Acousto-Optical Floquet Engineering

• Acousto-optical Floquet engineering is a technique that uses coherent acoustic drives to impart spatiotemporal modulations on optical systems, creating quasienergy ladders and directional resonances.
• Integrated platforms employ piezoelectrically driven interdigital transducers and resonant modulators to achieve phase modulation, sideband generation, and efficient intermodal frequency conversion.
• This approach paves the way for programmable, multi-tone and quantum photonic applications by enabling engineered bandstructures and effective nonlinear interactions.

Acousto-optical Floquet engineering denotes the use of acoustically generated periodic drives to control optical systems in the Floquet sense: a coherent acoustic excitation imposes a time-periodic, or more generally spatiotemporally periodic, perturbation on an optical phase, propagation constant, mode coupling, cavity resonance, or emitter transition energy, thereby generating quasienergy ladders, sideband structure, mode hybridization, and direction-dependent resonances. In integrated implementations, the acoustic drive is typically produced piezoelectrically by interdigital transducers and conveyed either as a traveling surface or bulk acoustic wave or as a resonant mechanical mode; in quantum-emitter settings it can act as a periodic detuning drive. The resulting phenomena range from phase modulation and sideband generation to intermodal frequency conversion, nonreciprocity, and double dressing of light–matter states (Kittlaus et al., 2020, Freedman et al., 11 Feb 2025, Groll et al., 11 Sep 2025).

1. Floquet framework and its acousto-optic specialization

The formal point of departure is the standard Floquet problem

$$i\hbar \frac{\partial}{\partial t}\psi(x,t)=H(x,t)\psi(x,t), \qquad H(x,t+T)=H(x,t),$$

with solutions of the form

$$\psi_n(x,t)=u_n(x,t)e^{-i\varepsilon_n t/\hbar}, \qquad u_n(x,t+T)=u_n(x,t),$$

and quasienergies defined modulo $\hbar\omega$, where $\omega=2\pi/T$. In spatially periodic media this extends to spatiotemporal Bloch-Floquet waves and quasienergy bands $\varepsilon_n(k)$, so periodic driving reshapes not only instantaneous spectra but the effective dispersion governing long-time transport (Holthaus, 2015).

In acousto-optic systems, the drive is supplied by a coherent acoustic field. For traveling-wave devices, a useful reconstruction of the induced perturbation is

$\delta n(z,t)=\delta n_0\cos(\Omega t-qz+\phi),$

or more generally $$\delta n(\mathbf r,t)\propto \cos(\Omega t-\mathbf K\!\cdot\!\mathbf r+\phi)$$. This is the canonical form of a spatiotemporal Floquet drive: $\Omega$ generates the quasienergy ladder $\omega\to\omega\pm m\Omega$, while the acoustic momentum $q$ or $$\psi_n(x,t)=u_n(x,t)e^{-i\varepsilon_n t/\hbar}, \qquad u_n(x,t+T)=u_n(x,t),$$0 supplies a synthetic momentum bias that can differentiate forward and backward propagation. In the limiting case $$\psi_n(x,t)=u_n(x,t)e^{-i\varepsilon_n t/\hbar}, \qquad u_n(x,t+T)=u_n(x,t),$$1, the same formalism reduces to nearly pure temporal modulation and a symmetric sideband ladder (Kittlaus et al., 2020). Resonant cavity and suspended-waveguide modulators realize the same Floquet structure in a temporally periodic form, with optical fields of the type

$$\psi_n(x,t)=u_n(x,t)e^{-i\varepsilon_n t/\hbar}, \qquad u_n(x,t+T)=u_n(x,t),$$2

so the acoustic drive directly couples optical frequency bins separated by $$\psi_n(x,t)=u_n(x,t)e^{-i\varepsilon_n t/\hbar}, \qquad u_n(x,t+T)=u_n(x,t),$$3 (Freedman et al., 11 Feb 2025).

This specialization is important because acoustics naturally supplies both a drive frequency and, in traveling-wave implementations, a signed wavevector. That combination is what distinguishes acousto-optical Floquet engineering from purely electro-optic phase modulation: the acoustic field can implement both temporal modulation and momentum-selective coupling within the same physical actuator (Kittlaus et al., 2020, Zhang et al., 2023).

