Finite-frequency magnetic common baths in ferromagnetic planar cavities

Abstract

We formulate the finite-frequency magnetic common-bath problem of two spin probes in a ferromagnetic planar cavity. The probes couple to the retarded magnetic Green tensor of the cavity, whose imaginary and real parts determine the collective decay rate \(γ_{12}\) and the coherent exchange coupling \(Ω_{12}\), respectively. We evaluate these quantities for a finite-thickness scalar transverse-electric (TE) cavity channel formed by ferromagnetic films on nonmagnetic conducting substrates, using a Polder-type scalar permeability \(μ_\perp(ω,B)\) as the magnetic input. The normalization is fixed by the free-space magnetic-dipole decay rate. In the \(ω\to0\) limit, the Green-tensor kernel reduces to the magnetostatic response, and the finite slab retains its static TE reflection amplitude. For a \(t=200\,\mathrm{nm}\) film with a representative Ni-like parameter set, the GHz transition of a spin probe at the mid-gap of a micron-scale cavity samples the spectral density of the body-assisted magnetic reservoir at millikelvin temperature. At the resonant bias, the cavity-induced contribution \(2|δγ_{12}^{(zz)}|\) to the collective linewidth splitting in the \(zz\) transition-moment projection is about two thirds of the local single-spin linewidth correction \(δγ_{11}^{(zz)}\) at \(ρ=3\,μ\mathrm m\). Together, the linewidth splitting and collective Lamb shift probe the dissipative and dispersive finite-frequency response, respectively, while the shared Green-tensor normalization connects these observables to the static TE coupling-frequency shift.

12 pages, 6 figures