3.7 GHz phononics and piezo-optomechanics

L1 Phenomenon

microwave → phonon → photon

In thin-film LN, sound and light can be confined in the same wavelength-scale structure. At 2 GHz the acoustic wavelength in LN is about 2 µm (3978 m/s / 2 GHz = 1.99 µm derived), comparable to the 0.70 µm wavelength of 1550 nm light inside the crystal (1550 nm / n_o, n_o = 2.211 P-db): a nanostructured beam can trap both and let a single phonon modulate the light. Because LN is also piezoelectric, electrodes on the same chip couple that phonon to a microwave circuit. The result is a chain microwave → phonon → photon (piezo-optomechanics), which is how superconducting quantum processors operating at GHz can be connected to optical fibers (microwave-to-optical transduction).

Raise the pump on the blue side: the efficiency climbs past 5% and diverges at C = 1 (phonon lasing); on the red side it peaks at the external efficiency at C = 1 and then falls. The microwave-to-phonon stage (η_m about 0.1%) is the smallest factor.

Microwave-to-optical efficiency of the Jiang et al. 2020 transducer vs intracavity photon number

η vs n_c; solid: red; dashed: blue (ends at C = 1)

Stage	factor	running
microwave → phonon (IDT, horn, waveguide), η_m	−30.0 dB	−30.0 dB
phonon → photon in the cavity, 4C/(1 ∓ C)²	+8.5 dB	−21.6 dB
cavity → waveguide, η_o = κ_e/κ	−1.8 dB	−23.4 dB
chip → fiber, η_oc	−1.9 dB	−25.3 dB

n_c: 3.98e+04
C = C₀n_c: 0.478
internal gain: +08.5 dB
η total: 2.98e−03
C₀ (what if): 1.20e−05
η_m (what if): 9.90e−04

Intracavity photons n_c3.98e+04

What if: C₀ × 10^k1.20e−05

What if: η_m × 10^k9.90e−04

The reported 5.5% is the highest stable efficiency in the linear regime, reached by reducing the detuning with the pump near 300 µW; it is marked at its value, not as a point on the n_c axis. η_oc η_o η_m = 4.26×10⁻⁴ vs fitted η_e = 4.24×10⁻⁴.

Jiang et al. 2020, Nat. Commun. 11, 1166 (2020), CC BY 4.0 reported model derived

Suspended LN chip: IDT, horn, waveguide, optomechanical cavity, fiber

electrodesphonon packet1550 nm light, false color, dashedpacket radius ∝ (surviving fraction)^0.12

Stage factors, dB (bars from −40 to +20 dB)

end-to-end efficiency: 0.2985 %
cooperativity C: 0.478

Pump: intracavity photons3.98e+04

Pump detuning

What if: C₀1.20e−05

What if: η_m9.90e−04

Jiang et al. 2020, CC BY 4.0 reportedmodel deriveddrawing schematic, not to scalereported 5.5%: highest stable efficiency in the linear regime (blue, detuning reduced, pump near 300 µW, internal gain ≈ 21 dB), not a point of this curveη_oc η_o η_m = 4.26×10⁻⁴ vs fitted η_e = 4.24×10⁻⁴

L2 Scales, platforms, the transduction model

Scales. A 2 GHz phonon carries hf = 8.3 µeV; a 1550 nm photon carries 0.80 eV, about 10⁵ times more. Thermal occupation n = 1/(exp(hf/k_BT) − 1) of a 2 GHz mode is about 3.1×10³ at 300 K and 6.8×10⁻⁵ at 10 mK derived.