2. Platform classes and control primitives

Current acousto-optical Floquet hardware spans traveling-wave waveguides, resonantly enhanced suspended modulators, cavity-based micro-rings, and programmable bulk acousto-optic modulators. The common control primitive is an externally prescribed RF or microwave waveform converted into a coherent acoustic perturbation.

Platform	Acoustic drive	Floquet-relevant demonstrated effect
SOI silicon waveguides with AlN IDTs	Electrically driven SAWs, 1–5 GHz	Intramodal phase modulation, single-sideband intermodal conversion, broadband nonreciprocity (Kittlaus et al., 2020)
GaN-on-sapphire waveguides	Traveling Rayleigh mode near 0.99–1.00 GHz	Near-unity optical conversion and unidirectional phase matching with isolation ratio above 10 dB (Zhang et al., 2023)
CMOS-fabricated SiN$$\psi_n(x,t)=u_n(x,t)e^{-i\varepsilon_n t/\hbar}, \qquad u_n(x,t+T)=u_n(x,t),$$4/AlN suspended circuit	Resonant breathing mode at 2.31 GHz	Visible-light phase modulation with $$\psi_n(x,t)=u_n(x,t)e^{-i\varepsilon_n t/\hbar}, \qquad u_n(x,t+T)=u_n(x,t),$$5 rad and $$\psi_n(x,t)=u_n(x,t)e^{-i\varepsilon_n t/\hbar}, \qquad u_n(x,t+T)=u_n(x,t),$$6 (Freedman et al., 11 Feb 2025)
Hybrid TFLN-ChG racetrack micro-ring	SAW-driven ring at 0.84 GHz and higher modes	$$\psi_n(x,t)=u_n(x,t)e^{-i\varepsilon_n t/\hbar}, \qquad u_n(x,t+T)=u_n(x,t),$$7 as small as $$\psi_n(x,t)=u_n(x,t)e^{-i\varepsilon_n t/\hbar}, \qquad u_n(x,t+T)=u_n(x,t),$$8 and up to fifth-order sidebands (Wan et al., 2024)

A distinct enabling primitive is the “acoustic-optical multiplexer,” which combines guided optics and a guided acoustic wave into a single co-guided output waveguide. In a suspended 220 nm Si/SiO$$\psi_n(x,t)=u_n(x,t)e^{-i\varepsilon_n t/\hbar}, \qquad u_n(x,t+T)=u_n(x,t),$$9 platform it was simulated to provide peak acoustic directionality of $\hbar\omega$0 dB, acoustic insertion loss of $\hbar\omega$1 dB at $\hbar\omega$2 GHz, and optical crosstalk below $\hbar\omega$3 dB, thereby addressing the hardware problem of feeding a traveling acoustic wave into the same cross-section as the optical modes (Dostart et al., 2020).

Programmable bulk AOMs provide a different primitive: direct transfer of RF phase and frequency to diffracted optical orders. The phase-transfer law

$\hbar\omega$4

together with the frequency shift

$\hbar\omega$5

makes the AOM a fast actuator for externally prescribed optical phase dynamics. This is not, by itself, full Floquet band engineering, but it furnishes a software-defined time-dependent phase element that can be embedded in interferometric or networked photonic architectures (Satapathy et al., 2012).

3. Traveling spatiotemporal modulation, intermodal conversion, and nonreciprocity

The clearest integrated realization of traveling-wave acousto-optical Floquet engineering in standard silicon photonics is the SAW-driven SOI platform with co-integrated AlN IDTs. In that system, silicon’s elasto-optic response is used to generate direct acousto-optic modulation inside ridge waveguides. Three regimes were demonstrated: a normal-incidence phase modulator with effectively $\hbar\omega$6 and symmetric $\hbar\omega$7 sidebands; an angled-SAW intermodal single-sideband modulator in which the acoustic momentum is lithographically selected; and a long serpentine traveling-wave modulator that yields direction-dependent phase matching. The reported device metrics include operation from 1–5 GHz, $\hbar\omega$8–$\hbar\omega$9 for short phase modulators, single-sideband suppression of about 30 dB in the intermodal device, 13.5% conversion in a 0.96 mm serpentine modulator, insertion loss below 0.6 dB, and nonreciprocal bandwidth above 100 GHz with about 15 dB contrast over 0.8 nm (Kittlaus et al., 2020).

