Atomic Spins Form Torus Revealing Attosecond Time Delays

A new method for measuring attosecond time delays using the spin of electrons ejected from atoms has been revealed. Xiaodan Mao and colleagues at Shanghai Jiao Tong University show that strong-field ionization with circularly polarized laser light creates a unique, toroidal spin texture in the momentum of photoelectrons. Their simulations and calculations demonstrate this spin texture encodes information about the timing of electron release from different atomic orbitals, offering a new approach to attosecond metrology that uses spin polarization as an internal reference.

Photoelectron spin textures are a valuable addition to conventional momentum spectroscopy, potentially advancing our understanding of ultrafast atomic processes. The rotation angle of a spin torus provides access to attosecond relative time delays associated with photoelectron wave packets released by tunnelling 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 strong internal degree of freedom for self-referenced attosecond metrology. In strong-field physics, the photoelectron momentum distribution (PMD) is a key probe of ionization dynamics, with a prominent example being the attoclock technique

The angular offset of Photoelectron Momentum Distributions (PMDs) generated by elliptically polarized light relates to an ionization time delay. This extracted time delay is inherently ambiguous, as the same angular offset also contains Coulomb-induced deflections accumulated during continuum propagation. Disentangling these contributions remains a central challenge in attosecond metrology. Electron spin offers a route around this ambiguity. For circularly polarized pulses, preferential ionization from counter-rotating p orbitals produces circular dichroism, which, together with bound-state spin-orbit coupling (SOC), gives rise to substantial longitudinal spin polarization.

Although continuum SOC is relativistically weak under typical strong-field ionization conditions, it can become relevant near the Cooper minimum in single-photon ionization, when correlated ionic dynamics are resolved, or when rescattering electrons enter the weakly relativistic regime. Consequently, for the nonrelativistic atomic strong-field ionization considered here, the Coulomb field strongly reshapes the outgoing momentum distribution while leaving the spin polarization along each trajectory largely intact. The resulting momentum-resolved spin polarization, known as the photoelectron spin texture (PST), therefore provides a complementary dynamical observable that, when combined with the PMD, forms a self-referenced probe of strong-field ionization dynamics.

Recent work has demonstrated that PSTs can exhibit vortex structures under linearly polarized driving, suggesting that topologically nontrivial PSTs exist more generally. This possibility is particularly intriguing in view of the broader importance of polarization textures in condensed-matter and optical systems, where topological durability against smooth deformations underpins a wide range of transport, magnetic, and optical phenomena. Investigations reveal whether such topologically nontrivial PSTs arise in circularly polarized fields by combining time-dependent Schrödinger equation (TDSE) simulations, spin-resolved classical-trajectory Monte Carlo (CTMC) calculations, and an extended spin-resolved strong-field approximation (eSFA) that includes intermediate excitation pathways.

A circularly polarized laser pulse induces strong-field ionization, producing a PST with toroidal topology in three-dimensional momentum space, which researchers refer to as a spin torus. The rotation angle of this torus, referenced to the attoclock offset angle, provides a self-referenced probe of relative ionization time delays, on the order of attoseconds, between wave packets emitted from the counter-rotating and co-rotating p-orbital channels. As a representative system, the tunneling ionization of Xe driven by a circularly polarized laser field in the x, y plane, A(t) = −A0 sin(ωt + ΦCEP), cos(ωt + ΦCEP), 0, is considered, where f(t) = sin2[ωt/(2N)] is the pulse envelope, N the number of optical cycles, ω the laser frequency, and ΦCEP the carrier-envelope phase (CEP). Owing to the relatively low ionization potential of the valence shell, the photoelectron signal is dominated by ionization from the outer 5p shell.

SOC splits this shell into the 5p1/2 and 5p3/2 sublevels; under the present conditions, the ionization signal is dominated by the 5p3/2 channel. Continuum SOC contributes only at relativistic order O(1/c2) and is neglected at leading order, an approximation further justified by the strong suppression of electron recolli-sion in long-wavelength circularly polarized fields. Under these conditions, the PST reduces to bilinear combinations of the p-orbital ionization amplitudes.

TDSE simulations using the Tong, Lin effective potential yield the PST shown. The PMD forms a toroidal support in momentum space on which the PST is defined. Representative cuts of the PST are displayed. Spin polarization points along the laser-propagation direction at pz = 0, with vanishing transverse components. Because the centroid of the PMD does not coincide with the spin nodal point, defined by ζ(p) = 0, the distribution acquires a net longitudinal polarization.

For finite pz, transverse components emerge: the polarization winds clockwise for pz > 0. In the x, z plane, the projected spin texture forms a vortex centred at the nodal point. The phase difference between the ionization amplitudes χ(+) and χ is shown, with the strong-field approximation result being purely radial, whereas the TDSE result exhibits a finite azimuthal component. CTMC simulations show that classical Coulomb drift in the continuum induces a finite spin rotation, in good agreement with the momentum offset angle.

Strong-field ionization of atoms in circularly polarized laser fields generates a photoelectron spin texture with toroidal topology in momentum space. Time-dependent Schrödinger equation simulations, spin-resolved classical-trajectory Monte Carlo calculations, and an extended spin-resolved strong-field approximation incorporating intermediate excitation pathways demonstrate that the rotation angle of this spin torus provides access to attosecond relative time delays associated with photoelectron wave packets released 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 strong internal degree of freedom for self-referenced attosecond metrology.

Toroidal mapping of electron spin enables self-referencing in attosecond time measurements

Researchers are refining the attosecond stopwatch, seeking ever more precise ways to measure the staggeringly short bursts of time governing electron behaviour. This new technique, mapping the spin of ejected electrons into a toroidal shape, offers a self-referencing method, sidestepping ambiguities that plague conventional approaches like attosecond streaking. By creating a self-referencing system through this mapping, the intrinsic time delays governing electron behaviour can be isolated, promising more accurate measurements at the attosecond scale.

A previously unseen toroidal pattern in the momentum of electrons ejected from atoms is revealed by mapping their spin, offering a new way to measure attosecond time differences. This “spin torus” arises when atoms are ionized using circularly polarized laser light, creating a distinct, measurable rotation linked to the timing of electron release from different atomic orbitals. In particular, this technique provides a self-referencing method, minimising errors caused by the electron’s subsequent movement away from the atom, a challenge for conventional methods.

Researchers demonstrated that strong-field ionization of atoms using circularly polarized laser fields generates a unique photoelectron spin texture with a toroidal shape. The rotation angle of this spin torus provides access to attosecond relative time delays, allowing for more precise measurements of electron behaviour. This method offers a self-referencing approach, improving upon conventional momentum spectroscopy by isolating intrinsic time delays and minimising measurement errors. The authors suggest this establishes photoelectron spin textures as a valuable source of dynamical information.