionths of a second. Understanding the dynamics of the ionized electrons and identifying the mechanisms of high-frequency radiation are essential steps toward interpreting and decoding the informations encrypted in the experimental measurements.In this thesis, atoms subjected to intense and elliptically polarized laser fields in the infrared regime are theoretically studied. Despite their fundamental quantal nature in atoms, electrons display some classical behaviors when subjected to intense laser pulses. We exploit these classical features to understand and picture, with the help of trajectories, the physical mechanisms at play in order to interpret experimental measurements. We show the interdependent role of the quantum tunnel ionization of the electron and its subsequent classical motion for interpreting measurements in attosecond science.After tunnel ionization of the electrons, the interplay between their interactions with the laser and their parent ion, by yielding their dynamics highly nonlinear, gives rise to rich and diverse ionization channels. Changing the ellipticity of the driving laser, which acts as a simple control knob in experiments, changes the prioritized ionization channel taken by the electrons. In this way, for instance, the electrons can probe different characteristics of the target atoms. The motion of the ionized electrons is analyzed using pertur- bative and nonperturbative techniques from nonlinear dynamics and Hamiltonian systems. This thesis work demonstrates the complementarity of quantum mechanics and nonlinear dynamics for understanding and illustrating the mechanisms involved when atoms are subjected to intense and elliptically polarized laser pulses.
... It is quite remarkable that this formula closely follows the results of the numerical 3D-TDSE calculations when we use the distance to the tunnelling exit (Ip/E0 given by the Keldysh theory 22 ) as the value for the electron impact parameter and the final electron speed (A0 =E0/ω given by the classical simpleman's theory) as its speed at the detector v∞. It also qualitatively explains the Ip and ω dependence of the attoclock offset angles reported in a recent systematic numerical study 23 . This provides a strong indication that the electron offset angles measured in an attoclock experiment originate mostly (or entirely) from the scattering of the continuum electrons on the Coulomb potential of a proton and not from any tunnelling time delays. ...
Tunnelling, one of the key features of quantum mechanics, ignited an ongoing debate about the value, meaning and interpretation of 'tunnelling time'. Until recently the debate was purely theoretical, with the process considered to be instantaneous for all practical purposes. This changed with the development of ultrafast lasers and in particular, the 'attoclock' technique that is used to probe the attosecond dynamics of electrons. Although the initial attoclock measurements hinted at instantaneous tunnelling, later experiments contradicted those findings, claiming to have measured finite tunnelling times. In each case these measurements were performed with multi-electron atoms. Atomic hydrogen (H), the simplest atomic system with a single electron, can be 'exactly' (subject only to numerical limitations) modelled using numerical solutions of the 3D-TDSE with measured experimental parameters and acts as a convenient benchmark for both accurate experimental measurements and calculations. Here we report the first attoclock experiment performed on H and find that our experimentally determined offset angles are in excellent agreement with accurate 3D-TDSE simulations performed using our experimental pulse parameters. The same simulations with a short-range Yukawa potential result in zero offset angles for all intensities. We conclude that the offset angle measured in the attoclock experiments originates entirely from electron scattering by the long-range Coulomb potential with no contribution from tunnelling time delay. That conclusion is supported by empirical observation that the electron offset angles follow closely the simple formula for the deflection angle of electrons undergoing classical Rutherford scattering by the Coulomb potential. Thus we confirm that, in H, tunnelling is instantaneous (with an upperbound of 1.8 as) within our experimental and numerical uncertainty.
... We conduct the socalled "numerical attoclock experiment" by solving the time-dependent Schrödinger equation (TDSE) driven by a short single-cycle circularly polarized laser pulse. Such simulations on atomic hydrogen have been conducted previously [5,6,10,13,14]. They showed their great utility by allowing for a very simple analysis and transparent physical interpretation. ...
We conduct a series of accurate numerical simulations of attosecond angular streaking by solving the time-dependent Schr\"odinger equation for atomic and molecular hydrogen driven by a single oscillation circularly polarized laser pulse. We compare results of these simulations, which we term "numerical attoclock", with predictions of a semiclassical model based on the saddle point method. This comparison allows us to highlight the essential physics behind the attoclock measurements. In particular, we further validate the Keldysh-Rutherford model of the attoclock [Phys. Rev. Lett 121, 123201 (2018)]. In molecular hydrogen, we observe a strong dependence of the width of the attoclock angular peak on the molecular orientation and attribute it to the two-center electron interference.
