ECAMP 15

Geometric magnetism addresses the geometric origin of enantio-sensitive observables in one or multiphoton ionization from the emergence of the propensity field $\vec{B}_{\vec{k}}$ [1]. We extend this approach to spin-resolved one-photon ionization, i.e., $\vec{B}_{\vec{k},\mu}=i\vec{D}_{\vec{k},\mu}^*\times\vec{D}_{\vec{k},\mu}$, where $\vec{D}_{\vec{k},\mu}$ is the spin-resolved transition dipole with $\mu=\pm\frac{1}{2}$. Its respective net value on the energy shell (also known as \textit{curvature}) is $\vec{\Omega}_{\mu}=\int d\Theta_k\vec{B}_{\vec{k},\mu}$, where $d\Theta_k$ corresponds to averaging over the orientations of photoelectron momentum $\vec{k}$ for fixed photoelectron energy $k^2/2$. It will be convenient to define the spin symmetric and antisymmetric vector quantities as follows: $\vec{A}_{\pm}=(\vec{A}_{\frac{1}{2}}\pm\vec{A}_{-\frac{1}{2}})/2$.

For one-photon ionization of an isotropic ensemble of chiral molecules by circularly polarized light $\vec{E}=E_\omega(\hat{x}+i\xi\hat{y})/\sqrt{2}$, we find that the enantio-sensitive orientation of a molecular cation $\vec{V}$ (e.g. permanent dipole) is locked to the spin of the detected electron $\hat{s}$. Specifially, the molecular vector orients itself along the direction orthogonal to both photoelectron spin $\hat{s}$ and photon spin $\hat{z}$:
\begin{equation}
\langle \vec{V} \rangle = \frac{\xi| E _ {\omega} |^2} {12} [(\hat{\sigma} _ {\frac{1}{2}}\times\vec{\Omega} _ -)\cdot\vec{V}] (\hat{s}\times\hat{z}),
\end{equation}
wherein the pseudoscalar $[(\hat{\sigma}_{\frac{1}{2}}\times\vec{\Omega}_-)\cdot\vec{V}]$ has opposite signs for opposite enantiomers. This suggests that if opposite enantiomers were oriented in the same way, then their spins would have opposite orientations, hence, equivalently oriented left and right enantiomers ionized by circularly polarized light should result in the ejection of photoelectrons with opposite spins -- an effect which could be regarded as one of the manifestations of chirality induced spin selectivity [2].

We also show that the net photoelectron current acquires and enantiosensitive component in the plane of polarization of light, i.e.,
\begin{equation}
\vec{j} =\vec{j} _ 0\ + \left{ \frac{\xi|E _ {\omega}|^2}{12} \int d\Theta_k (\hat{\sigma} _ {\frac{1}{2}} \cdot \vec{B} _ {\vec{k},-}^\perp) \right} \left( \hat{s} \times \hat{z} \right)
\end{equation}
where, $ \vec{j}_{0}$ is the PECD (photoelectron circular dicrhoism) current and $\vec{B}_{\vec{k},-}^\perp = \vec{k}\times\vec{B}_{\vec{k},-}$. The second term of $\vec{j}$ is a spin polarization vortex which rotates in opposite direction for opposite enantiomers. This observable arises from the ``coupling'' of the \textit{propensity field} to spin, and can lead to high spin polarization even for very small spin-orbit interaction. This current reproduces earlier predictions of Ref. [3].

Our results are then illustrated for synthetic chiral matter. We construct chiral superpositions of electronic states in Argon, and perform \textit{ab initio} simulations of its spin dynamics [4].

[1] A F Ordonez \textit{et al.}, Comm Phys \href{https://www.nature.com/articles/s42005-023-01358-y} {{\bf 6}, 257} (2023).
[2] F Evers \textit{et al.}, Adv Mater \href{https://advanced.onlinelibrary.wiley.com/doi/10.1002/adma.202106629} {{\bf 34}, 2106629} (2022).
[3] N A Cherepkov, J Phys B \href{https://iopscience.iop.org/article/10.1088/0022-3700/16/9/013/meta}{{\bf 16}, 1543} (1983).
[4] S Carlstr\"om \textit{et al.}, Phys Rev A \href{https://journals.aps.org/pra/abstract/10.1103/PhysRevA.106.042806}{{\bf 106} 042806} (2022).