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This paper is available on arxiv under CC 4.0 license. Authors: (1) Terence Blésin, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL); (2) Wil Kao, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL); (3) Anat Siddharth, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL); (4) Alaina Attanasio, OxideMEMS lab, Purdue University; (5) Hao Tian, OxideMEMS lab, Purdue University; (6) Sunil A. Bhave, OxideMEMS lab, Purdue University; (7) Tobias J. Kippenberg, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL). Table of Links Introduction Results A. Physics and design B. Device characterization C. Bidirectional microwave-optical transduction D. Pulsed transduction Discussion Acknowledgments Appendix A: Theory Appendix B: Fabrication process flow Appendix C: Optical characterization Appendix D: Characterization of acoustic resonances Appendix E: Characterization of bidirectional transduction Appendix F: Optimization of optical extraction efficiency Appendix F: Optimization of optical extraction efficiency An optimal optical external coupling can be chosen to maximize the total conversion efficiency. Equation E9 can be rewritten in terms of quality factors by using the relation κ = ω/Q, which yields For the optics, we have Qo = Qint + Qext, where we separate the intrinsic quality factor Qint, oftentimes fabricationlimited and hence not easily adjustable, from the external coupling quality factor Qex that can be readily engineered through coupler design. The optical extraction efficiency can thus be written in the form where R = Qint/Qex and the total optical quality factor The total efficiency in the low-cooperativity regime is then given by where For a given F, the optimal efficiency is achieved at the critical coupling condition Qex = Qint, or R = 1. [1] H. Haus and W. Huang, Proceedings of the IEEE 79, 1505 (1991). [2] T. Blésin, H. Tian, S. A. Bhave, and T. J. Kippenberg, Phys. Rev. A 104, 052601 (2021). [3] A. Rueda, W. Hease, S. Barzanjeh, and J. M. Fink, npj Quantum Inf. 5, 1 (2019). [4] C. Zhong, Z. Wang, C. Zou, M. Zhang, X. Han, W. Fu, M. Xu, S. Shankar, M. H. Devoret, H. X. Tang, and L. Jiang, Phys. Rev. Lett. 124, 010511 (2020). [5] S. Krastanov, H. Raniwala, J. Holzgrafe, K. Jacobs, M. Lončar, M. J. Reagor, and D. R. Englund, Phys. Rev. Lett. 127, 040503 (2021). [6] L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, Nature 414, 413 (2001). [7] M. H. P. Pfeiffer, J. Liu, A. S. Raja, T. Morais, B. Ghadiani, and T. J. Kippenberg, Optica 5, 884 (2018). [8] J. Liu, G. Huang, R. N. Wang, J. He, A. S. Raja, T. Liu, N. J. Engelsen, and T. J. Kippenberg, Nat. Commun. 12, 2236 (2021). [9] H. Tian, J. Liu, B. Dong, J. C. Skehan, M. Zervas, T. J. Kippenberg, and S. A. Bhave, Nat. Commun. 11, 3073 (2020). [10] A. Siddharth, A. Anastasio, G. Lihachev, J. Zhang, Z. Qiu, S. Kenning, R. N. Wang, S. A. Bhave, J. Riemensberger, and T. J. Kippenberg, Hertz-linewidth and frequency-agile photonic integrated extended-DBR lasers (2023), arxiv:2306.03184 [physics]. [11] H. Tian, J. Liu, A. Siddharth, R. N. Wang, T. Blésin, J. He, T. J. Kippenberg, and S. A. Bhave, Nat. Photon. 15, 828 (2021). [12] M. Gao, Q.-F. Yang, Q.-X. Ji, H. Wang, L. Wu, B. Shen, J. Liu, G. Huang, L. Chang, W. Xie, S.-P. Yu, S. B. Papp, J. E. Bowers, T. J. Kippenberg, and K. J. Vahala, Nat. Commun. 13, 3323 (2022). [13] V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, in Laser Applications, Vol. 2097 (SPIE, 1994) pp. 283–288. [14] A. Schliesser, P. Del’Haye, N. Nooshi, K. J. Vahala, and T. J. Kippenberg, Phys. Rev. Lett. 97, 243905 (2006). [15] K. Leong and J. Mazierska, IEEE Trans. Microw. Theory Tech. 50, 2115 (2002).