Device	Platform	Key numbers (reported)
Piezo-optomechanical crystal (Jiang et al. 2019, Optica) reported	suspended TFLN photonic-crystal nanobeam with electrodes	g₀/2π ≈ 120 kHz; cooperativity C > 1 with phonon lasing; microwave-to-mechanics quantum efficiency ≈ 10⁻⁸; phononic bandgap observed by piezoelectric drive and optical readout
Bidirectional piezo-optomechanical transducer (Jiang et al. 2020, Nat. Commun.) reported	all-LN suspended film: IDT, horn taper (SH2 to L1), 1.3 µm waveguide, optomechanical crystal; room temperature	f_m = 1.85 GHz; g₀/2π = 70 kHz (backaction), 84 kHz (from C₀); loaded optical Q 1.6×10⁵; IDT 1.9 mS, 3.36 MHz; acousto-optic V_π = 0.02 V; efficiency 10⁻⁵ at 3.3 µW red-detuned, 5.5% at 323 µW blue-detuned (internal gain ≈ 21 dB); C₀ = 1.2×10⁻⁵, η_e = 4.24×10⁻⁴, η_m = 9.9×10⁻⁴
Wavelength-scale waveguide transducer (Dahmani et al. 2020, PR Applied) reported	suspended TFLN, 1 µm waveguide	1.7 GHz; −12 dB conversion, limited by material loss
Nanobenders (Jiang et al. 2020, Commun. Phys.) reported	piezoelectric bending beams driving a sliced photonic-crystal cavity	0.01 nm/V standard vs tens of nm/V for nanobenders; (0.01 nm/V) R²U scaling (Eq. 1); ≈ 5 nm/V (0.6 THz/V) small-signal at \|U\| < 0.5 V; 1520 to 1560 nm (≈ 400 linewidths) within 4 V
GHz phononic integrated circuits (Mayor et al. 2021, PR Applied) reported	TFLN on sapphire, no suspension (index guiding)	50 Ω matched transducer; 1 µm waveguides; delay lines, racetrack resonators, meanders; Q ≈ 50,000 at 4 K; phononic four-wave mixing
Optically heralded microwave photon addition (Jiang et al. 2023, Nat. Phys.) reported	290 nm X-cut MgO:LN transfer-printed on 220 nm SOI (Si optomechanical crystal + LN piezo), NbTiN microwave chip, < 10 mK	5% continuous conversion (4.9 ± 0.5%); 15 Hz heralding; thermal noise < 2 microwave photons; f_m ≈ 3.6 GHz; g₀/2π ≈ 410 kHz

Numbers from arXiv versions (abstract or v1 text); journal versions may differ in the last digit.

η_{total} = η_{oc} η_{o} η_{m} \frac{4 C}{{(1 \pm C)}^{2}}, C = C_{0} n_{c}, C_{0} = \frac{4 g_{0}^{2}}{κ γ}

(1)

Model of the transduction figure (Jiang et al. 2020, Eq. 4 and App. A reported) at pump detuning Δ = ±ω_m: + for a red-detuned pump (beam-splitter interaction; ideal conversion at C = 1), − for a blue-detuned pump (two-mode squeezing; gain diverges at C = 1, phonon lasing). η_oc = 65.2 ± 0.4% (fiber-to-chip), η_o = κ_e/κ = 66 ± 1%.

Heralding. Step through the published protocol: detecting one optical photon adds exactly one photon to the microwave resonator.

Optically heralded microwave photon addition (Jiang et al. 2023)

Setup below 10 mK: LN piezo block on a Si optomechanical crystal, NbTiN microwave resonator chip
A blue-detuned optical pulse drives the optomechanical cavity (two-mode squeezing)
The interaction creates a correlated optical photon and a mechanical phonon near 3.6 GHz
A click on the optical single-photon detector heralds that one phonon was added
Piezoelectric coupling turns the phonon into a microwave photon in the resonator

continuous conversion efficiency: 4.9 ± 0.5 %
heralding rate: 15 Hz
added thermal noise: < 2 microwave photons
f_m, g₀/2π: ≈ 3.6 GHz, ≈ 410 kHz
n_th at 3.6 GHz: 300 K / 10 mK: 1.7×10³ / 3.1×10⁻⁸

Plot area for the published curves (Fig. 2, Fig. 3, Extended Data Figs. 1, 2, 3, 5; Zenodo 10.5281/zenodo.7903643, CC BY 4.0). It holds no data.

