Ultrafast Ionization Dynamics Encoded in a Photoelectron Spin Torus

Abstract

We demonstrate that strong-field ionization of atoms in circularly polarized laser fields generates a photoelectron spin texture with toroidal topology in momentum space. Using time-dependent Schrödinger equation simulations, spin-resolved classical-trajectory Monte Carlo calculations, and an extended spin-resolved strong-field approximation including intermediate excitation pathways, we show that the rotation angle of this spin torus provides access to attosecond relative time delays associated with photoelectron wave packets released by tunneling from the counter-rotating and co-rotating \(p\)-orbital channels. When intermediate-state dynamics become significant, the torus develops a clear splitting. These results establish photoelectron spin textures as a complementary source of dynamical information beyond conventional momentum spectroscopy, and identify spin polarization as a robust internal degree of freedom for self-referenced attosecond metrology.

Figures (6)

• Figure 1: Toroidal structure of the PST. (a) Three-dimensional visualization of the PST; green arrows indicate the spin-polarization direction, and the red curve marks the rotation direction of the laser electric field. (b1,b2) Spin polarization on two representative momentum-space cuts: (b1) the $x$--$y$ plane at $p_z=-0.02$ a.u. and (b2) the $x$--$z$ plane at $p_y=0$ a.u. The background colormap shows the corresponding PMD. Laser wavelength: 1030 nm; peak intensity: $10^{14}$ W/cm$^2$; number of optical cycles: $N=4$; averaged over the CEP.
• Figure 2: Origin of the spin torus and its connection to attoclock dynamics. (a) Phase difference between the ionization amplitudes $\chi^{(+)}$ and $\chi^{(0)}$. (b) Radial $(\zeta_r)$ and azimuthal $(\zeta_\phi)$ spin-polarization components. The SFA result is purely radial, whereas the TDSE result exhibits a finite azimuthal component. (c) Spin-rotation angle $\theta_s$ and Coulomb-induced angular shift $\Phi_c$ as functions of radial momentum $p_r$, obtained from TDSE and CTMC calculations. (d) Difference in $\theta_s$ between the TDSE and CTMC results, together with the corresponding eSFA contribution. Here, $\boldsymbol{e}_r=(\cos\phi_{\boldsymbol{p}},\sin\phi_{\boldsymbol{p}},0)$ and $\boldsymbol{e}_\phi=(-\sin\phi_{\boldsymbol{p}},\cos\phi_{\boldsymbol{p}},0)$ are the radial and azimuthal unit vectors, respectively. Laser parameters are the same as those in Fig. \ref{['FigPSTTorus']}.
• Figure 3: Splitting of the spin torus induced by intermediate excited-state dynamics. (a) Spin vortex in the $x$--$z$ plane; green arrows indicate the local spin-polarization direction. (b) Relative phase angles $\theta^{+}$ and $\theta^{-}$, together with the spin-rotation angle $\theta_s$, as functions of $p_r$. The shaded regions denote the partial photoelectron yields $|\chi^{(+)}|^2$ and $|\chi^{(-)}|^2$, plotted on the right axis. Laser wavelength: 400 nm; intensity: $I=10^{14}$ W/cm$^2$; number of optical cycles: $N=10$; CEP: $\Phi_{\rm CEP}=0$.
• Figure 4: Extraction of the relative emission time delay from the PST in (a) the low-energy region dominated by the counter-rotating channel, where the delay corresponds to the $(+,0)$ channels, and (b) the high-energy region dominated by the co-rotating channel, where the delay corresponds to the $(0,-)$ channels. Solid black curves show the exact results obtained directly from the TDSE wave functions, while dashed blue curves show the estimates from Eq. \ref{['eq:td_est']}. The shaded regions indicate the longitudinal spin polarization $\zeta_z$ (right axis). Laser parameters are the same as those in Fig. \ref{['FigPSTTorus']}, except that $\Phi_{\rm CEP}=0$.
• Figure 5: Average spin-rotation angle $\langle \theta_s \rangle$ and attoclock offset angle $\langle \theta_{\mathrm{atto}} \rangle$ as functions of the peak laser intensity. A finite mismatch between the two angles persists throughout the investigated intensity range. Laser wavelength: 1030 nm; number of optical cycles: $N=4$; CEP: $\Phi_{\rm CEP}=0$.