Bidirectional Microwave-optical Transduction Based on Integration: Appendix F: Optimization

This paper is available on arxiv under CC 4.0 license. Authors: (1) Terence Blésin, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL); (2) Wil Kao, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL); (3) Anat Siddharth, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL); (4) Alaina Attanasio, OxideMEMS lab, Purdue University; (5) Hao Tian, OxideMEMS lab, Purdue University; (6) Sunil A. Bhave, OxideMEMS lab, Purdue University; (7) Tobias J. Kippenberg, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL). Table of Links Introduction Results A. Physics and design B. Device characterization C. Bidirectional microwave-optical transduction D. Pulsed transduction Discussion Acknowledgments Appendix A: Theory Appendix B: Fabrication process flow Appendix C: Optical characterization Appendix D: Characterization of acoustic resonances Appendix E: Characterization of bidirectional transduction Appendix F: Optimization of optical extraction efficiency Appendix C: Optical characterization 1. Resonance linewidths Fitting the hybridized transmission spectrum in Fig. 2c to the coupled mode theory presented in Appendix A 1 and in Ref. [2] yields κr/(2π) = 154 MHz and κl/(2π) = 190 MHz, with κex,l/(2π) = κex,r/(2π) = 60 MHz. 2. Hybridization of optical resonances via integrated thermo-optic heaters For the purpose of rapid characterization of the photonic molecules at room temperature, the bottom electrode of the piezoelectric actuator also serves as an integrated heater to control the relative detuning between the microrings by thermo-optic effect. As seen in Fig. 2a, we pattern Mo to make three electrical connections to the bottom electrode. The top and bottom connections are connected to ground, whereas the central connection is biased to a constant voltage. The finite resistance of Mo at room temperature leads to Joule heating as current flows through the bottom electrode, modifying the refractive index of the Si3N4 waveguide buried directly underneath. 3. Thermal response characterization through cavity-enhanced photothermal spectroscopy. present transducer due to optical pump heating. Specifically, thermally induced refractive index change causes a shift in resonance frequency of the Si3N4 micro-ring, which is detected using cavity-enhanced pump-probe spectroscopy [12]. The intensity-modulated pump addresses a TE resonance, while the probe measures the resulting side-of-fringe modulation of another TE mode. To ascertain the physical mechanism of each cross phase modulation (XPM) process, we sweep the pump modulation frequency to leverage the separation of time scales. Seen in Supplementary Fig. 2, the response exhibits three plateaus. We associate the process at modulation frequencies ≳ 1 MHz with Kerr-induced XPM. The two slower processes at ∼ 100 kHz and ∼ 1 kHz are characteristic of photothermal XPM in quasi-freestanding microresonators such as spheres [13] and toroids [14]. Here, the former “local” time scale can be attributed to thermalization of the mode volume with the acoustic resonator, while the thermalization of the suspended structure with the rest of the chip constitutes the latter “global” time scale. Therefore, it is expected that for future cryogenic operations, employing a pulse-on time (τon) shorter than the local time scale to gate the optical pump will significantly reduce thermal occupancy in the cladding acoustic mode. The pulse repetition rate (frep) and hence pulse-off time can then be chosen based on the available cooling power.

Bidirectional Microwave-optical Transduction Based on Integration: Appendix C: Optical

This paper is available on arxiv under CC 4.0 license. Authors: (1) Terence Blésin, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL); (2) Wil Kao, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL); (3) Anat Siddharth, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL); (4) Alaina Attanasio, OxideMEMS lab, Purdue University; (5) Hao Tian, OxideMEMS lab, Purdue University; (6) Sunil A. Bhave, OxideMEMS lab, Purdue University; (7) Tobias J. Kippenberg, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL). Table of Links Introduction Results A. Physics and design B. Device characterization C. Bidirectional microwave-optical transduction D. Pulsed transduction Discussion Acknowledgments Appendix A: Theory Appendix B: Fabrication process flow Appendix C: Optical characterization Appendix D: Characterization of acoustic resonances Appendix E: Characterization of bidirectional transduction Appendix F: Optimization of optical extraction efficiency Appendix E: Characterization of bidirectional transduction 1. Estimation of cooperavity and single-photon coupling rate Given the measured off-chip efficiency, the single-photon coupling rate can be estimated. The off-chip efficiency is given by where ηprobes and ηfiber-chip denote the microwave and optical insertion losses, respectively. The on-chip efficiency can be written as The internal efficiency in the low-cooperativity regime, where the plus and minus signs correspond to the anti-Stokes and Stokes configurations, respectively. The optical and microwave extraction efficiencies The single-photon cooperativity C0 is enhanced by the intracavity photons where the pump detuning is assumed to be small. The input power in the bus waveguide Pwg is related to the power in the input fiber Pin by the fiber-chip coupling efficiency The off-chip efficiency with respect to the microwave probe and optical fiber is thus 2. Estimation of on-chip and internal transduction efficiencies We have achieved η tot = −48 dB at an input pump power of 21 dBm. With the losses quoted in Appendix E 1, the on-chip and internal efficiencies can be estimated. For conversion in either directions, the signal goes through the respective input and output ports exactly once, acquiring an attenuation of η probesη fiber−chip = −7 dB. We therefore have η oc = 7.9×10−5 . Knowing the extraction efficiencies ηo = 35% and ηm = 11%, we further obtain η int = 2×10−3 .

