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New Journal of Physics

Effect of spin polarization on tunneling delay in attoclock settings

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

Published Mar 1, 2026• 1 citation
Publisher

Abstract

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.

DOI
10.1088/1367-2630/ae4f1a

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Physical Review Letters•Oct 18, 2022•23 citations
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.
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.
Tunnelling lies at the heart of quantum mechanics and is a fundamental process in attosecond science, molecular biology, and quantum devices. Whether tunnelling takes time and how a microscopic particle transits through a barrier have been debated since the early days of quantum mechanics. The time required for an electron to tunnel through an atomic potential barrier has been measured with attosecond angular streaking (attoclock), and a recent work on the hydrogen atom claimed that electron tunnelling is instantaneous. However, the time required for Rb atoms to tunnel through an optical potential barrier has been measured to be on the order of milliseconds with a recent Larmor clock measurement. The essence of electron and atom tunnelling is identical, but the reason for the contradictory conclusions remains unknown. Here, we demonstrate that the sub-barrier potential interaction is the root of the nonzero tunnelling delay time. We reveal that the wave-particle duality of a tunnelling electron must be fully taken into account when decoding the tunnelling delay time from the attoclock measurements. Based on energy-resolved attoclock measurements, we show that the tunnelling delay time of an electron ranges from 24 to 58 attoseconds, and the counterintuitive result that an electron with a lower energy may spend less time in the barrier is consistent with the velocity-dependent tunnelling time of atoms in Larmor clock measurements. Our results unify the tunnelling time of microscopic particles by highlighting the classically forbidden sub-barrier potential interactions of matter waves.
Attosecond chronoscopy represents a major breakthrough in the study of ultrafast phenomena and has the potential to revolutionize our understanding of the fundamental physics of matter. We theoretically investigate the spin-orientation-resolved attosecond chronoscopy for the first time by the circular RABBIT technique for Kr atoms. Due to the spin-orbit interaction and the sensitivity of ionization in circularly polarized fields to the sense of electron rotation in the initial state, the spin-resolved ionization rates of photoelectrons emitted from 2P1/2^2P_{1/2}2P1/2​ and 2P3/2^2P_{3/2}2P3/2​ channels can be expressed via mmm-resolved ionization rates of initial state, where mmm is the orbital magnetic quantum number. We demonstrate that the yields difference between spin-up and spin-down photoelectrons from each channel are closely associated with the different behaviors of corresponding Wigner time delay. We find that the Wigner time delay between spin-up and spin-down photoelectrons in the polarization plane can reach several tens of attoseconds in the co-rotating geometry, but a few attoseconds in the counter-rotating geometry. Our approach opens up a new avenue for probing the spin-dependent behavior of Wigner time delay, and lays the foundation for spin-orientation-resolved attosecond chronoscopy, which can be verified by the current experimental techniques.
The measurement of the tunneling time delay in attosecond experiments, termed attoclock, sparked a heated debate about tunneling time and the role of time in quantum mechanics. We show a model that explains the experimental result of the attoclock in the nonadiabatic field calibration, where we find good agreement with the experimental data of Hofmann et al. (2019). We confirm our result with the numerical integration of the time-dependent Schrödinger equation. We also found that at a field strength F≤Fa (Fa is the atomic field strength), the model always predicts a time delay with respect to the quantum (lower) limit τa at F=Fa. For an adiabatic tunneling, the saturation at the limit (F=Fa) explains the well-known Hartman effect or Hartman paradox.
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.
Attosecond time-resolved electron tunneling dynamics have been investigated by using attosecond angular streaking spectroscopy, where a clock reference to the laser field vector is required in atomic strong-field ionization and the situation becomes complicated in molecules. Here we reveal a resonant ionization process via a transient state by developing an electron-tunneling-site-resolved molecular attoclock in Ar-Kr^{+}. Two distinct deflection angles are observed in the photoelectron angular distribution in the molecular frame, corresponding to the direct and resonant ionization pathways. We find the electron is temporally trapped in the Coulomb potential wells of the Ar-Kr^{+} before finally releasing into the continuum when the electron tunnels through the internal barrier. By utilizing the direct tunneling ionization as a self-referenced arm of the attoclock, the time delay of the electron trapped in the resonant state is revealed to be 3.50±0.04 fs. Our results give an impetus to exploring the ultrafast electron dynamics in complex systems and also endow a semiclassical presentation of the electron trapping dynamics in a quantum resonant state.