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arXiv:2101.08096 · Physical Review A

Role of reflections in the generation of a time delay in strong-field ionization

Daniel Bakucz Canário, Michael Klaiber, Karen Z. Hatsagortsyan, Christoph H. Keitel

Published Sep 7, 2021• 7 citations
arXivHTMLPDF

Abstract

The problem of time delay in tunneling ionization is revisited. The origin of time delay at the tunnel exit is analyzed, underlining the two faces of the concept of the tunneling 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 wave function, exists as a matter of principle and arises due to the sub-barrier interference of the reflected and transmitted components of the tunneling electronic wave packet. We exemplify this by describing the tunneling ionization of an electron bound by a short-range potential within the strong-field approximation in a ``deep tunneling'' regime. If sub-barrier reflections are extracted from this wave function, then the time delay of the peak is shown to vanish. Thus, we assert that the disturbance of the tunneling wave packet by the reflection from the surface of the barrier causes a time delay in the neighborhood of the tunnel exit.

DOI
10.1103/physreva.104.033103

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The problem of time delay in tunneling ionization is revisited. The origin of time delay at the tunnel exit is analyzed, underlining the two faces of the concept of the tunneling 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 wave function, exists as a matter of principle and arises due to the sub-barrier interference of the reflected and transmitted components of the tunneling electronic wave packet. We exemplify this by describing the tunneling ionization of an electron bound by a short-range potential within the strong-field approximation in a ``deep tunneling'' regime. If sub-barrier reflections are extracted from this wave function, then the time delay of the peak is shown to vanish. Thus, we assert that the disturbance of the tunneling wave packet by the reflection from the surface of the barrier causes a time delay in the neighborhood of the tunnel exit.
Abstract Quantum tunnelling is a fundamental phenomenon in quantum physics, yet the question of how long it takes for a particle to tunnel through a barrier remains a topic of ongoing debate. Recent advances in attoclock experimental techniques in strong-field physics have renewed interest in this topic. While various definitions of tunneling time have been proposed, a consensus has yet to be reached. In this study, we adopt a definition of tunnelling time based on the ratio of the barrier width to the expected value of the velocity operator within the barrier. This approach reduces to the Büttiker-Landauer time (τBL\tau_{BL}τBL​) in the case of an opaque barrier and offers a clear physical interpretation for very thin barriers, where τBL\tau_{BL}τBL​ may not be well-defined. Additionally, by utilizing a virtual detector at the tunnel exit, we calculate the time delay between the peak of the electric field and the moment of ionization for long wavelengths ($\geq$6 μ\muμm). Our findings indicate that the time delay stabilizes around 190 as, remaining independent of the laser wavelength. Furthermore, through our quantum tunnelling time definition, this time delay can be interpreted as the time required for the electron to tunnel from a point within the barrier to the exit.
A clear consensus on how long it takes a particle to tunnel through a potential barrier has never been so urgently required, since the electron dynamics in strong-field ionization can be resolved on attosecond time-scale in experiment and the exact nature of the tunneling process is the key to trigger subsequent attosecond techniques. Here a general picture of tunneling time is suggested by introducing a quantum travel time, which is defined as the ratio of the travel distance to the expected value of the velocity operator under the barrier. Specially, if applied to rectangular barrier tunneling, it can retrieve the Büttiker-Landauer time τBLτ_{BL}τBL​ in the case of an opaque barrier, and has a clear physical meaning in the case of a very thin barrier wherein τBLτ_{BL}τBL​ can not be well defined. In the case of strong-field tunneling process, with the help of the newly defined time, the tunneling delay time measured by attoclock experiment can be interpreted as a travel time spent by the electron to tunnel from a point under the barrier to the tunnel exit. In addition, a peculiar oscillation structure in the wavelength dependence of tunneling delay time in deep tunneling regime is observed, which is beyond the scope of adiabatic tunneling picture. This oscillation structure can be attributed to the interference between the ground state tunneling channel and the excited states tunneling channels.