Bidirectional Microwave-optical Transduction Based on Integration: Appendix E: Characterization

This paper is available on arxiv under CC 4.0 license. Authors: (1) Terence Blésin, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL); (2) Wil Kao, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL); (3) Anat Siddharth, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL); (4) Alaina Attanasio, OxideMEMS lab, Purdue University; (5) Hao Tian, OxideMEMS lab, Purdue University; (6) Sunil A. Bhave, OxideMEMS lab, Purdue University; (7) Tobias J. Kippenberg, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL). Table of Links Introduction Results A. Physics and design B. Device characterization C. Bidirectional microwave-optical transduction D. Pulsed transduction Discussion Acknowledgments Appendix A: Theory Appendix B: Fabrication process flow Appendix C: Optical characterization Appendix D: Characterization of acoustic resonances Appendix E: Characterization of bidirectional transduction Appendix F: Optimization of optical extraction efficiency Appendix D: Characterization of acoustic resonances The microwave reflection measurements (also known as S11) shown here were measured at room temperature using custom probes (GGB industries nickel-alloy 40A-GS-135-PC-N Picoprobe) and a vector network analyzer (Rhode&Schwarz ZNB-20). The spectrum shown in Fig. 2d is corrected using a calibration substrate (GGB industries CS-8) to remove the phase delay and attenuation introduced by coaxial cables and probes. However, this calibration procedure was only used to get the S-parameter, and was not applied during the total conversion efficiency measurements shown in Fig. 3 and Fig. 4. In a previous version of the transducers and in Ref. [11] where the Band-Aid process (Appendix B) is not employed, the piezoelectric actuator extends beyond the suspended cladding and covers the electrical feedline, causing two detrimental effects for transduction. First, the resulting stray capacitance leads to excess microwave insertion loss. Given the small size of the suspended area (∼ 1000 µm2 ) compared to that of the feedline (≳ 10 000 µm2 ), this effect is readily seen in Supplementary Fig. 3, where the spectrum without the Band-Aid process exhibits reduced background reflection. Second, the part of the actuator on top of the unreleased cladding leads to resonant excitation of HBAR modes extending over the entire Si substrate. These modes have a smaller free spectral range (FSR) of around 19 MHz and a lower microwave extraction efficiency. The theory presented in Appendix A does not correspond to the case of such a multimode system as it assumes being in the sideband-resolved regime, implying the high density of substrate modes is inappropriate for low-noise frequency conversion. The feedline Band-Aid, discussed in Appendix B and shown in Supplementary Fig. 3, enables the removal of the piezoelectric layer above the substrate, leaving only HBAR modes well confined in the oxide cladding. We fit the measured complex reflection to obtain the mechanical quality factor Qm = 284 and microwave extraction efficiency κex,m/κm = 0.11 for the transduction mode [15]. In addition, we simulate microwave reflection of the designed stack geometry using finite-element method (FEM). Fitting the simulated spectrum leads to Qm = 205 and κex,m/κm = 0.61. The discrepancy is attributed to difference in clamping losses and piezoelectric coefficients between the fabricated and simulated devices. We extract the effective mass of the transduction HBAR mode from FEM simulations as

Bidirectional Microwave-optical Transduction Based on Integration: Appendix D: Characterization

