Floquet Spin Waves in Magnonic Systems

• Floquet spin waves are coherent collective excitations in magnetic systems induced by time-periodic drives, exhibiting distinct quasienergy band structures.
• They are generated through nonlinear three-wave splitting processes in vortex disks and spin-phonon-coupled insulators, leading to observable frequency combs and mode interactions.
• Floquet engineering allows precise control over magnonic band structures, paving the way for on-chip frequency conversion, coherent magnon control, and topological magnonic circuits.

Floquet spin waves are coherent collective excitations in magnetically ordered materials, shaped by time-periodic or driven perturbations such as electromagnetic fields, phonon-driven distortions, or intrinsic dynamical backgrounds. The defining property of these states is their quasienergy (Floquet) band structure—a direct consequence of time-periodic Hamiltonians—resulting in frequency combs and novel dynamical effects. Floquet engineering offers control over magnon band structure, coherence, topological properties, and mode interactions, with implications across magnetic vortex systems, spin-phonon-coupled insulators, and magnonic quantum Hall phases.

1. Physical Mechanisms of Floquet Spin Wave Generation

The formation of Floquet spin waves involves the interplay of nonlinear magnon interactions, mode splitting, and periodic dynamical backgrounds. In magnetic vortex-state disks, forced excitation of azimuthal spin-wave modes (typically $m = \pm 1$, $f_\mathrm{az} \sim 5$–8 GHz) by in-plane radio-frequency fields induces three-wave processes: the high-frequency eigenmode $|m=-1\rangle$ splits into a vortex gyration magnon $|g\rangle$ ($m=+1$, $f_\mathrm{gyr} \sim 0.6$ GHz) and a Floquet spin wave $|F\rangle$ ($m=-2$, $f_F$). The conservation laws are

$$f_\mathrm{az} = f_\mathrm{gyr} + f_F,\qquad m_\mathrm{az} = m_\mathrm{gyr} + m_F.$$

This "three-wave splitting" initiates a large-amplitude gyration, modulating the local magnetic environment and reorganizing the spin-wave spectrum into Floquet spin waves with a frequency comb spacing given by the gyration frequency (Devolder et al., 13 Nov 2025). Similar Floquet band formation arises in spin-phonon coupled Mott insulators (Sutcliffe et al., 2024) through high-frequency phonon drives, as well as in bilayer antiferromagnets under circularly-polarized light, where photon-assisted magnon processes induce Dzyaloshinskii-Moriya interactions and magnonic quantum spin Hall effects (Owerre, 2018).

2. Floquet Band Structure: Dispersion and Comb Formation

Floquet analysis applies to linearized spin-wave equations with time-periodic backgrounds. If

$$i \frac{\partial}{\partial t} \psi(\mathbf{r},t) = \mathcal{L}(t) \psi(\mathbf{r},t),\quad \mathcal{L}(t+T)=\mathcal{L}(t),$$

Floquet's theorem yields solutions with quasienergies $\varepsilon_\alpha$ and time-periodic envelopes. The resulting Floquet spectrum displays combs: $$f_n = f_\mathrm{rf} + n f_\mathrm{gyr},\quad n \in \mathbb{Z},$$ with $\Delta f = f_\mathrm{gyr}$ the comb spacing (Devolder et al., 13 Nov 2025, Philippe et al., 26 Jul 2025). In vortex disks, each azimuthal spin-wave mode spawns sidebands offset by integer multiples of the gyration frequency. Sideband intensity, linewidth, and threshold depend on drive power, vortex core dynamics, and nonlinear coupling. For driven Mott insulators (Sutcliffe et al., 2024), steady-state Floquet sidebands appear at $\omega_{n} = \omega_\mathrm{mode} + n\Omega$; the spectral weight is redistributed by the time-periodic phonon-induced terms.

3. Nonlinear Effects, Thresholds, and Hysteresis

Floquet spin-wave mode population is strongly nonlinear. There exists a critical drive amplitude $P_c$, beyond which three-wave splitting and comb generation commence. The onset is not instantaneous; rather, a stochastic incubation delay $\tau$ diverges at threshold, with $\tau \sim (P - P_c)^{-\alpha}$ and $\alpha\approx 1$ (Devolder et al., 13 Nov 2025). Real-time measurements confirm that the population build-up of gyration magnons and the strongest Floquet mode occur nearly simultaneously after $\tau$, with minimum observed delays as short as 3 ns.

In nanopillar disks, coupled-mode and collective coordinate theory show that gyration radius $R_\mathrm{ss}$ reflects nonlinear feedback between magnon and vortex core modes. The Thiele equation for vortex dynamics incorporates nonlinear magnon feedback, resulting in multiple steady-state solutions: $\dot{R} = -\Gamma_g R + f(R),$ where $f(R)$ encapsulates three-wave interactions coupling sideband amplitudes to orbit size (Philippe et al., 26 Jul 2025). Nonmonotonic $f(R)$ enables hysteretic switching—bifurcation and saddle-node transitions between stable and unstable gyration states—tunable via drive power, frequency, and disk geometry.

