Classical and quantum theory of magnonic and magnetoelastic nonlinear dynamics in continuum geometries

TL;DR

The paper develops a comprehensive classical and quantum theory of nonlinear magnon–phonon dynamics in continuum geometries, focusing on a ferromagnetic film on a substrate supporting surface acoustic waves. By merging the Landau–Lifshitz–Gilbert formalism with a magnetoelastic Hamiltonian, it derives coupled nonlinear equations for magnon and phonon amplitudes, including three- and four-magnon processes and magnetoelastic interactions, and provides explicit expressions for all coupling rates. The authors then quantize the model with Heisenberg–Langevin equations and analyze quantum fluctuations within a mean-field framework, revealing how parametric instabilities modify magnetization fluctuations and enabling quantum control of magnons via acoustic driving. The framework reproduces experimental observations of phonon‑to‑magnon down-conversion, identifies dominant nonlinear pathways, and offers a practical path toward quantum magnonics in continuum systems without relying on qubits.

Abstract

We provide a theory of spin and acoustic wave coupled nonlinear dynamics in continuum systems. Combining the Landau-Lifshitz-Gilbert equations with the magnetoelastic Hamiltonian, we derive classical equations of motion for the magnetization and acoustic wave amplitudes, that include magnonic nonlinearity -- both three- and four-magnon processes -- as well as linear and nonlinear magnetoelastic interactions. We focus on two-dimensional magnetic films sustaining surface acoustic waves, a geometry where our model successfully reproduces our recent experimental observation of phonon-to-magnon down-conversion under acoustic drive. We provide analytical expressions for all the rates in our equations, which make them particularly suitable for quantization. We then quantize our model, deriving Heisenberg-Langevin equations of motion for magnon and phonon operators, and show how to compute quantum expectation values in the mean field approximation. Our work paves the way toward acoustic control of magnons in the quantum regime.

Figures (9)

• Figure 1: We consider an infinitely extended ferromagnetic thin film with thickness $d$ in the presence of a uniform magnetic field $\mathbf{B}_0$ which leads to saturation of the magnetization. Magnons in the thin film couple, both linearly and nonlinearly, to surface acoustic waves sustained in the whole film and susbtrate system.
• Figure 2: (a) Dispersion relation of lowest-band ($n=0$) phonon and magnon modes propagating parallel to the external magnetic field, for the parameters of Table \ref{['tab:parameters_1']}. Background colors designate the different regimes of phonon mode types, i.e. bulk, hybrid, or SAW within the film (f) or substrate (s). Below the black dotted line modes are SAWs in the substrate, i.e., they decay exponentially inside the substrate. The pink dotted lines mark the driving frequency $\omega_\text{d}/2\pi=6.11\,\mathrm{GHz}$ and wave vector amplitude $k_0=2\pi/625\,\mathrm{nm}^{-1}$ of the acoustic mode we use for our results in Sec. \ref{['subsec:parametric_excitations_threshold_ordering']}. (b) A schematic depiction of the difference between bulk (above) and SAW (below) phonon modes. (c) Schematic visualization of phonon mode families. Modes with polarization index $\sigma=1$ (orange) are purely transversal $\mathbf{u}_{1,k}(\mathbf{r})\cdot\mathbf{k}=0$, while modes with $\sigma=2$ have a longitudinal component, $\mathbf{u}_{2,k}(\mathbf{r})\cdot\mathbf{k}\ne0$.
• Figure 3: Schematic depiction of the different (a) three-magnon ($T^{(1,2)}$) and (b) four-magnon ($F^{(1,2,3)}$) interactions appearing in the equations of motion Eq. \ref{['eq:magnon_eom']}. (c) Coupling rate density $T^{(1)}$ corresponding to a magnon of wavev ector $\mathbf{k}_0=(0,0,2\pi/625)\,\mathrm{nm^{-1}}$ scattering into two magnons ($\mathbf{k}=k(0,\sin\phi_k ,\cos\phi_k)$ and $\mathbf{k}_0-\mathbf{k}$), as a function of modulus and propagation angle of magnon $\mathbf{k}$. System parameters are listed in Tab. \ref{['tab:parameters_1']}.
• Figure 4: Resonant magnetoelastic linear coupling rates $\Omega_{\text{r},\sigma}$ [Eq. \ref{['EqRabi']}] vs mode wave vector and direction, for the $\sigma=1$ (a) and $\sigma=2$ (b) phonon families, respectively. System parameters are listed in Tab. \ref{['tab:parameters_1']}.
• Figure 5: (a) Schematic depiction of the types of nonlinear magnetoelastic interactions appearing in the equations of motion (red/green lines indicate magnons and phonons, respectively). (b) Nonlinear coupling strength density $G^{(3)}$ corresponding to a phonon of mode $\{\sigma=1,\mathbf{k}_0=(0,0,2\pi/625)\,\mathrm{nm^{-1}}\}$ scattering into two magnons [$\mathbf{k}=k(0,\sin\phi_k ,\cos\phi_k)$ and $\mathbf{k}_0-\mathbf{k}$], as a function of modulus and propagation angle of magnon $\mathbf{k}$. System parameters are listed in Tab. \ref{['tab:parameters_1']}.