Cavity-Modified Zeeman Effect via Spin-Polariton Formation

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.

💡 Research Summary

In this work the authors investigate how the Zeeman splitting of an electronic spin‑½ system is altered when the spin is simultaneously coupled to a low‑frequency optical cavity and subjected to a static external magnetic field. Starting from the minimal‑coupling Pauli‑Fierz Hamiltonian, they perform a first‑order quasi‑degenerate perturbative reduction that retains the leading beyond‑dipole contributions of the cavity magnetic field. The resulting effective Hamiltonian consists of three parts: the conventional Zeeman term (gₑμ_B S_z B_z), a “cavity Zeeman” term (gₑμ_B S·B̂_c) that couples the spin to the quantized magnetic field of the cavity, and the bare cavity photon energy. The cavity magnetic‑field operator is expressed as B̂_c ∝ (k × e_λ)(b̂_λ − b̂†_λ), where the wave‑vector k and polarization vectors e_λ determine the direction of the field.

Two polarization scenarios are examined. For a cavity mode polarized along the z‑axis (k‖x, e_z), the cavity Zeeman term reduces to i g₀ gₑμ_B/(c√(2ω_c)) σ_y (b̂_z − b̂†_z). This term flips the spin while creating or annihilating a photon, thereby coupling the product states |↑,0⟩ and |↓,1⟩. In the basis {|↑,0⟩,|↓,1⟩,|↓,0⟩,|↑,1⟩} the Hamiltonian becomes block‑diagonal with a 2×2 “polariton” block H_p and a spectator block H_s. When the static field satisfies the resonance condition B* = ω_c/(gₑμ_B), the two bare states become degenerate and a Rabi splitting ˜Δ_Rabi = g₀ gₑμ_B √(ω_c/2)/c emerges. The resulting eigenstates are spin‑polariton hybrids |±⟩ = (|↑,0⟩ ∓ |↓,1⟩)/√2. Their energies are shifted by ±˜Δ_Rabi relative to the bare cavity frequency, leading to a cavity‑modified Zeeman splitting ˜Δ_Zee = ω_c − ˜Δ_Rabi. Consequently the effective g‑factor becomes ˜gₑ = ˜Δ_Zee/(μ_B B*), which reduces to the free‑electron value gₑ when the light‑matter coupling g₀→0.

In contrast, a mode polarized along the y‑axis yields a cavity Zeeman term proportional to σ_z, which does not flip the spin and therefore does not generate polariton states; it merely shifts both spin components equally. When both z‑ and y‑polarized modes are present, the y‑mode acts as a spectator, adding uncoupled branches to the spectrum while the z‑mode still produces polaritons. This two‑mode situation enriches the EPR spectrum with additional lines and modifies the apparent g‑factor in a more complex way.

The authors also analyze limiting cases. In the zero‑field limit (B_z→0) the cavity‑modified splitting vanishes because the cavity magnetic field only couples states with different photon numbers; states with identical photon occupation remain uncoupled. In the strong‑field regime (B_z≫B*) the polariton energies revert to the ordinary Zeeman splitting, and higher‑photon manifolds (neglected in the present model) would become relevant.

For experimental relevance, the paper proposes parameters compatible with a 94 GHz (≈3.35 T) EPR spectrometer and a cavity frequency ω_c≈3.1 cm⁻¹ (≈93 GHz). By tuning the collective light‑matter coupling g₀, the system can enter the strong‑coupling regime defined by η = ˜Δ_Rabi/ω_c ≤ 0. In this regime the EPR spectrum is predicted to display a doublet separated by the Rabi splitting, providing a direct optical signature of spin‑polariton formation. Moreover, measuring the shifted Zeeman splitting ˜Δ_Zee allows extraction of the cavity‑renormalized g‑factor, offering a route to control magnetic properties via cavity QED.

Overall, the study presents a rigorous theoretical framework that incorporates the magnetic component of the cavity field beyond the dipole approximation, predicts the emergence of spin‑polariton quasiparticles, and quantifies how strong light‑matter coupling can modify the Zeeman effect and electronic g‑factor. These insights open new avenues for manipulating spin dynamics in molecular magnets, quantum information platforms, and cavity‑enhanced spectroscopies.