4. Floquet Spin Waves in Spin-Phonon-Coupled Mott Insulators

In spin-$S>1/2$ Mott insulators, the SU(3) spin-phonon Hamiltonian incorporates quadrupolar modes linearly coupled to phonons. Periodic drives of the Eg phonon doublet generate Floquet states by two mechanisms (Sutcliffe et al., 2024):

• Chiral drive ($Q_{xz}, Q_{yz}$): Yields a Floquet-Zeeman field $h_\mathrm{eff} = \lambda^2 Q_0^2/(2\Omega)$, inducing a uniform magnetization in both antiferromagnetic (AFM) and quadrupolar paramagnetic (QPM) phases.
• Linear drive ($Q_{xz}$): Realizes Floquet-Ising anisotropy, modifying exchange and single-ion anisotropy terms, breaking $O(2)\to Z_2$ symmetry, and gapping out the Goldstone mode—a process detectable through the emergence of gapped peaks in inelastic neutron spectra.

The Floquet spin-wave structure includes copies of both transverse and longitudinal eigenmodes at $\omega_\mathrm{mode} + n\Omega$, with observables such as mode gaps, frequency shifts, and induced magnetization directly linked to drive parameters.

5. Floquet Topological Magnons and Quantum Spin Hall Edge States

In honeycomb bilayer antiferromagnets, circularly-polarized electric fields induce time-dependent Peierls phases on magnon hopping (Owerre, 2018). Floquet-Magnus expansion yields an effective static Hamiltonian augmented by photoinduced Dzyaloshinskii–Moriya (DM) interactions: $$\Delta\mathcal{H} = D_F \sum_{\langle\langle i j \rangle\rangle, \ell} \nu_{ij} \mathbf{S}_{i,\ell}\!\cdot\!(\mathbf{S}_{j,\ell}\times \hat{z}),$$ with $D_F$ determined by laser amplitude and frequency. The bilayer's combined space-time symmetry $\mathcal{PT}$ protects a $\mathbb{Z}_2$ Floquet topological invariant. Diagonalization in the bosonic BdG framework yields helical magnon edge states, spin-filtered transport, and a nonzero Floquet magnon spin-Nernst effect ($\alpha^s_{xy}$), while thermal Hall response vanishes. The phase diagram is tunable by light amplitude and polarization, with trivial and $\mathbb{Z}_2$ nontrivial regions set by gap-closing transitions at Dirac points.

Experimental realization requires THz-range off-resonant irradiation and sub-Kelvin temperatures in systems like bilayer CrI$_3$. Detection schemes include Brillouin or inelastic neutron scattering for band gap measurement, nonlocal spin Hall voltage monitoring, and ultrafast Kerr rotation for Floquet gap tracking.

6. Experimental Methods and Observational Signatures

Floquet spin waves are probed and quantified through diverse experimental approaches:

• Tunnel-junction voltage demodulation: Time-resolved, single-shot I/Q measurements with 3 ns temporal resolution for direct tracking of mode populations and delays (Devolder et al., 13 Nov 2025).
• Spectrum analysis: Frequency combs resolved with bandwidth $\sim$20 GHz and line widths $\leq$10 MHz, supporting coherence times up to 100 ns (Devolder et al., 13 Nov 2025).
• Inelastic neutron scattering & pump-probe: Reveals mode gaps and Floquet sidebands at $\omega_0 \pm n\Omega$ (Sutcliffe et al., 2024).
• Magneto-optical Kerr effect: Real-time measurement of induced magnetization and Floquet Zeeman field build-up.
• Terahertz spectroscopy: Tracks shifts in magnon resonance corresponding to Floquet modulation.
• Micromagnetic simulation and SU(N) dynamics: Lattice models with explicit time-periodic terms, Monte Carlo, and molecular dynamics for spectrum and mode evolution (Sutcliffe et al., 2024, Philippe et al., 26 Jul 2025).

7. Implications for Floquet Magnonics and Device Applications

The precise temporal and spectral control enabled by Floquet spin-wave engineering opens routes for:

• On-chip magnonic frequency conversion: Generation of tunable frequency combs for GHz-to-MHz down-conversion (Devolder et al., 13 Nov 2025, Philippe et al., 26 Jul 2025).
• Coherent magnon population control: Shaped pulses for sub-nanosecond switching, population inversion, and magnon storage (Devolder et al., 13 Nov 2025).
• Multistate vortex devices: Frequency fingerprints of distinct gyration radii for reservoir computing or nonvolatile magnonic memory (Philippe et al., 26 Jul 2025).
• Floquet-topological magnon bands: Light-induced nonreciprocal propagation, edge states, and spin-Nernst transport in quantum magnonic circuits (Owerre, 2018).
• Floquet-gated information transfer and waveform programming: Time-domain magnonics leveraging dynamic background modulation.

Further explorations are anticipated in thicker magnetic disks, low-damping garnet materials, Floquet lattice design, and integration with spin-phonon control in complex oxides. The combination of robust Floquet control, nontrivial topology, and tunable nonlinearity defines a versatile landscape for both fundamental and applied magnonics.