Phonon Floquet Engineering

• Phonon Floquet Engineering is a method employing coherent lattice vibrations to periodically drive and dress electronic states, creating Floquet replicas and dynamical gaps.
• The technique uses phonon amplitude, frequency, and polarization as tuning knobs to modify material properties and induce topological transitions.
• Realizations span from graphene to trapped-ion arrays, with applications in Floquet-Bloch spectroscopy, quantized transport, and symmetry-controlled phase changes.

Searching arXiv for recent and foundational papers on phonon Floquet engineering. Phonon Floquet engineering is the deliberate use of a coherent lattice vibration to dress and thereby reshape the electronic structure of a solid in much the same way that a strong light field does in photon-Floquet protocols. In its minimal form, a coherently excited phonon provides a periodic coordinate $Q(t)=Q_0\cos(\Omega t)$ that enters the Hamiltonian as a time-periodic bosonic perturbation, generating Floquet replicas, dynamical gaps, and, when time-reversal symmetry is broken, topological band inversions. The same logic has been extended to Floquet-Bloch spectroscopy, magnons, quantum emitters, nanowires, trapped-ion phonons, and defect-based phonon lasing, so the term now denotes a broader class of periodically driven matter in which coherent vibrational motion acts as the drive (Hübener et al., 2018, Tai et al., 29 Jun 2026).

1. Foundational formulation

The foundational construction starts from an electronic system coupled to a single coherent phonon mode of frequency $\Omega$. In second-quantized form,

$H(t)=H_0+H_{\rm ph}+H_{\rm el-ph}(t),$

with

$$H_0=\sum_{k,\alpha}\varepsilon_{k,\alpha}c^\dagger_{k,\alpha}c_{k,\alpha},$$

and, in the large-amplitude coherent-state limit,

$$Q(t)=Q_0\cos(\Omega t)\Rightarrow H_{\rm el-ph}(t)=g\,Q(t)\sum_{k,\alpha,\beta}M_{\alpha\beta}(k)\,c^\dagger_{k,\alpha}c_{k,\beta}.$$

Because $H(t+2\pi/\Omega)=H(t)$, the problem is cast into Floquet form through

$H_F=H(t)-i\hbar\partial_t,$

with solutions

$$|\Psi_\alpha(t)\rangle=e^{-i\varepsilon_\alpha t/\hbar}|\Phi_\alpha(t)\rangle,\qquad |\Phi_\alpha(t+T)\rangle=|\Phi_\alpha(t)\rangle.$$

Expanding $$|\Phi_\alpha(t)\rangle=\sum_m e^{-im\Omega t}|u_\alpha^m\rangle$$ converts the time-dependent Schrödinger equation into a static infinite-matrix eigenvalue problem whose block form is

$$(H_F)^{mn}=H_0\delta_{mn}+m\hbar\Omega\,\delta_{mn}+\frac{gQ_0}{2}M(\delta_{m,n+1}+\delta_{m,n-1}),$$

so that sidebands at $\Omega$0 are hybridized by matrix elements proportional to $\Omega$1 (Hübener et al., 2018).

This formalism is not restricted to electrons. One formulation in the low-frequency regime writes a monochromatic drive as $\Omega$2 and replaces the high-frequency expansion by the self-consistent effective operator

$\Omega$3

In that language, phonon modes enter as classical periodic fields $\Omega$4 coupled through $\Omega$5, with Fourier components at $\Omega$6 (Rodriguez-Vega et al., 2020).

The immediate significance of this formulation is that the lattice coordinate itself becomes a control parameter in quasienergy space. Frequency $\Omega$7, amplitude $\Omega$8, and polarization trajectory of $\Omega$9 act as the central tuning knobs, directly analogous to the role played by amplitude, frequency, and polarization in optical Floquet protocols (Hübener et al., 2018).

