Micromagnetic Lagrangian and Hamiltonian Normal-Mode Formulation of Cavity Magnon-Photon Interactions

Abstract

We investigate magnon-photon interactions in a resonant cavity starting from the coupled Maxwell and Landau-Lifshitz equations linearized about an arbitrary micromagnetic ground state. By introducing an electric vector potential and a suitable interaction Lagrangian, the formulation avoids the cancellation of large magnetostatic contributions that arises in the conventional magnetic-vector-potential description and becomes inaccurate under modal truncation. The magnetization and cavity fields are expanded in the normal modes of the uncoupled magnetic and electromagnetic subsystems, with coupling coefficients given by overlap integrals of the corresponding eigenfunctions. The resulting multimode Lagrangian has a noncanonical structure due to the gyrotropic terms, and the corresponding Hamiltonian formulation is obtained using the Faddeev-Jackiw approach. The radiative magnon-photon interaction is encoded in the resulting Poisson-bracket structure, while the cavity-induced modification of the magnetostatic field enters explicitly in the Hamiltonian. The coupled system is diagonalized through a generalized Bogoliubov transformation, yielding canonical polariton variables that provide a natural basis for quantization. Application to a perpendicularly magnetized YIG disk in a rectangular microwave cavity shows hybridization between the cavity mode and several magnetostatic modes, together with appreciable cavity-wall-induced frequency shifts when the sample is placed close to the conducting walls.

The paper addresses how to derive reduced multimode Hamiltonians for cavity magnonics from Maxwell–Landau–Lifshitz dynamics without double-counting magnetostatic fields. In a conventional magnetic-vector-potential expansion, recovering the relevant driving field requires cancellation of large magnetostatic terms, which can fail under cavity-mode truncation. The formulation linearizes about an arbitrary stable, potentially nonuniform micromagnetic equilibrium and separates radiative fields from magnetostatic fields, including their cavity-wall correction. An electric vector potential eliminates the problematic cancellation before modal expansion. Couplings are overlap integrals of uncoupled micromagnetic and empty-cavity eigenfunctions. A Faddeev–Jackiw construction places radiative coupling in the noncanonical Poisson structure and wall-induced magnetostatic energy explicitly in the Hamiltonian. That correction is proved positive semidefinite; generalized normal-mode diagonalization gives real polariton frequencies and canonical variables for bosonic quantization. A numerical example retains 15 magnetic modes of a perpendicularly magnetized YIG disk and one rectangular-cavity mode at $4\,\mathrm{GHz}$. Three bright modes have coupling strengths $|g|/2\pi\approx10.1$, $4.8$, and $3.2\,\mathrm{MHz}$. At $0.2\,\mathrm{mm}$ from a wall, shielding shifts magnetic resonances by roughly $20$–$30\,\mathrm{MHz}$ and crossing fields by up to $1.1\,\mathrm{mT}$, while changing gaps by at most about $11\%$. Shifts fall below $2\,\mathrm{MHz}$ at $2\,\mathrm{mm}$. The model assumes a nonconducting magnet, negligible dielectric response, perfectly conducting closed walls, and linear dynamics. The example additionally neglects exchange and anisotropy and restricts thickness dependence. Dissipation, ports, driving, open boundaries, dielectric loading, and nonlinear interactions are discussed as extensions rather than implemented.