In Floquet terms, the intermodal condition

$\omega=2\pi/T$0

is an indirect interband transition driven by one temporal quantum $\omega=2\pi/T$1 and one momentum quantum $\omega=2\pi/T$2. Because only one sideband channel is phase matched, the modulation is single-sideband rather than symmetric phase modulation. The serpentine nonreciprocal device then uses the same traveling modulation to make the forward and backward resonance conditions different, with the backward direction suppressed by the nonreciprocal phase mismatch

$\omega=2\pi/T$3

so the observed nonreciprocity is a spatiotemporal phase-matching effect rather than a magneto-optic one (Kittlaus et al., 2020).

The GaN-on-sapphire traveling-wave platform pushes this logic into a higher-conversion regime. A 1-$\omega=2\pi/T$4m-thick GaN waveguide on sapphire supports both optical and acoustic confinement without suspension, and an electrically launched Rayleigh mode near 0.99–1.00 GHz drives TE$\omega=2\pi/T$5$\omega=2\pi/T$6TE$\omega=2\pi/T$7 intermodal conversion over an interaction length of 3 mm. The platform reaches near-unity optical conversion efficiency at 560 mW RF power, shows an optical modulation bandwidth of 3.5 nm, and demonstrates nonreciprocal propagation with isolation ratio above 10 dB under unidirectional phase matching (Zhang et al., 2023).

The central kinematic relations there are

$\omega=2\pi/T$8

for Stokes conversion and

$\omega=2\pi/T$9

for anti-Stokes conversion. Because the TE$\varepsilon_n(k)$0 mode has larger wavevector than TE$\varepsilon_n(k)$1, the Stokes process dominates in the chosen geometry. This directly realizes momentum-biased Floquet coupling in an integrated waveguide. A plausible implication is that such high-conversion traveling-wave devices are well suited to strong avoided crossings in synthetic quasienergy–momentum space, although the paper itself is framed as an intermodal mode converter and nonreciprocal modulator rather than a full quasiband experiment (Zhang et al., 2023).

4. Temporal modulation, sideband ladders, and cavity-enhanced frequency control

Not all acousto-optical Floquet engineering requires traveling-wave momentum bias. A second major branch uses resonantly enhanced temporal modulation to generate coherent sideband ladders and strong phase driving. In a visible-light CMOS-fabricated SiN$\varepsilon_n(k)$2/AlN suspended circuit, a piezoelectrically driven breathing mode modulates the optical propagation constant of a 2 mm device at 2.31 GHz. The measured modulation depth reaches $\varepsilon_n(k)$3 rad with 15 mW applied microwave power, the extracted figure of merit is $\varepsilon_n(k)$4, and first- and second-order sidebands can exceed the remaining carrier power. The paper is careful that this is a phase modulator rather than a nonreciprocal or traveling-wave Floquet lattice, but the coherent coupling among frequency bins $\varepsilon_n(k)$5 is directly measured and matches the Bessel-function prediction for pure sinusoidal phase modulation (Freedman et al., 11 Feb 2025).

A hybrid thin-film lithium-niobate/chalcogenide racetrack resonator shows the cavity-enhanced variant of the same principle. A SAW launched by an IDT periodically modulates the optical resonance, and electrode engineering is used to obtain constructive double-arm modulation in the two straight sections of the racetrack. The reported best performance is $\varepsilon_n(k)$6, acousto-optic coupling strength $\varepsilon_n(k)$7 at 0.84 GHz, microwave-to-optical conversion efficiency of 0.05%, and optical sidebands up to fifth order at 12 dBm RF power (Wan et al., 2024). In Floquet language, this is a periodically driven cavity of the form

$\varepsilon_n(k)$8

with resolved or marginally resolved sideband structure because the modulation frequencies 0.84, 1.48, and 1.52 GHz are comparable to or larger than the optical linewidths implied by the measured loaded $\varepsilon_n(k)$9 values.