2. Electronic band dressing and Floquet-Bloch spectra

The paradigmatic example is graphene near a Dirac point $H(t)=H_0+H_{\rm ph}+H_{\rm el-ph}(t),$0, where one may take

$H(t)=H_0+H_{\rm ph}+H_{\rm el-ph}(t),$1

A planar optical phonon enters as an effective gauge field $H(t)=H_0+H_{\rm ph}+H_{\rm el-ph}(t),$2, yielding a time-dependent Dirac Hamiltonian whose Floquet spectrum shows replicas of the equilibrium Dirac dispersion displaced by $H(t)=H_0+H_{\rm ph}+H_{\rm el-ph}(t),$3,

$H(t)=H_0+H_{\rm ph}+H_{\rm el-ph}(t),$4

At crossings satisfying $H(t)=H_0+H_{\rm ph}+H_{\rm el-ph}(t),$5, dynamical gaps open. To leading order near such a crossing, the gap is $H(t)=H_0+H_{\rm ph}+H_{\rm el-ph}(t),$6, and the spectral weight of the $H(t)=H_0+H_{\rm ph}+H_{\rm el-ph}(t),$7 Floquet sideband is approximately given by Bessel-function factors $H(t)=H_0+H_{\rm ph}+H_{\rm el-ph}(t),$8 with $H(t)=H_0+H_{\rm ph}+H_{\rm el-ph}(t),$9 (Hübener et al., 2018).

This establishes the core phenomenology of phonon-dressed electronic states: replica bands, avoided crossings, and sideband-weight redistribution. It also clarifies that a coherent phonon drive can imprint electron-phonon coupling directly onto the measured electronic spectrum, rather than merely broadening it or shifting it perturbatively.

Time-domain evidence for this picture was reported for graphene-covered Ir(111), where laser-excited coherent phonons drove Floquet-Bloch states that were tracked by time-resolved multiphoton photoemission combined with quantum beat spectroscopy. In that experiment, the coherent phonon had $$H_0=\sum_{k,\alpha}\varepsilon_{k,\alpha}c^\dagger_{k,\alpha}c_{k,\alpha},$$0, corresponding to $$H_0=\sum_{k,\alpha}\varepsilon_{k,\alpha}c^\dagger_{k,\alpha}c_{k,\alpha},$$1, and the beat signal indicated sideband structure with the coherent-phonon frequency as its fundamental period. The beats persisted for several picoseconds, with coherent-phonon lifetime exceeding $$H_0=\sum_{k,\alpha}\varepsilon_{k,\alpha}c^\dagger_{k,\alpha}c_{k,\alpha},$$2, which is one to two orders of magnitude longer than typical light-driven Floquet states (Tai et al., 29 Jun 2026).

A common misconception is that Floquet signatures require direct optical overlap in the photoemission final state. The quantum-beat measurements on graphene/Ir(111) explicitly separated phonon-driven sideband coherence from final-state Volkov replicas: the observed oscillations occurred at $$H_0=\sum_{k,\alpha}\varepsilon_{k,\alpha}c^\dagger_{k,\alpha}c_{k,\alpha},$$3, were phase-locked to the coherent phonon, and persisted on picosecond timescales, whereas Volkov replicas appear only during optical overlap and at photon-energy spacing (Tai et al., 29 Jun 2026).

3. Topology, magnetism, and symmetry transmutation

When time-reversal symmetry is broken by the phonon trajectory, phonon Floquet engineering becomes a route to topological phase control. In graphene, a circular phonon formed by superposing two degenerate optical modes with a $$H_0=\sum_{k,\alpha}\varepsilon_{k,\alpha}c^\dagger_{k,\alpha}c_{k,\alpha},$$4 phase shift produces a Floquet block of Haldane type,

$$H_0=\sum_{k,\alpha}\varepsilon_{k,\alpha}c^\dagger_{k,\alpha}c_{k,\alpha},$$5

with induced mass $$H_0=\sum_{k,\alpha}\varepsilon_{k,\alpha}c^\dagger_{k,\alpha}c_{k,\alpha},$$6. The Berry curvature

$$H_0=\sum_{k,\alpha}\varepsilon_{k,\alpha}c^\dagger_{k,\alpha}c_{k,\alpha},$$7

and Chern number

$$H_0=\sum_{k,\alpha}\varepsilon_{k,\alpha}c^\dagger_{k,\alpha}c_{k,\alpha},$$8

then jump from $$H_0=\sum_{k,\alpha}\varepsilon_{k,\alpha}c^\dagger_{k,\alpha}c_{k,\alpha},$$9 to $$Q(t)=Q_0\cos(\Omega t)\Rightarrow H_{\rm el-ph}(t)=g\,Q(t)\sum_{k,\alpha,\beta}M_{\alpha\beta}(k)\,c^\dagger_{k,\alpha}c_{k,\beta}.$$0 when $$Q(t)=Q_0\cos(\Omega t)\Rightarrow H_{\rm el-ph}(t)=g\,Q(t)\sum_{k,\alpha,\beta}M_{\alpha\beta}(k)\,c^\dagger_{k,\alpha}c_{k,\beta}.$$1 changes sign, and the driven steady state becomes a nontrivial Chern insulator (Hübener et al., 2018).

