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Physical Review A

Effect of spin-orbit coupling in laser-induced ionization of atoms

Yibo Hu, Kunlong Liu, Klaus Renziehausen, Yidian Tian, Qingbin Zhang, Min Li, Yueming Zhou, Peixiang Lu

Published Aug 8, 2023• 5 citations
Publisher

Abstract

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 $p$ 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.

DOI
10.1103/physreva.108.023113

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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↑{p}_{+}^{\ensuremath{\uparrow}}p+\ensuremath↑​ and p+\ensuremath↓{p}_{+}^{\ensuremath{\downarrow}}p+\ensuremath↓​ (as well as the p\ensuremath−\ensuremath↑{p}_{\ensuremath{-}}^{\ensuremath{\uparrow}}p\ensuremath−\ensuremath↑​ and p\ensuremath−\ensuremath↓){p}_{\ensuremath{-}}^{\ensuremath{\downarrow}})p\ensuremath−\ensuremath↓​) 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.
Abstract Spin-dependent effects in atomic processes can be caused by the spin–orbit interaction or/and by the requirement that the wave function of identical electrons is antisymmetric. Such effects are usually neglected in strong-field physics. We show two examples, supported by theoretical results and numerical calculations, in which these effects are important. The first one is based on strong-field ionization of Xe atoms by a bicircular field. The corresponding momentum distribution of spin-polarized electrons emitted in the above-threshold ionization process exhibits wavelength-dependent fast oscillations. For longer wavelengths and small electron emission angle there is a wide photoelectron kinetic energy region in which the spin asymmetry parameter changes continuously from large positive to large negative values. In addition, the emission time of such electrons in high-order above-threshold ionization is determined on the attosecond time scale. The second process considered is high-order above-threshold ionization of excited Li + ions. We have found that in this case, even in the absence of spin–orbit coupling, the photoelectron momentum distribution strongly depends on the initial spin state. In the singlet state we have characteristic minima, which are caused by the destructive interference of the direct and exchange rescattering amplitudes. Such minima are absent in the triplet state.
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.
We derive the analytical formula of the ratio of the ionization rates of degenerate valence ${\ensuremath{\pi}}_{\ifmmode\pm\else\textpm\fi{}}$ orbitals of prealigned linear molecules in strong circularly polarized (CP) laser fields. Interestingly, our theory shows that the ionization ratio for molecular orbitals with opposite azimuthal quantum numbers $\ifmmode\pm\else\textpm\fi{}|m|$ (e.g., ${\ensuremath{\pi}}_{\ifmmode\pm\else\textpm\fi{}}$) is identical to that for atomic orbitals with the same $\ifmmode\pm\else\textpm\fi{}|m|$ (e.g., ${p}_{\ifmmode\pm\else\textpm\fi{}}$). In general, the electron counter-rotating to the CP laser field tunnels more easily, not only for atoms but also for linear molecules. Our theoretical predictions are then verified by numerically solving the three-dimensional time-dependent Schr\"odinger equation for the ionization of the prealigned nitric oxide (NO) molecule in strong CP laser fields. Due to the spin-orbital coupling in the electronic ground state of NO and the sensitivity of ionization to the sense of electron rotation, the ionization of NO in CP fields can produce spin-polarized photoelectrons with high controllability of spin polarization up to 100100%100.
The energy levels of atoms and molecules exposed to intense, short, high-frequency laser pulses undergo time-dependent distortion known as dynamic Stark shift (DSS). These level shifts are induced by the coupling with nonresonant (nonessential) states and follow the temporal intensity profile of the laser pulse. Owing to the different DSSs of the individual atomic levels, transient resonance suppressions and enhancements significantly modify the multiphoton ionization pathway, leaving clear fingerprints (asymmetry, splitting, and shifting) in the energy spectrum of emitted photoelectrons. Here we investigate this phenomenon by solving the time-dependent Schr\"odinger equation of the valence electron of Na and Li in increasing levels of complexity: (i) developing minimal models for two-photon transition and (2+12+12+1)-photon ionization, (ii) applying a spectral method without continuum-continuum couplings, and (iii) propagating the electron wave packet on a large spatial grid accounting for all possible couplings. We show that appropriately detuned transform limited pulses provide a high level of control in strong-field excitation, when the atomic levels are dynamically shifted. Furthermore, we demonstrate the role of DSSs in the multipeak structure of the Autler-Townes doublet, found when several Rabi oscillations are induced during the strong-field resonant ionization.
Via a non-perturbative method we study the population dynamics and photoelectron spectra of Cs atoms subject to intense chirped laser pulses, with Gaussian beams. We include above threshold ionization spectral peaks. The frequency of the laser is near resonance with the 6s-7p transition. Dominant couplings are included exactly, weaker ones accounted for perturbatively. We calculate the relevant transition matrix elements, including spin-orbit coupling. The pulse is taken to be a hyperbolic secant in time and the chirping a hyperbolic tangent. This choice allows the equations of motions for the probability amplitudes to be solved analytically as a series expansion in the variable , where is a measure of the pulse length. We find that the chirping changes the ionization dynamics and the photoelectron spectra noticeably, especially for longer pulses of the order of au. The peaks shift and change in height, and interference effects between the 7p levels are enhanced or diminished according to the amount of chirping and its sign. The integrated ionization probability is not strongly affected.