Jiang et al. 2023, arXiv abstract and text reported thermal occupation derived data record reported

Exploded stack, drawn schematically

active in this stepoptical path, false color, dashed

Protocol

Step 0/4: Setup below 10 mK: LN piezo block on a Si optomechanical crystal, NbTiN microwave resonator chip

Step

Reported

conversion efficiency, continuous: 4.9 ± 0.5 %
heralding rate: 15 Hz
added thermal noise: < 2 photons
thermal phonons at 3.6 GHz, 300 K / 10 mK: 1.7×10³ / 3.1×10⁻⁸
f_m, g₀/2π: ≈ 3.6 GHz, ≈ 410 kHz

Plot area for the published curves (Zenodo 10.5281/zenodo.7903643, CC BY 4.0). It holds no data.

Jiang et al., Nat. Phys. 19, 1423 (2023) reporteddata record CC BY 4.0 reported

Nanobenders. Stretch the beam tenfold and the bender's displacement grows a hundredfold while the plain plate does not move more: geometry, not material, supplies the gain.

Displacement per volt vs aspect ratio R = L/w (scaling law, prefactor 0.01 nm/V)

R: 10.0
plate, nm/V: 1.0e−02
transverse extension, nm/V: 1.0e−01
bender, nm/V: 1.0e+00
displacement at U, selected type: 1.0e+00 nm

Reported optical tuning (Jiang et al. 2020)	value
small-signal slope, linear fit at \|U\| < 0.5 V	≈ 5 nm/V (0.6 THz/V)
span within 4 V (nonlinear curve; not a slope)	1520 to 1560 nm, ≈ 400 linewidths

Aspect ratio R 10.0

Voltage U1.00 V

Jiang et al., Commun. Phys. 3, 156 (2020), CC BY 4.0, Eq. 1 reported d₂₂, d₁₅ from Kushibiki et al. 1999 derived lines are order-of-magnitude scaling, not FEM

Bending beam with a field reversing across its width; curvature drawn ×40

electrodes on and along the beamLN beam; drawn length grows with R

Displacement per volt at R = 10.0

nm/V on a log scale from 10⁻³ to 10³; prefactor 0.01 nm/V (Jiang et al. 2020 Eq. 1)

Aspect ratio R 10.0

Voltage U1.00 V

Jiang et al. 2020, CC BY 4.0 reportedreported tuning: ≈ 5 nm/V small-signal, 1520 to 1560 nm within 4 Vbars are order-of-magnitude scaling, not FEMreference: d₂₂ = 20.3 pm/V, d₁₅ = 66.4 pm/V derived

Other groups. Microwave-to-optical conversion with thin-film LN acoustic resonators (Shao et al. 2019 reported; room temperature, up to 4.5 GHz); an integrated LN-on-Si microwave-to-optics interface reaching 0.9% at 1 µW (Weaver et al. 2024 reported). Same lab, quantum acoustics: phonon-number resolution with a qubit coupled to LN phononic-crystal resonators (Arrangoiz-Arriola et al. 2018, 2019 S S), entangled mechanical resonators (Wollack et al. 2022 S), loss channels of LN phononic-crystal resonators at mK (Wollack et al. 2021 S); a perspective on integrated phononics (Safavi-Naeini et al. 2019 S).

L3 Coupling, bookkeeping, material limits

Optomechanical coupling. g₀ in LN combines photoelastic and moving-boundary terms; the photoelastic term depends on in-plane orientation through the rotated p tensor (3.6 L3), with the optimum stated for X-cut by Jiang et al. 2019 in that paper's own in-plane angle convention, which is not mapped to the convention used here; the X-cut optimum also inherits the unresolved p₁₄/p₄₁ sign pairing.

Efficiency bookkeeping. Total = optical coupling × optical external ratio × microwave-to-mechanics × internal optomechanical term; in the Jiang et al. 2020 transducer the microwave-to-mechanics factor (η_m ≈ 0.1%) dominates the loss. The “what if” sliders of the transduction figure move C₀ and η_m by up to a decade.