This paper is available on arxiv under CC 4.0 license. Authors: (1) Terence Blésin, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL); (2) Wil Kao, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL); (3) Anat Siddharth, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL); (4) Alaina Attanasio, OxideMEMS lab, Purdue University; (5) Hao Tian, OxideMEMS lab, Purdue University; (6) Sunil A. Bhave, OxideMEMS lab, Purdue University; (7) Tobias J. Kippenberg, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL). Table of Links Introduction Results A. Physics and design B. Device characterization C. Bidirectional microwave-optical transduction D. Pulsed transduction Discussion Acknowledgments Appendix A: Theory Appendix B: Fabrication process flow Appendix C: Optical characterization Appendix D: Characterization of acoustic resonances Appendix E: Characterization of bidirectional transduction Appendix F: Optimization of optical extraction efficiency Appendix B: Fabrication process flow The piezoelectric actuators are monolithically integrated on Si3N4 waveguides, fabricated using the photonic Damascene process [7, 8]. The 850-nm-thick Si3N4 film is deposited using low-pressure chemical vapor deposition into the Damascene preform, with 2.8 µm thermal oxide below the waveguides. After annealing to drive hydrogen impurities out of the Si3N4 layer, 2.1-µm-thick TEOS oxide and 1.0-µm-thick low-temperature oxide top claddings are deposited and subsequently annealed. The metallic and piezoelectric films—95 nm of Mo, 1.0 µm of AlN and another 95 nm of Mo—are sputtered through foundry services provided by Plasma-Therm [9]. They are then patterned using deep reactive ion etching (DRIE) to form the actuators, ground planes and integrated heaters. We employ the same BandAid process as in Ref. [10] to localize the electrodes atop the suspended cladding. In particular, when connecting the Al feedline to the top electrode by a lift-off process, the bottom electrode is etched back using XeF2 to avoid shortcircuiting the electrodes. The process to suspend the cladding in the vicinity of the actuator is combined with the chip singulation steps of the Damascene process. First, a hole is opened in the middle of the donut-shaped actuator. The chip facets are simultaneously defined where trenches in the SiO2 are etched by C4F8 DRIE between neighboring chips. A second photolithography is performed to protect the facets while leaving the etched holes exposed for further processing, resulting in narrower trenches between chips. The Si substrate is then isotropically etched with SF6 until the SiO2 cladding below the actuator is suspended [11]. Following cladding suspension, the Si substrate is etched anisotropically until the desired chip thickness of 250 µm is reached. A final isotropic Si etch removes the parts of the substrate protruding from the SiO2 facet, providing proper access to the bus waveguide nanotapers with lensed fibers, or even using the butt-coupling scheme. To facilitate this last step, we employ ion-beam etching at a 20◦ tilt angle to remove the passivation layer formed on the hole sidewalls during the Si DRIE. The chips are ultimately released by grinding the back side of the wafer. This convoluted process ensures that all the actuators on the wafer are fully suspended, while the chip facets do not become exceedingly fragile from the undercut.

Bidirectional Microwave-optical Transduction Based on Integration: Appendix B: Fabrication

