Spin-cavity interactions in relativistic Jahn-Teller systems under strong light-matter coupling

Abstract: We extend our recent work on the cavity-modified spin Zeeman effect of an effective spin-1/2-system[J. Chem. Phys. 163, 174307 (2025)] to a relativistic Jahn-Teller scenario under strong light-matter coupling. Here, the effective spin-1/2-system is realized via a single electron or a single hole in a doubly-degenerate molecular orbital system of trigonal symmetric transition metal complexes. Both single-particle and single-hole systems are subject to both vibronic and spin-orbit coupling (SOC) augmented by interactions with a quantized cavity field via the cavity Zeeman interaction. Methodologically, we combine the relativistic $E\times e$-Jahn-Teller model with a recently introduced effective Hamiltonian formalism based on quasi-degenerate perturbation theory, which treats the cavity-spin interaction in leading order beyond the dipole approximation. We derive analytic expressions for Kramers pair energies in weak and strong SOC regimes as well as related cavity-modified effective electronic g-factors. We find cavity-induced modifications of the electronic g-factor to become relevant in the weak SOC regime for both single-particle and single-hole systems while being effectively quenched under strong SOC. Alternating signs of the cavity-Zeeman correction render single-particle and single-hole scenarios distinct in their response to the cavity field from a g-factor perspective.

• The paper demonstrates that cavity-induced modifications to effective g-factors are strongly dependent on the interplay of spin-orbit and vibronic couplings.
• It employs an augmented Pauli-Fierz Hamiltonian with a z-polarized cavity mode and quasi-degenerate perturbation theory for analytic solutions.
• Numerical analysis reveals that observable spin-cavity effects emerge under weak SOC, guiding experimental strategies in molecular spintronics.

Spin-Cavity Interactions in Relativistic Jahn-Teller Systems Under Strong Light-Matter Coupling

Introduction

This study develops a comprehensive theoretical framework for describing spin-cavity interactions in molecular systems that are both relativistically affected (via spin-orbit coupling, SOC) and exhibit vibronic coupling as per the $E \times e$ Jahn-Teller (JT) effect. It critically extends earlier work that characterized the cavity-modified Zeeman effect on a single-electron spin [fischer2025zeeman] by incorporating realistic molecular JT scenarios, treating both “single-particle” (one electron in a doubly degenerate orbital) and “single-hole” systems (one hole in such an orbital), as realized in trigonal transition metal complexes, notably Mo(III) and Mo(V) systems. The principal innovation is the explicit modeling of the cavity Zeeman interaction’s influence on effective electronic $g$-factors in both weak and strong SOC regimes, illuminating how the magnetic component of quantized cavity fields perturbs spin properties beyond the dipole approximation.

Theoretical Model

The relativistic JT Hamiltonian (RJT) for each molecular motif consists of SOC, Zeeman, and vibronic coupling terms. For both single-particle and single-hole systems, the degenerate $^2E$ ground state provides a platform for nonadiabatic (JT) distortions and SOC. Crucially, SOC introduces either antiferromagnetic or ferromagnetic spin-orbital interaction depending on the electron configuration: antiferromagnetic for single-particle, ferromagnetic for single-hole.

The canonical Pauli-Fierz Hamiltonian is augmented with a $z$-polarized quantized cavity mode, capturing the leading-order cavity Zeeman interaction: $$\hat{H}_\mathrm{cZee} = i \frac{g_0 g_e\mu_B}{2c} \sqrt{\frac{\hbar\omega_c}{2}} \sigma_y \left(\hat{b}_z^\dagger - \hat{b}_z \right),$$ where the cavity-induced spin-photon coupling allows spin-flip transitions via the creation or annihilation of a cavity photon, yielding spin-polariton formation. This term fundamentally arises beyond the dipole approximation, thus including magnetic field effects neglected in standard cavity quantum chemistry.

Vibronic and spin-cavity Hamiltonians are decomposed into block-diagonal “polariton” and “spectator” manifolds, reflecting both the selection rules of photon-mediated spin transitions and the absence of inter-manifold coupling. This structure is central to the analytic tractability and the subsequent application of quasi-degenerate perturbation theory (QDPT).

Analytic Solution and Key Results

Exact diagonalization within the lower-dimensional (3D) truncated manifolds—where the high-lying states are well separated—provides a reliable route to the energies of the Kramers pairs under weak external $B_z$-fields. The analytic expressions for the effective $g$-factor corrections in various coupling limits are central outcomes.

For weak SOC ($\xi \ll F\rho$), the cavity-modified effective $g$-factor is: $$\tilde{g}_\mathrm{eff} = g_e \mp \frac{\xi}{F\rho} \pm \frac{g_0^2 g_e^2\mu_B^2}{8c^2} \frac{1}{F\rho} + \mathcal{O}(\xi^3),$$ where the sign alternates between single-particle and single-hole cases, reflecting their opposing spin-orbit interactions.

Conversely, in the strong SOC regime ($g$0),

$g$1

Here, the cavity correction is efficiently quenched for both configurations.

The study unambiguously demonstrates:

• In the weak SOC regime, cavity-induced $g$2-factor modifications are relevant, with the sign depending on the electron/hole realization.
• In the strong SOC limit, the cavity-induced corrections are suppressed by $g$3 and thus are practically negligible.
• The fundamental coupling mechanism is encoded in the interplay of vibronic, SOC, and cavity Zeeman interactions.
• The cavity correction is independent of the cavity frequency in the weak field regime (since it is set by energy denominators $g$4 much larger than molecular scales).

Numerical and Physical Implications

Numerical analysis using realistic parameters (e.g., $g$5 cm$g$6, $g$7 molecules, and $g$8) illustrates that collective spin-cavity interactions can produce observable changes in electronic $g$9-factors under weak SOC, rendering cavity QED effects relevant for ensembles accessible to cavity-based EPR experiments.

These findings delineate experimental regimes where manipulation of molecular spin dynamics via cavity quantum electrodynamics may be possible. The theoretical construction can guide future experiments aiming to probe cavity-induced spin phenomena in transition metal complexes or radical systems, especially with tunable SOC and vibronic strengths.

Future Directions

Several promising extensions follow directly from this work:

• Inclusion of higher-lying electronic states and additional orbital/spin angular momentum channels would introduce further couplings, potentially enabling control of complex spin textures via strong light-matter coupling.
• Consideration of deviations from the idealized junction between metal and ligand orbitals would allow for the study of more chemically realistic complexes, where spin localization and delocalization can compete.
• Extensions to models incorporating higher-order (beyond linear) magnetic interactions with the cavity field, which are largely unexplored, may reveal further controlling principles relevant to cavity-modified spin chemistry.

These trajectories are relevant in the domains of molecular spintronics, quantum information, and polaritonic chemistry, where precise control of molecular magnetism is central.

Conclusion

This work formulates and solves the relativistic $^2E$0 Jahn-Teller problem for realistic molecular motifs under strong spin-cavity coupling, deriving analytic results for cavity-induced $^2E$1-factor modifications and dissecting their dependence on SOC and vibronic regimes. The study establishes that molecular spin properties can be non-trivially tailored by the quantized cavity field, particularly in systems with weak SOC. These insights may inform future experimental and theoretical investigations into cavity-controlled molecular magnetism and photonic manipulation of spin degrees of freedom.