Material limits named by the papers. Optical absorption heating the mechanics at mK (Jiang et al. 2023: added thermal noise), two-level-system and other loss channels in LN phononic crystals at cryogenic temperature (Wollack et al. 2021), material loss limiting waveguide transduction (Dahmani et al. 2020).

References

3.7

Source	Reference	Link
Jiang et al. 2019	Jiang et al., Optica 6 845 (2019). Lithium niobate piezo-optomechanical crystals	doi:10.1364/OPTICA.6.000845
Arrangoiz-Arriola et al. 2019	Arrangoiz-Arriola et al., Nature 571 537-540 (2019). Resolving the energy levels of a nanomechanical oscillator	doi:10.1038/s41586-019-1386-x
Jiang et al. 2020	Jiang et al., Nature Communications 11 1166 (2020). Efficient bidirectional piezo-optomechanical transduction between microwave and optical frequency	doi:10.1038/s41467-020-14863-3
Dahmani et al. 2020	Dahmani et al., Physical Review Applied 13 024069 (2020). Piezoelectric Transduction of a Wavelength-Scale Mechanical Waveguide	doi:10.1103/PhysRevApplied.13.024069
Jiang et al. 2020	Jiang et al., Communications Physics 3 156 (2020). Nanobenders as efficient piezoelectric actuators for widely tunable nanophotonics at CMOS-level voltages	doi:10.1038/s42005-020-00412-3
Mayor et al. 2021	Mayor et al., Physical Review Applied 15 014039 (2021). Gigahertz Phononic Integrated Circuits on Thin-Film Lithium Niobate on Sapphire	doi:10.1103/PhysRevApplied.15.014039
Sarabalis et al. 2021	Sarabalis et al., Optica 8 477 (2021). Acousto-optic modulation of a wavelength-scale waveguide	doi:10.1364/OPTICA.413401
Wollack et al. 2021	Wollack et al., Applied Physics Letters 118 123501 (2021). Loss channels affecting lithium niobate phononic crystal resonators at cryogenic temperature	doi:10.1063/5.0034909
Jiang et al. 2023	Jiang et al., Nature Physics 19 1423-1428 (2023). Optically heralded microwave photon addition	doi:10.1038/s41567-023-02129-w
Jiang et al. 2023	Jiang et al., Zenodo (2023). Optically heralded microwave photon addition	doi:10.5281/zenodo.7903643
Shao et al. 2019	Shao et al., Optica 6 1498 (2019). Microwave-to-optical conversion using lithium niobate thin-film acoustic resonators	doi:10.1364/OPTICA.6.001498
Weaver et al. 2024	Weaver et al., Nature Nanotechnology 19 166-172 (2024). An integrated microwave-to-optics interface for scalable quantum computing	doi:10.1038/s41565-023-01515-y
Arrangoiz-Arriola et al. 2018	Arrangoiz-Arriola et al., Physical Review X 8 031007 (2018). Coupling a Superconducting Quantum Circuit to a Phononic Crystal Defect Cavity	doi:10.1103/PhysRevX.8.031007
Wollack et al. 2022	Wollack et al., Nature 604 463-467 (2022). Quantum state preparation and tomography of entangled mechanical resonators	doi:10.1038/s41586-022-04500-y
Safavi-Naeini et al. 2019	Safavi-Naeini et al., Optica 6 213 (2019). Controlling phonons and photons at the wavelength scale: integrated photonics meets integrated phononics	doi:10.1364/OPTICA.6.000213
Zelmon et al. 1997	Zelmon, Small & Jundt, JOSA B 14 3319 (1997)	doi:10.1364/JOSAB.14.003319
Kushibiki et al. 1999	Kushibiki et al., IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control 46 1315-1323 (1999). Accurate measurements of the acoustical physical constants of LiNbO3 and LiTaO3 single crystals	doi:10.1109/58.796136