This paper is available on arxiv under CC 4.0 license. Authors: (1) Terence Blésin, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL); (2) Wil Kao, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL); (3) Anat Siddharth, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL); (4) Alaina Attanasio, OxideMEMS lab, Purdue University; (5) Hao Tian, OxideMEMS lab, Purdue University; (6) Sunil A. Bhave, OxideMEMS lab, Purdue University; (7) Tobias J. Kippenberg, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL). Table of Links Introduction Results A. Physics and design B. Device characterization C. Bidirectional microwave-optical transduction D. Pulsed transduction Discussion Acknowledgments Appendix A: Theory Appendix B: Fabrication process flow Appendix C: Optical characterization Appendix D: Characterization of acoustic resonances Appendix E: Characterization of bidirectional transduction Appendix F: Optimization of optical extraction efficiency Appendix A: Theory 1. Coupled mode theory for optical modes hybridization Using a pair of optical resonances increases the intracavity photon number for a given input optical power, resulting in an enhanced optomechanical interaction. Such an optical doublet is implemented here by evanescently coupling two micro-rings, forming a photonic molecule. Consider two bare optical modes with resonance frequencies ωl and ωr that correspond to the left and right micro-rings, respectively. The two modes couple with a rate J, which yields the Hamiltonian [1] and the equations of motion With Laplace transform, they can be rewritten in frequency domain as 2. Transducer Hamiltonian and input-output relations In terms of bare micro-ring modes, the transducer Hamiltonian is given by Note that the piezo-coupled acoustic mode interacts with only one of the micro-rings. To compute quantities directly related to the experiments, we rewrite the Hamiltonian in terms of the supermodes (Appendix A 1) as Finally, by defining effective optomechanical coupling rates and going into a rotating frame defined by the operator we have in the interaction picture. The time evolution of the bosonic modes can be computed using the Heisenberg-Langevin equations given by We can move to the frequency domain by using the Fourier transform along with the properties We then have 3. Conversion efficiency and added noise For the following analysis, we consider the triply resonant configuration where ω+ −ω− = ωm and the optical pump is on resonance with one of the optical supermodes. First, we consider the anti-Stokes configuration where ∆− = 0 and ∆+ = −ωm. From Eq. A24, we have the input-output relation The equations of motion is given by 4. Estimation of correlated microwave-optical pair generation rates Equations A40 and A43 suggest that entangled pairs of microwave and optical photons can be generated in the Stokes configuration since it gives rise to an effective two-mode squeezing Hamiltonian [3–5]. In order to verify that the HBAR transducer can generate entangled pairs of microwave and optical photons, we compute the on-chip pair generation rate, expected to be equal to the output photon flux from the lower frequency optical mode as in standard spontaneous parametric down-conversion (SPDC) scheme. Integrating over the transduction bandwidth, we have In practice, the pump laser should be filtered for the optical output to be dominated by heralding photons, and other losses from the measurement setup can further reduce the effective rate. For η oc = 7.9×10−5 , κ− = 2π×300 MHz and κm = 2π × 10 MHz, this would corresponds to an on-chip pair generation rate of 190 Hz. Losses in the measurement chain will have to be accounted for to estimate the count rate at the detector. In order to measure entanglement between the microwave and optical fields, the on-chip rate must be higher than the thermal decoherence rate of the acoustic mode κmnth,m(ωm), where the thermal occupancy nth,m(ωm) is given by the Bose-Einstein distribution. For ωm = 2π×3.5 GHz and κm = 2π×10 MHz, the decoherence rate at 800 mK is 43 MHz and 0.5 Hz at 10 mK. Therefore, the transducer needs to be operated in the mixing chamber of the dilution refrigerator, with a pulsed pump to reduce heat load. The total rate of optical heralding is hence gated by the pump duty cycle.

Bidirectional Microwave-optical Transduction Based on Integration: Appendix A: Theory