A third temporally driven branch uses resonant guided-mode sensitivity in the mid-infrared. In a planar semiconductor structure operated in the Otto configuration, a 1 GHz longitudinal acoustic wave modulates both prism permittivity and air-gap thickness, thereby shifting the coupling to a guided or surface phonon-polariton resonance. For optimized positive-permittivity semiconductor waveguides, the calculated modulation coefficient reaches 100% at $\delta n(z,t)=\delta n_0\cos(\Omega t-qz+\phi),$0m with nanometric acoustic boundary motion; the mechanism is resonantly enhanced switching of prism-to-guided-mode coupling rather than bulk acousto-optic diffraction (Sopko et al., 2019). This suggests a route to acousto-optical Floquet control in wavelength ranges where conventional bulk acousto-optics is weak, although the paper itself remains a resonant modulation study rather than a sideband-resolved Floquet analysis.

5. Programmable waveforms, multi-tone control, and quantum extensions

A key feature of acousto-optical platforms is that the acoustic drive is electronically synthesized. That makes waveform design, phase programming, and multi-tone interference central to the field. The programmable AOM study on interferometric phase noise is exemplary: RF phase jumps with resolution down to $\delta n(z,t)=\delta n_0\cos(\Omega t-qz+\phi),$1 and intervals as short as 500 ns are transferred to the optical phase, producing order-dependent visibility

$\delta n(z,t)=\delta n_0\cos(\Omega t-qz+\phi),$2

with explicit forms

$\delta n(z,t)=\delta n_0\cos(\Omega t-qz+\phi),$3

This is not periodic Floquet driving in the narrow sense, but it establishes that digitally engineered RF waveforms can implement fast, externally prescribed optical phase trajectories, including stochastic and non-Gaussian drives that are useful for studying driven systems with controlled dephasing (Satapathy et al., 2012).

Transferable Floquet design rules for multi-tone modulation are supplied by cold-atom lattice studies. Two-frequency phase modulation of a one-dimensional lattice shows that commensurate tones create coherent sums of resonant pathways, allowing separate and simultaneous control over gap closing and reopening in the folded Floquet spectrum. In particular, $\delta n(z,t)=\delta n_0\cos(\Omega t-qz+\phi),$4 driving can give asymmetric gap control through interference, while $\delta n(z,t)=\delta n_0\cos(\Omega t-qz+\phi),$5 can close both inversion-point gaps simultaneously; the effective couplings are Bessel-dressed and phase sensitive (Sandholzer et al., 2021). A later experiment on anomalous one-dimensional Floquet topology shows that adding a second tone with controlled relative phase can set the sign configuration of the $\delta n(z,t)=\delta n_0\cos(\Omega t-qz+\phi),$6- and $\delta n(z,t)=\delta n_0\cos(\Omega t-qz+\phi),$7-gap windings, so multi-frequency control becomes a direct knob on gap-resolved topology rather than only on band gaps (Zhao et al., 2 Mar 2026). These are not acousto-optic experiments, but they provide the general Floquet design grammar most relevant to future acousto-optic multi-tone systems.

Periodic linear mode mixing can also engineer effective nonlinear optical interactions. In a generic two-mode nonlinear device, a repeated fast periodic sequence of $\delta n(z,t)=\delta n_0\cos(\Omega t-qz+\phi),$8-type mixing pulses yields the effective Hamiltonian

$\delta n(z,t)=\delta n_0\cos(\Omega t-qz+\phi),$9

which contains induced pair hopping,

$$\delta n(\mathbf r,t)\propto \cos(\Omega t-\mathbf K\!\cdot\!\mathbf r+\phi)$$0

and, in the classical limit, four-wave-mixing terms of the form $$\delta n(\mathbf r,t)\propto \cos(\Omega t-\mathbf K\!\cdot\!\mathbf r+\phi)$$1 and $$\delta n(\mathbf r,t)\propto \cos(\Omega t-\mathbf K\!\cdot\!\mathbf r+\phi)$$2 (Goldman, 2022). A plausible implication is that acoustically driven intermode couplers could synthesize analogous effective nonlinearities in photonic cavities and waveguides if the required periodic basis rotations are implemented acoustically.

