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Physical Review A

Relativistic spin-flip effect in the attoclock

Igor A. Ivanov, Anatoli S. Kheifets, Alexandra S. Landsman

Published Jul 18, 2025• 1 citation
Publisher

Abstract

We study the effect of spin flip on tunneling ionization in the attoclock settings. Our study is based on a fully relativistic approach in which we seek a numerical solution of the time-dependent Dirac equation. We find that spin flip during strong field ionization significantly increases the attoclock offset angle, indicating longer tunneling delay relative to electrons with no spin flip. To interpret our numerical findings, we conduct additional modeling within the perturbative Breit-Pauli approach, which allows us to elucidate the role of different relativistic mechanisms. We show that the spin-orbit interaction is mainly responsible for the spin flip, rather than interaction with the magnetic component of the laser field. We further discuss the dynamic mechanisms of the spin-flip effect, suggesting that it takes place while the electron is still bound. Our findings indicate that the attoclock can be used to investigate electron spin-flip dynamics during the tunnel ionization process.

DOI
10.1103/pfxy-2j96

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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

•Jan 1, 2014•0 citations
We study the effect of spin flip on tunneling ionization in the attoclock settings. Our study is based on a fully relativistic approach in which we seek a numerical solution of the time-dependent Dirac equation. We find that spin flip during strong field ionization significantly increases the attoclock offset angle, indicating longer tunneling delay relative to electrons with no spin flip. To interpret our numerical findings, we conduct additional modeling within the perturbative Breit-Pauli approach, which allows us to elucidate the role of different relativistic mechanisms. We show that the spin-orbit interaction is mainly responsible for the spin flip, rather than interaction with the magnetic component of the laser field. We further discuss the dynamic mechanisms of the spin-flip effect, suggesting that it takes place while the electron is still bound. Our findings indicate that the attoclock can be used to investigate electron spin-flip dynamics during the tunnel ionization process.
Abstract We investigate how spin polarization of the initial atomic state affects the tunneling delay in the attoclock settings. Using the time dependent Schrödinger equation and the Breit–Pauli Hamiltonian we demonstrate that spin polarization can produce measurable corrections to tunneling delays in attoclock spectroscopy. Our findings indicate that spin–orbit coupling introduces an additional spin-dependent degree of freedom into under-the- barrier motion, suggesting new opportunities for spin-resolved attosecond measurements. We show that the electron’s spin orientation during tunneling produces a contribution to the attoclock offset angle. Our results are interpreted within the framework of the strong field approximation and the imaginary time methods. We show that this approach provides a rare opportunity to get an insight into the details of the sub-barrier electron motion during the tunneling process.
Summary form only given. Spin plays a fundamental role in the electronic structure of matter, from single atoms and molecules through to condensed matter. However, spin dynamics and the spin response of electrons to ultra-short, strong laser pulses has remained largely unexplored until today. In 2011 and 2013, pioneering theoretical work [1, 2] extended the fundamental PPT theory [3] to treat ionization of p+ and p” electrons by circular fields. This work predicted the sensitivity of ionization probabilities in strong field ionization using circular fields to the magnetic quantum number, showing that counter rotating electrons can tunnel ionize easier. It also predicted the possibility to obtain spin polarized electrons from strong, circularly polarized fields with a high degree of efficiency. The preference of ionization was later confirmed experimentally by [4], however, the experiment in [4] did not provide a confirmation of the spin polarization predictions.
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.
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 experimental result in the nonadiabatic field calibration, where one reaches a good agreement with the experimental data of Hofmann et al. (J. of Mod. Opt. {\bf 66}, 1052, 2019). Furthermore, we confirm the result with the numerical integration of the time-dependent Schrödinger equation. The model is appealing because it offers a clear picture of the multiphoton and tunneling field-ionization regimes. In the nonadiabatic case (the nonadiabatic field calibration), the ionization is mainly driven by multiphoton absorption. Surprisingly, at a field strength F≤FaF \le F_aF≤Fa​ (FaF_aFa​ is the atomic field strength) the model always predicts a time delay with respect to the quantum limit τaτ_aτa​ at F=FaF=F_aF=Fa​. For an adiabatic tunneling the saturation at the limit (F=FaF=F_aF=Fa​) explains the well-known Hartman effect or Hartman paradox.
Based on the expansion of the time-dependent Dirac equation in the second order on the reciprocal of the speed of light, we derive a simplified second-order time-dependent Pauli equation (second-TDPE) for single-active-electron systems, in which the time-dependent spin-orbit interaction is taken into account. A numerical method is presented for calculating the observables for atoms interacting with the electric fields of laser pulses. It is shown that the energy-level splitting between the ${p}_{+}^{\ensuremath{\uparrow}}$ and ${p}_{+}^{\ensuremath{\downarrow}}$ (as well as the ${p}_{\ensuremath{-}}^{\ensuremath{\uparrow}}$ and ${p}_{\ensuremath{-}}^{\ensuremath{\downarrow}})$ orbitals can be reproduced by the spin-orbit coupling (SOC) term of the second-TDPE. By numerically solving the second-TDPE, we demonstrate the SOC effect in single-photon, multiphoton, and tunneling ionization of atomic Kr with valence ppp orbitals. In particular, a few counterintuitive strong-field ionization phenomena are discussed and analyzed. The present work provides a convenient way to explore the SOC effect in strong-field interactions.
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