Closely related electronic physics was developed for a honeycomb lattice with circularly polarized phonons. There, the Floquet-Magnus commutator generates an effective next-nearest-neighbor hopping exactly analogous to Haldane’s model. Near the valleys,

$$Q(t)=Q_0\cos(\Omega t)\Rightarrow H_{\rm el-ph}(t)=g\,Q(t)\sum_{k,\alpha,\beta}M_{\alpha\beta}(k)\,c^\dagger_{k,\alpha}c_{k,\beta}.$$2

with $$Q(t)=Q_0\cos(\Omega t)\Rightarrow H_{\rm el-ph}(t)=g\,Q(t)\sum_{k,\alpha,\beta}M_{\alpha\beta}(k)\,c^\dagger_{k,\alpha}c_{k,\beta}.$$3, and the Chern number is

$$Q(t)=Q_0\cos(\Omega t)\Rightarrow H_{\rm el-ph}(t)=g\,Q(t)\sum_{k,\alpha,\beta}M_{\alpha\beta}(k)\,c^\dagger_{k,\alpha}c_{k,\beta}.$$4

The trivial-to-Chern transition occurs when $$Q(t)=Q_0\cos(\Omega t)\Rightarrow H_{\rm el-ph}(t)=g\,Q(t)\sum_{k,\alpha,\beta}M_{\alpha\beta}(k)\,c^\dagger_{k,\alpha}c_{k,\beta}.$$5. In the same framework, circularly polarized phonons generate orbital magnetization and, with Rashba spin-orbit coupling, a finite out-of-plane spin magnetization; numerics give $$Q(t)=Q_0\cos(\Omega t)\Rightarrow H_{\rm el-ph}(t)=g\,Q(t)\sum_{k,\alpha,\beta}M_{\alpha\beta}(k)\,c^\dagger_{k,\alpha}c_{k,\beta}.$$6 per unit cell for realistic parameters (Yao et al., 9 Jun 2026).

An important qualification is that not every coherent phonon changes topology. In a magnonic setting based on \textit{ab initio} spin-lattice coupling for monolayer CrI$$Q(t)=Q_0\cos(\Omega t)\Rightarrow H_{\rm el-ph}(t)=g\,Q(t)\sum_{k,\alpha,\beta}M_{\alpha\beta}(k)\,c^\dagger_{k,\alpha}c_{k,\beta}.$$7, linearly polarized phonons leave the spectrum unchanged, whereas circular and elliptical phonons carrying finite phonon angular momentum induce chiral interactions that open and tune gaps at Dirac points. The induced effective Dzyaloshinskii-Moriya scale obeys

$$Q(t)=Q_0\cos(\Omega t)\Rightarrow H_{\rm el-ph}(t)=g\,Q(t)\sum_{k,\alpha,\beta}M_{\alpha\beta}(k)\,c^\dagger_{k,\alpha}c_{k,\beta}.$$8

and the Dirac-point gap scales as

$$Q(t)=Q_0\cos(\Omega t)\Rightarrow H_{\rm el-ph}(t)=g\,Q(t)\sum_{k,\alpha,\beta}M_{\alpha\beta}(k)\,c^\dagger_{k,\alpha}c_{k,\beta}.$$9

The Chern numbers satisfy $H(t+2\pi/\Omega)=H(t)$0, so reversing phonon handedness flips the topological magnon phase (Weißenhofer et al., 27 May 2026).

Phonon driving can also alter topology through space-time symmetry rather than through a conventional mass term. In two-dimensional class D and class AIII models, a coherently excited reflection-odd phonon promotes a purely spatial reflection symmetry into a time-glide symmetry,

$H(t+2\pi/\Omega)=H(t)$1

In extended Floquet space this flips the algebraic relation between reflection and the non-spatial symmetries, allowing mirror-graded invariants that were forbidden in the static model. The result is a Floquet second-order topological phase with co-dimension-two corner modes at reflection-symmetric boundaries (Chaudhary et al., 2019).

This symmetry-based mechanism clarifies another frequent misunderstanding: a nontrivial phonon-induced Floquet phase does not always require a large leading-order modification of the static band Hamiltonian. In the high-frequency treatment of the class D and AIII examples, the leading commutator correction vanishes at $H(t+2\pi/\Omega)=H(t)$2, yet the enlarged Floquet-space symmetry still changes the topological classification (Chaudhary et al., 2019).