The explicitly quantum acousto-optical Floquet case is the acoustically modulated single-photon emitter. A two-level emitter under strong optical driving and periodic acoustic detuning is described by

$$\delta n(\mathbf r,t)\propto \cos(\Omega t-\mathbf K\!\cdot\!\mathbf r+\phi)$$3

with $$\delta n(\mathbf r,t)\propto \cos(\Omega t-\mathbf K\!\cdot\!\mathbf r+\phi)$$4. Optical driving first produces the usual dressed states $$\delta n(\mathbf r,t)\propto \cos(\Omega t-\mathbf K\!\cdot\!\mathbf r+\phi)$$5; the acoustic modulation then Floquet-dresses those dressed states again. The resulting resonance-fluorescence spectrum shows odd-harmonic anticrossings, even-harmonic crossings, and central-line suppressions governed by symmetry and interference, while the optically dressed splitting is renormalized by a phonon-induced Bloch-Siegert shift

$$\delta n(\mathbf r,t)\propto \cos(\Omega t-\mathbf K\!\cdot\!\mathbf r+\phi)$$6

The accompanying feasibility study concludes that bulk acoustic waves interfaced with quantum dots are especially promising for experimental acousto-optical Floquet engineering (Groll et al., 11 Sep 2025).

6. Limits, misconceptions, and trajectories of the field

A recurrent misconception is that any acousto-optic sideband generation is already full Floquet engineering. The present literature shows a hierarchy instead. Temporal phase modulators and programmable AOMs clearly realize periodically driven or waveform-driven optical Hamiltonians, but they do not by themselves constitute a modulated lattice, a synthetic gauge field, or a topological quasienergy bandstructure (Freedman et al., 11 Feb 2025, Satapathy et al., 2012). Traveling-wave intermodal devices in silicon and GaN go further by adding acoustic momentum bias and direction-selective phase matching, yet even these works are better described as strong Floquet primitives than as completed topological Floquet media (Kittlaus et al., 2020, Zhang et al., 2023).

A second misconception concerns nonreciprocity. In the integrated waveguide experiments discussed here, nonreciprocity is not magneto-optic. It arises from spatiotemporal modulation with a directed acoustic wavevector and from the fact that forward and backward processes satisfy different momentum-conservation conditions. The nonreciprocal contrast is therefore a phase-matching effect of a traveling Floquet drive, not an equilibrium material asymmetry (Kittlaus et al., 2020, Zhang et al., 2023).

The main constraints are equally generic. In driven optical lattices, an optimal Floquet window requires

$$\delta n(\mathbf r,t)\propto \cos(\Omega t-\mathbf K\!\cdot\!\mathbf r+\phi)$$7

so that one avoids both low-frequency intraband resonances and high-frequency excitation into unwanted bands (Sun et al., 2018). Although integrated acousto-optic devices are not identical to shaken many-body lattices, this suggests an analogous engineering principle: the acoustic frequency should exceed the relevant low-energy optical scale one wants to average over, yet remain far enough from parasitic higher-order or higher-manifold resonances that uncontrolled mode conversion, excess loss, or heating-like spectral leakage do not dominate.

Open-system control is another frontier. Floquet-Born-Markov analysis of driven optical lattices coupled to a structured bosonic bath shows that carefully engineered bosonic mode spectra can suppress unwanted Floquet-Umklapp processes and stabilize target steady states (Schnell et al., 2023). This is only indirectly relevant to acousto-optical photonics, but it indicates that future hybrid phononic–photonic Floquet devices may need to treat acoustic modes not only as coherent drives but also as structured environments.

The near-term trajectory is therefore clear but still incomplete. The field already has CMOS-compatible traveling-wave SAW modulators in silicon, near-unity intermodal converters in GaN, visible-light GHz phase modulators with $$\delta n(\mathbf r,t)\propto \cos(\Omega t-\mathbf K\!\cdot\!\mathbf r+\phi)$$8 rad at milliwatt RF powers, nonsuspended micro-ring AOMs with $$\delta n(\mathbf r,t)\propto \cos(\Omega t-\mathbf K\!\cdot\!\mathbf r+\phi)$$9 in the millivolt-centimeter regime, and an explicit quantum-emitter theory of acousto-optical double dressing (Kittlaus et al., 2020, Zhang et al., 2023, Freedman et al., 11 Feb 2025, Wan et al., 2024, Groll et al., 11 Sep 2025). What remains to be demonstrated at scale are phase-coherent arrays of such elements, multi-tone and multi-site control with programmable relative phases, explicit quasienergy-band spectroscopy in photonic networks, and topological or anomalous Floquet phases built from acoustically programmable spatiotemporal media.