Finite-frequency magnetic common baths in ferromagnetic planar cavities

We formulate the finite-frequency magnetic common bath 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 kernel $\gamma_{12}$ and the Lamb-shift kernel $\Omega_{12}$. We evaluate these kernels for a finite-thickness scalar TE channel formed by a ferromagnetic film on a nonmagnetic conducting substrate, using the transverse diagonal Polder permeability $\mu_\perp(\omega,B)$ as the magnetic input. The normalization is fixed by the free-space magnetic-dipole decay rate. In the $\omega\to0$ limit, the constant-reflection benchmark reproduces the static image-series reference, while the finite slab retains the corresponding static TE reflection amplitude. For a $t=200\,\mathrm{nm}$ film described by a representative Ni-like parameter set in a micron-scale mid-gap cavity, the GHz probe transition samples the positive-frequency response of the body-assisted magnetic reservoir at millikelvin temperature. On resonance, the off-diagonal linewidth splitting reaches about two thirds of the single-spin scattering linewidth at $\rho=3\,\mu\mathrm m$. The resulting linewidth splitting and collective Lamb shift provide finite-frequency counterparts of the static TE coupling-frequency shift.