4. Quantum-optical, phononic, and mechanical realizations

Phonon Floquet engineering has moved well beyond electronic bands in crystals. In a driven two-level emitter under simultaneous coherent acoustic modulation and strong optical driving, the Hamiltonian in the rotating frame reads

$H(t+2\pi/\Omega)=H(t)$3

The resonance-fluorescence spectrum becomes a sum of Lorentzians indexed by Floquet quasifrequencies and acoustic sidebands, with a structure of crossings, anticrossings, and line suppressions. When $H(t+2\pi/\Omega)=H(t)$4, odd $H(t+2\pi/\Omega)=H(t)$5 gives anticrossings and complete suppression of the zero-phonon line at exact resonance, whereas even $H(t+2\pi/\Omega)=H(t)$6 yields protected crossings and dimming by a selection rule (Groll et al., 11 Sep 2025).

In defect-based solid-state membranes, Floquet control can engineer not only gain but squeezed gain. For two electronic spins coupled to a mechanical mode and periodic spin control, the leading-order effective Hamiltonian takes the form

$H(t+2\pi/\Omega)=H(t)$7

where $H(t+2\pi/\Omega)=H(t)$8 is a Bogoliubov-transformed squeezed mode. The squeezed-lasing threshold is reached when

$H(t+2\pi/\Omega)=H(t)$9

and the reduced squeezed-quadrature fluctuation saturates at

$H_F=H(t)-i\hbar\partial_t,$0

The same scheme can also break the $H_F=H(t)-i\hbar\partial_t,$1 phase symmetry of the lasing manifold and produce phase-locked squeezed phonon lasing (Molinares et al., 3 Jun 2026).

In trapped-ion arrays, phonon Floquet engineering is realized directly on motional degrees of freedom. Local parametric modulations $H_F=H(t)-i\hbar\partial_t,$2 dress phonon hopping amplitudes and imprint Peierls phases. The resulting static Floquet Hamiltonian has effective couplings

$H_F=H(t)-i\hbar\partial_t,$3

thereby controlling both the magnitude and complex phase of long-range phonon hopping. Experiments demonstrated steering, sign reversal of phonon flow by phase jumps, and two-path interference in a two-dimensional array (Kiefer et al., 2019).

A closely related synthetic-gauge-field architecture uses surface-acoustic-wave cavities and auxiliary qubits. There a longitudinally driven qubit induces effective inter-cavity hopping

$H_F=H(t)-i\hbar\partial_t,$4

so that the drive phase becomes a Peierls phase for phonon transport. On a triangle, the gauge-invariant accumulated phase

$H_F=H(t)-i\hbar\partial_t,$5

acts like a magnetic flux, enabling circulator transport and Aharonov-Bohm interference for acoustic waves (Wang et al., 2019).

Taken together, these realizations show that the phrase “phonon Floquet engineering” now covers at least three distinct but mathematically aligned settings: phonons driving electrons, phonons as the driven quasiparticles, and phonons as intermediate carriers that encode synthetic phases or squeezed-mode structure into an effective Hamiltonian.

5. Steady states, dissipation, and transport

A central issue in any Floquet protocol is heating. For interacting driven systems coupled to phonons, the relevant kinetic description is the Floquet-Boltzmann equation

$H_F=H(t)-i\hbar\partial_t,$6

where the collision integrals contain both normal processes and Floquet-Umklapp channels with $H_F=H(t)-i\hbar\partial_t,$7. In a one-dimensional driven electronic model coupled to an acoustic phonon bath, an insulator-like steady state with low excitation density emerges when phonon-mediated cooling dominates phonon-assisted Floquet-Umklapp heating and electron-electron Auger-type heating. An especially favorable regime occurs when the phonon bandwidth satisfies $H_F=H(t)-i\hbar\partial_t,$8, so normal cooling is allowed but phonon-Umklapp heating remains forbidden (Seetharam et al., 2018).

The broader implication is that phonons play a dual role in Floquet matter. A coherent mode can act as the drive, while incoherent phonons can simultaneously act as the bath that stabilizes the desired Floquet population. This dual use is particularly important in phonon-based protocols because the drive and the relaxation channel are naturally embedded in the same material platform.