A numerical solution to the time evolution equation of the Wigner distribution function (WDF) with an accuracy necessary to simulate the passage of a wave packet past a barrier is developed, where quantum effects require high accuracy and fine discretization. A wave packet incident on a barrier, a portion of which tunnels through, demonstrates behavior that can define various characteristic transmission and reflection delay (TARD) times useful in the simulation of electron emission. A model for the TARD times is proposed that relies only on the asymptotic maxima of the position $\ensuremath{\rho}(x,t)$ and wave number $\ensuremath{\rho}(k,t)$ densities given by the WDF and applied to a ballistic trajectory model for the (faster) transmitted and (slower) reflected parts. The dependence of the TARD times on barrier width, symmetry, and abruptness is analyzed. For symmetrical barriers with characteristics similar to field emission barrier heights and widths, TARD times are on the order of a fraction of a femtosecond. The TARD times for when tunneling predominates are contrasted to tunneling times in the literature. Use of the TARD times in simulations of field emission in nanogap devices or to model ultrashort pulses generated under rapidly changing conditions for electron sources are proposed.
Tunnel-ionization is investigated in the framework of relativistic quantum mechanics. \nFor an arbitrary constant electromagnetic field a gauge invariant energy \noperator is introduced in order to identify the classically forbidden region for \ntunnel-ionization. Furthermore, relativistic features of tunnel-ionization are explored. \nA one-dimensional intuitive picture predicts that the ionized electron wave \npacket in the relativistic regime experiences a momentum shift along the laser’s \npropagation direction. This is shown to be consistent with the well-known strong \nfield approximation. Furthermore, spin dynamics in tunnel-ionization process \nis discussed in the standard as well as in the dressed strong field approximation. \nNext, the tunneling time delay is investigated for tunnel-ionization by extending \nthe definition of the Wigner time delay. Later, this concept is redefined in terms of \nthe phase of the fixed energy propagator. The developed formalism is applied to \nthe deep-tunneling and the near-threshold-tunneling regimes. It is shown that in \nthe latter case signatures of the tunneling time delay can be measurable at remote \ndistance. Finally, the path-dependent formulation of gauge theory is discussed. It \nis demonstrated that this equivalent formulation of gauge theory leads to a canonical \ngauge fixing, in which the Feynman path integral becomes more intuitive and \nthe calculation of the quasiclassical propagator is considerably simplified
Laser-induced electron tunneling ionization from atoms and molecules plays as the trigger for a broad class of interesting strong-field phenomena in attosecond community. Understanding the time of electron tunneling ionization is vital to achieving the ultimate accuracy in attosecond metrology. We propose a novel attosecond photoelectron interferometer, which is based on the interference of the direct and near-forward rescattering electron wave packets, to determine the time information characterizing the tunneling process. Adding a weak perturbation in orthogonal to the strong fundamental field, the phases of the direct and the near-forward rescattering electron wave packets are modified, leading to the shift of the interferogram in the photoelectron momentum distributions. By analyzing the response of the interferogram to the perturbation, the real part of the ionization time, which denotes the instant when the electron exits the potential barrier, and the associated rescattering time are precisely retrieved. Moreover, the imaginary part of the ionization time, which has been interpreted as a quantity related to electron motion under the potential barrier, is also unambiguously determined.
The problem of time delay in one-dimensional quantum-mechanical scattering by a potential barrier is studied in the framework of time-dependent scattering theory. It is shown how the concept of sojourn time can be used to define three separate time delays: the time delay for transmission through the barrier, the time delay for reflection, and the total time delay (i.e., averaged over transmission and reflection). In addition to the anticipated dependence on the S matrix and its energy derivative, the time delays depend on certain parameters which can be interpreted as describing positions of detectors in a gedanken-experiment measurement. The problem of defining separate time delays for transmission and reflection appears to be rooted in the foundations of quantum theory and its interpretation.