Operating regimes and materials limits of nonlinear magnon dynamics in Co-doped YIG
Magnonic information processing uses Kerr-type four-magnon anharmonicity to bound parametrically driven excitations. Here we show that this anharmonicity is constrained by material and mode properties. For the uniform mode of a saturated ferrimagnet, the Holstein--Primakoff reduction gives $U=ω_a/(2N_{\rm eff}Ω)$, where $ω_a$ is the anisotropy-field frequency, $Ω$ the mode frequency, and $N_{\rm eff}$ the net spin content fixed by saturation magnetization and mode volume. Requiring the drive to exceed Gilbert damping while keeping the saturated occupation below one gives $αN_{\rm eff}<ω_a/Ω$, reducing to $N_{\rm eff}<1/α$ at $Ω=ω_a$. We test this criterion for Co-doped yttrium iron garnet using Lindblad calculations with thermal operators. Thermal stability permits low-damping garnet at 50 GHz ($α<9.7\times10^{-5}$) and 200 GHz ($α<3.9\times10^{-4}$), but this is not sufficient: the admissible drive window is nearly adiabatic and four orders of magnitude weaker than required. For a $20\times20\times10$ nm$^3$ cell, $U=6.6\times10^{-5}$, three orders below the range showing Fock-state confinement. One mechanism survives: the pair-phase transfer coefficient remains $-1.000$ down to $U=0.01$, with phase deviations of $22.4^\circ$--$40.7^\circ$ ordered by thermal occupation. These results give a quantitative materials criterion for nonlinear magnonic devices and identify phase encoding as the mechanism accessible to a garnet film.