Transport proposals make this stabilization logic explicit. In a metallic quantum nanowire driven by a coherent terahertz phonon wave, the Floquet spectrum satisfies

$H_F=H(t)-i\hbar\partial_t,$9

If exactly one Floquet band is filled over one reduced zone, the steady current becomes

$$|\Psi_\alpha(t)\rangle=e^{-i\varepsilon_\alpha t/\hbar}|\Phi_\alpha(t)\rangle,\qquad |\Phi_\alpha(t+T)\rangle=|\Phi_\alpha(t)\rangle.$$0

The proposed steady state is maintained by coupling the Floquet eigenstates to incoherent phonons at low temperature and to electron-electron scattering, which relaxes carriers into the target filled Floquet band (Yang et al., 2024).

This transport result should not be confused with an adiabatic Thouless pump. The non-adiabatic regime described for the nanowire allows quantization even when $$|\Psi_\alpha(t)\rangle=e^{-i\varepsilon_\alpha t/\hbar}|\Phi_\alpha(t)\rangle,\qquad |\Phi_\alpha(t+T)\rangle=|\Phi_\alpha(t)\rangle.$$1, and the mechanism depends explicitly on the finite phonon momentum $$|\Psi_\alpha(t)\rangle=e^{-i\varepsilon_\alpha t/\hbar}|\Phi_\alpha(t)\rangle,\qquad |\Phi_\alpha(t+T)\rangle=|\Phi_\alpha(t)\rangle.$$2 and on phonon-assisted relaxation processes (Yang et al., 2024).

6. Control regimes, materials design, and low-frequency nonlinear phononics

At the level of design principles, the central generalization is simple: any coherent bosonic mode that can be treated classically as $$|\Psi_\alpha(t)\rangle=e^{-i\varepsilon_\alpha t/\hbar}|\Phi_\alpha(t)\rangle,\qquad |\Phi_\alpha(t+T)\rangle=|\Phi_\alpha(t)\rangle.$$3 and coupled bilinearly as $$|\Psi_\alpha(t)\rangle=e^{-i\varepsilon_\alpha t/\hbar}|\Phi_\alpha(t)\rangle,\qquad |\Phi_\alpha(t+T)\rangle=|\Phi_\alpha(t)\rangle.$$4 generates Floquet replicas at $$|\Psi_\alpha(t)\rangle=e^{-i\varepsilon_\alpha t/\hbar}|\Phi_\alpha(t)\rangle,\qquad |\Phi_\alpha(t+T)\rangle=|\Phi_\alpha(t)\rangle.$$5 coupled by $$|\Psi_\alpha(t)\rangle=e^{-i\varepsilon_\alpha t/\hbar}|\Phi_\alpha(t)\rangle,\qquad |\Phi_\alpha(t+T)\rangle=|\Phi_\alpha(t)\rangle.$$6, dynamical gaps $$|\Psi_\alpha(t)\rangle=e^{-i\varepsilon_\alpha t/\hbar}|\Phi_\alpha(t)\rangle,\qquad |\Phi_\alpha(t+T)\rangle=|\Phi_\alpha(t)\rangle.$$7 at resonances, and, if the drive breaks time-reversal symmetry, topological band inversions (Hübener et al., 2018).

Two operating regimes are repeatedly distinguished. In the off-resonant case, with $$|\Psi_\alpha(t)\rangle=e^{-i\varepsilon_\alpha t/\hbar}|\Phi_\alpha(t)\rangle,\qquad |\Phi_\alpha(t+T)\rangle=|\Phi_\alpha(t)\rangle.$$8 well above the band width, one obtains a clean effective Hamiltonian

$$|\Psi_\alpha(t)\rangle=e^{-i\varepsilon_\alpha t/\hbar}|\Phi_\alpha(t)\rangle,\qquad |\Phi_\alpha(t+T)\rangle=|\Phi_\alpha(t)\rangle.$$9

