Cavity-Modified Zeeman Effect via Spin-Polariton Formation

TL;DR

The paper addresses how a low-frequency cavity with a quantized magnetic field component alters the electronic spin Zeeman effect for a spin-$\tfrac{1}{2}$ system under a static field. It develops an effective spin-polariton Hamiltonian from the Pauli-Fierz formalism beyond the dipole approximation using first-order quasi-degenerate perturbation theory and analyzes both single- and two-mode cavity scenarios. The results show cavity-induced spin-polariton formation that modifies the Zeeman splitting, provides a resonance condition $B_z^{\star}=\hbar\omega_c/(g_e\mu_B)$, and yields a cavity-modified Zeeman splitting $\tilde{\Delta}_\mathrm{Zee}$ and Rabi splitting $\tilde{\Delta}_\mathrm{Rabi}$, along with a derivation of a cavity-modified electronic g-factor $\tilde{g}_e$. The findings suggest observable signatures in EPR spectroscopy and highlight a mechanism to tailor electronic spin properties via cavity fields, with potential extensions to spin-orbit coupling effects in polaritonic chemistry.

Abstract

We study the electronic spin Zeeman effect for an effective spin-1/2-system subject to both strong coupling to a low-frequency optical cavity and an external static magnetic field. Specifically, we address the interplay between the cavity magnetic field component in a cavity Zeeman interaction and the canonical spin Zeeman interaction from the perspective of an effective spin-polariton Hamiltonian. The latter is derived from the minimal coupling Pauli-Fierz Hamiltonian beyond the common dipole approximation via first-order quasi-degenerate perturbation theory. We find the spin Zeeman effect to be modified in the presence of the cavity field due to the formation of spin-polariton states, which result from an intricate interplay of cavity and external magnetic fields in our model. Spin-polariton signatures are discussed in the context of electron paramagnetic resonance (EPR) spectroscopy along with cavity-induced modifications of the electronic g-factor.

Figures (2)

• Figure 1: Eigenvalues of the spin-polariton Hamiltonian for a single cavity mode polarized along the $z$-axis as function of static B-field strength, $B_z$. Spin-polariton branches related to eigenvalues $\varepsilon^p_\mp$ in Eq.\ref{['eq.spin_polariton_energies']} (colored in red) and spectator state branches related to eigenvalues $\varepsilon^s_\mp$ in Eq.\ref{['eq.spectator_energies']} (colored in blue). Additionally shown is the canonical Zeeman splitting, $\Delta_\mathrm{Zee}$, besides its cavity-modified counterpart, $\tilde{\Delta}_\mathrm{Zee}$, and the spin-polariton Rabi splitting, $\tilde{\Delta}_\mathrm{Rabi}$; the latter two for the resonance static B-field value $B^\star_z$ given in Eq.\ref{['eq.res_B_field']}. The inset shows a sketch of the cavity frame for the single-mode scenario and the orientation of an external static B-field.
• Figure 2: Eigenvalues of the spin-polariton Hamiltonian for two cavity modes polarized along $y$- and $z$-axis as function of static B-field strength, $B_z$. Spin-polariton branches (colored in red) and spectator state branches (colored in blue) for the two-mode scenario described by the spin-polariton Hamiltonian in Eq.\ref{['eq.two_modes_spin_polariton_hamilton']}. In comparison to the single-mode scenario presented in Fig.\ref{['fig.cavity_zeeman']}, two new branches appear with energies, $\varepsilon^s_\downarrow$ and $\varepsilon^p_\uparrow$, which correspond to unaltered zero-order product states, $\ket{\downarrow,1_y,0_z}$ and $\ket{\uparrow,1_y,0_z}$, containing one photon in the $y$-polarized cavity mode (cf. Appendix \ref{['sec.no_polariton_scenario']}). The inset shows a sketch of the cavity frame for the two-mode scenario and the orientation of an external static B-field.