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arXiv:2606.13830

A Collective-Spin Derivation of the Uniform Magnon Hamiltonian in Cavity Magnonics

Tomas Aguiar, Marcos Cesar de Oliveira

cond-mat.mes-hallquant-phSubmitted Jun 11, 2026• Announced Jun 15, 2026
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Abstract

We present a direct collective-spin derivation of the effective uniform-mode Hamiltonian used in cavity magnonics. Starting from a nearest-neighbor Heisenberg ferromagnet coupled to long-wavelength magnetic fields, we show that the relevant dynamics can be restricted to the fully symmetric spin sector, where the exchange interaction contributes only a constant energy shift and the ferromagnet behaves as a macrospin of length $Ns$. Applying the Holstein--Primakoff transformation directly to this total spin yields the usual uniform magnon mode and its leading nonlinear corrections without first introducing site-resolved bosonic operators. This collective formulation makes explicit the interpretation of the ferromagnet as a synthetic large-spin atom and provides a compact route to the effective Hamiltonians used in driven and Floquet cavity magnonics. As a physical consequence, the leading nonlinear correction produces an occupation-dependent reduction of the effective magnon--photon coupling, providing a simple signature of finite-spin saturation under strong uniform-mode driving.

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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Aug 17, 2010•2 citations
We present a direct collective-spin derivation of the effective uniform-mode Hamiltonian used in cavity magnonics. Starting from a nearest-neighbor Heisenberg ferromagnet coupled to long-wavelength magnetic fields, we show that the relevant dynamics can be restricted to the fully symmetric spin sector, where the exchange interaction contributes only a constant energy shift and the ferromagnet behaves as a macrospin of length NsNsNs. Applying the Holstein--Primakoff transformation directly to this total spin yields the usual uniform magnon mode and its leading nonlinear corrections without first introducing site-resolved bosonic operators. This collective formulation makes explicit the interpretation of the ferromagnet as a synthetic large-spin atom and provides a compact route to the effective Hamiltonians used in driven and Floquet cavity magnonics. As a physical consequence, the leading nonlinear correction produces an occupation-dependent reduction of the effective magnon--photon coupling, providing a simple signature of finite-spin saturation under strong uniform-mode driving.
We develop a perturbative input-output formalism to compute the reflectance and transmittance spectra of cavity magnonics systems subject to a Floquet modulation. The method exploits the strong hierarchy between the magnetic-dipole couplings transverse (drive field) and parallel (modulation field) to the static bias field, which naturally introduces the small parameter ε=(2Ns)−1/2ε= (2Ns)^{-1/2}ε=(2Ns)−1/2 associated with the total spin NsNsNs of the ferromagnet. By organizing the cavity and magnon fields in a systematic expansion in εεε, we obtain compact analytic expressions for the spectra up to second order. Using these results, we reproduce the characteristic sideband structure observed in recent Floquet cavity electromagnonics experiments. Furthermore, accounting for the Zeeman interaction between the modulation field and the fully polarized ground state - a contribution typically neglected in previous treatments - we predict an additional magnon detuning of approximately 0.8 GHz0.8\,\mathrm{GHz}0.8GHz, independent of both modulation frequency and sample size and determined solely by the spatial volume occupied by the modulation field. This identifies a measurable and previously overlooked shift relevant for the interpretation and design of cavity magnonics experiments.
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 Holstein-Primakoff boson representation of quantum spins and associated large-SSS expansion have been the standard framework for describing the spin wave excitations in magnetically order phases of quantum spin systems. However, we will show that the omission of projection operators and normal-ordering in this representation can produce incorrect magnon hamiltonians for finite SSS. We will present the exact normal-ordered forms of the finite-SSS projection operators and projected Holstein-Primakoff boson representations of spin and quadrupole operators, which can produce exact two-magnon interaction terms under ferromagnetic or fully polarized states. We will also discuss the difficulties of applying this projected representation to antiferromagnetic spin wave theory.
Cavity magnonics is moving beyond the conventional paradigm of coherent spin--photon hybridization toward the controlled engineering of dissipative and coherent interactions, Floquet dynamics, and quantum fluctuations. This Perspective examines how these ingredients reshape collective dynamics and open new avenues for quantum state engineering, transport, and sensing. We discuss the role of dissipative coupling and anti-PT\mathcal{PT}PT symmetry in enabling long-lived collective modes and enhanced spectral response, and highlight Floquet engineering as a means of engineering nonreciprocal transport and phase-controlled interactions. We then review recent progress in quantum magnonics, ranging from squeezed and entangled magnon states, including the squeezing of thermal magnons, to prospective schemes for preparing magnonic Schrödinger-cat states. We also discuss how parametric driving and active gain strongly modify hybrid susceptibilities and amplify magnon--photon response. These advances expand the scope of cavity magnonics beyond conventional polariton physics, establishing it as a versatile platform for sensing, transduction, spin transport, and hybrid quantum technologies.
We show how the decoherence that occurs in an entangling atomic spin–light interface can be simply modeled as the dynamics of a bosonic mode. Although one seeks to control the collective spin of the atomic system in the permutationally invariant (symmetric) subspace, diffuse scattering and optical pumping are local, making an exact description of the many-body state intractable. To overcome this issue we develop a generalized Holstein–Primakoff approximation for collective states which is valid when decoherence is uniform across a large atomic ensemble. In different applications the dynamics is conveniently treated as a Wigner function evolving according to a thermalizing diffusion equation, or by a Fokker–Planck equation for a bosonic mode decaying in a zero-temperature reservoir. We use our formalism to study the combined effect of Hamiltonian evolution, local and collective decoherence, and measurement backaction in preparing nonclassical spin states for application in quantum metrology.
We predict that strong coupling is feasible between photons and a ferromagnetic nanomagnet, due to exchange interactions that cause very large numbers of spins to coherently lock together with a significant increase in oscillator strength while still maintaining very long coherence times. The interaction of a ferromagnetic nanomagnet with a single photonic mode of a cavity is analyzed in a fully quantum-mechanical treatment. Exceptionally large quantum-coherent magnet-photon coupling with coupling terms in excess of several THz are predicted to be achievable in a spherical cavity of ~ 1 mm radius with a nanomagnet of ~ 100 nm radius and ferromagnet resonance frequency of ~ 200 GHz. This should substantially exceed the coupling observed in solids between orbital transitions and light. Eigenstates of the nanomagnet-photon system correspond to entangled states of spin orientation and photon number over 105 values of each quantum number. Initial coherent state of definite spin and photon number evolve dynamically to produce large coherent oscillations in the microwave power with exceptionally long dephasing times of few seconds. In addition to dephasing, several decoherence mechanisms including elementary excitation of magnons and crystalline magnetic anisotropy are investigated and shown to not substantially affect coherence upto room temperature. The optimal nanomagnet size is predicted to be just below the threshold for failure of the macrospin approximation.