with small absorption. In the on-resonant case, with $$|\Phi_\alpha(t)\rangle=\sum_m e^{-im\Omega t}|u_\alpha^m\rangle$$0 an interband transition, one expects large sidebands, strong mixing, and dynamical gaps at $$|\Phi_\alpha(t)\rangle=\sum_m e^{-im\Omega t}|u_\alpha^m\rangle$$1 satisfying $$|\Phi_\alpha(t)\rangle=\sum_m e^{-im\Omega t}|u_\alpha^m\rangle$$2. The drive amplitude $$|\Phi_\alpha(t)\rangle=\sum_m e^{-im\Omega t}|u_\alpha^m\rangle$$3 is chosen so that $$|\Phi_\alpha(t)\rangle=\sum_m e^{-im\Omega t}|u_\alpha^m\rangle$$4 matches the desired gap scale, with $$|\Phi_\alpha(t)\rangle=\sum_m e^{-im\Omega t}|u_\alpha^m\rangle$$5–$$|\Phi_\alpha(t)\rangle=\sum_m e^{-im\Omega t}|u_\alpha^m\rangle$$6 stated as typical, while the coupling $$|\Phi_\alpha(t)\rangle=\sum_m e^{-im\Omega t}|u_\alpha^m\rangle$$7 is set by deformation potential, phonon eigenvector overlap, or dipole matrix elements (Hübener et al., 2018).

In the low-frequency and nonlinear-phononics regime, the drive may be rectified into an effectively static lattice distortion. For a resonantly driven infrared-active phonon $$|\Phi_\alpha(t)\rangle=\sum_m e^{-im\Omega t}|u_\alpha^m\rangle$$8 coupled cubically to a Raman mode $$|\Phi_\alpha(t)\rangle=\sum_m e^{-im\Omega t}|u_\alpha^m\rangle$$9 through

$$(H_F)^{mn}=H_0\delta_{mn}+m\hbar\Omega\,\delta_{mn}+\frac{gQ_0}{2}M(\delta_{m,n+1}+\delta_{m,n-1}),$$0

the time average of $$(H_F)^{mn}=H_0\delta_{mn}+m\hbar\Omega\,\delta_{mn}+\frac{gQ_0}{2}M(\delta_{m,n+1}+\delta_{m,n-1}),$$1 produces a static shift

$$(H_F)^{mn}=H_0\delta_{mn}+m\hbar\Omega\,\delta_{mn}+\frac{gQ_0}{2}M(\delta_{m,n+1}+\delta_{m,n-1}),$$2

which in turn modifies couplings such as

$$(H_F)^{mn}=H_0\delta_{mn}+m\hbar\Omega\,\delta_{mn}+\frac{gQ_0}{2}M(\delta_{m,n+1}+\delta_{m,n-1}),$$3

A detailed case study is bilayer CrI$$(H_F)^{mn}=H_0\delta_{mn}+m\hbar\Omega\,\delta_{mn}+\frac{gQ_0}{2}M(\delta_{m,n+1}+\delta_{m,n-1}),$$4, where the relevant IR-active modes include a layer-shear mode at $$(H_F)^{mn}=H_0\delta_{mn}+m\hbar\Omega\,\delta_{mn}+\frac{gQ_0}{2}M(\delta_{m,n+1}+\delta_{m,n-1}),$$5 and a Raman shear mode at $$(H_F)^{mn}=H_0\delta_{mn}+m\hbar\Omega\,\delta_{mn}+\frac{gQ_0}{2}M(\delta_{m,n+1}+\delta_{m,n-1}),$$6. For $$(H_F)^{mn}=H_0\delta_{mn}+m\hbar\Omega\,\delta_{mn}+\frac{gQ_0}{2}M(\delta_{m,n+1}+\delta_{m,n-1}),$$7 and $$(H_F)^{mn}=H_0\delta_{mn}+m\hbar\Omega\,\delta_{mn}+\frac{gQ_0}{2}M(\delta_{m,n+1}+\delta_{m,n-1}),$$8, the review reports $$(H_F)^{mn}=H_0\delta_{mn}+m\hbar\Omega\,\delta_{mn}+\frac{gQ_0}{2}M(\delta_{m,n+1}+\delta_{m,n-1}),$$9, with $\Omega$00 and $\Omega$01; under appropriate displacement the interlayer exchange can switch sign, driving an antiferromagnet into a transient ferromagnetic state (Rodriguez-Vega et al., 2020).

These design rules imply that phonon Floquet engineering is best regarded not as a single protocol but as a hierarchy of strategies. In one limit, the coherent phonon is a direct periodic drive that creates Floquet replicas and topological masses. In another, it acts through nonlinear mode coupling to produce a rectified structural coordinate that renormalizes exchange or hopping. Across both limits, the decisive material parameters are the phonon frequency, coherence time, polarization, anharmonic coupling, and the microscopic matrix elements linking lattice motion to electronic, spin, or bosonic observables (Hübener et al., 2018, Rodriguez-Vega et al., 2020).