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

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.