This paper is available on arxiv under CC 4.0 license. Authors: (1) Terence Blésin, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL); (2) Wil Kao, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL); (3) Anat Siddharth, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL); (4) Alaina Attanasio, OxideMEMS lab, Purdue University; (5) Hao Tian, OxideMEMS lab, Purdue University; (6) Sunil A. Bhave, OxideMEMS lab, Purdue University; (7) Tobias J. Kippenberg, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL). Table of Links Introduction Results A. Physics and design B. Device characterization C. Bidirectional microwave-optical transduction D. Pulsed transduction Discussion Acknowledgments Appendix A: Theory Appendix B: Fabrication process flow Appendix C: Optical characterization Appendix D: Characterization of acoustic resonances Appendix E: Characterization of bidirectional transduction Appendix F: Optimization of optical extraction efficiency III. DISCUSSION Non-classically correlated microwave-optical photon pairs can be generated through spontaneous parametric down-conversion in our transducer. These correlated photon pairs constitute a key ingredient toward entangling distant quantum processors via the Duan-LukinCirac-Zoller protocol [4, 45]. With the current device parameters, the pair generation rate on chip amounts to 190 Hz, which is well above the thermal decoherence rate for 3.5 GHz at 10 mK as explained in Appendix A 4. However, losses in the measurement setup specific to such an experiment, as well as the gating of the optical pump required to alleviate the effective heat load, are expected to limit further the final heralding rate. Quasi-free-standing structures, such as 1D OMCs, do not readily thermalize and are thus more susceptible to heating effects. Even though our transducer also utilizes a suspended acoustic resonator, it may be feasible to operate at a higher duty cycle than what is presented in Fig. 4. It has been shown that a buffer gas environment can facilitate thermalization of a 1D OMC, at the cost of increased damping of the mechanical breathing mode [46]. On the contrary, the HBAR mode of interest here should be relatively insensitive to such viscous damping, as the acoustic wave propagates predominantly along the longitudinal direction inside the cladding. Therefore, a sample cell affixed to the mixing chamber flange and filled with buffer superfluid helium—an inviscid fluid that is both an excellent thermal conductor and electrical insulator—could potentially improve the heralding rate without hampering the conversion efficiency [47]. The present design is also suited for tasks beyond quantum state transfer. The transducer can be a key component of photonic interconnects for superconducting qubits [7, 8], where laser lights routed through optical fibers are used to encode microwave signals directly inside the dilution refrigerator during qubit readout. It may be challenging for traveling-wave EOMs to attain the requisite half-wave voltage to be competitive in noise performance against conventional all-electrical readout scheme utilizing high-electron-mobility transistor amplifiers [8]. Our triply resonant approach affords a path towards a scalable optically controlled cryogenic waveform generator for both qubit readout and control. The multimode nature of the transducer is particularly suited for frequency-multiplexed dispersive readout. Transduction leveraging multiple HBAR modes is already demonstrated in Fig. 3e. One can additionally utilize line-type instead of point-type dimer coupling for the photonic molecule, which gives rise to a dispersion in the supermode splitting [48]. In combination with the multitude of acoustic overtones, a single transducer can therefore support several spectrally distinguishable triply resonant systems that serve as qubit multiplex channels. Finally, since the transducer requires no superconducting element to function, it may be pertinent for applications in classical microwave photonics. Optimization of design and fabrication should engender further improvement on device performance. The conversion efficiency, proportional to microwave extraction efficiency (Eq. 2), can be improved by replacing AlN with scandium-doped AlN [49], which features a piezoelectric coefficient d33 about five times larger than AlN [50, 51] while preserving CMOS compatibility. There is likewise additional upside on the photonics (Appendix C 1). Bending radiation loss can be significantly reduced by increasing the micro-ring radius r. In particular, intrinsic quality factor exceeding 107 has been demonstrated with an r approximately ten times larger using the same fabrication process [52]. It may be anticipated that the efficiency gain here may be chiefly compensated by the reduction in optomechanical coupling rate g0 ∼ 1/ √ r. Nevertheless, by thinning the waveguide to 200 nm that approaches half of the acoustic wavelength, g0 is expected to remain unchanged. As in the thick-core waveguide fabricated using the photonic Damascene process, such a thin-core waveguide on a subtractive manufacturing platform is expected to exhibit negligible bending loss [53]. Assuming critical coupling conditions (Appendix F), we therefore expect another 1.5- time gain in η tot. Finally, optical insertion loss can be nearly eliminated by employing a spot-size converter to facilitate mode-matching with lensed fiber [54–56]. Combining these improvements, our transducer could achieve a total efficiency η tot = 6 × 10−3 . In conclusion, we have designed, fabricated, and characterized a compact piezo-optomechanical microwaveoptical transducer that integrates wafer-scale, CMOScompatible HBAR and Si3N4 photonic technologies. Free of any superconducting components, this triply resonant transducer attains a bidirectional off-chip photon number conversion efficiency of 1.6 × 10−5 and a bandwidth of 25 MHz, with an input pump power of 21 dBm at room temperature. Multimode transduction leveraging distinct HBAR modes have been demonstrated, suggesting prospects for designing frequency-multiplexed photonic interconnects for superconducting qubits. Realistic design and fabrication improvements may even allow access to experiments in the quantum regime.

Bidirectional Microwave-optical Transduction Based on Integration: Discussion

This paper is available on arxiv under CC 4.0 license. Authors: (1) Terence Blésin, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL); (2) Wil Kao, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL); (3) Anat Siddharth, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL); (4) Alaina Attanasio, OxideMEMS lab, Purdue University; (5) Hao Tian, OxideMEMS lab, Purdue University; (6) Sunil A. Bhave, OxideMEMS lab, Purdue University; (7) Tobias J. Kippenberg, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL). Table of Links Introduction Results A. Physics and design B. Device characterization C. Bidirectional microwave-optical transduction D. Pulsed transduction Discussion Acknowledgments Appendix A: Theory Appendix B: Fabrication process flow Appendix C: Optical characterization Appendix D: Characterization of acoustic resonances Appendix E: Characterization of bidirectional transduction Appendix F: Optimization of optical extraction efficiency ACKNOWLEDGMENTS The authors would like to thank Amirali Arabmoheghi, Nils Johan Engelsen and Junyin Zhang for experimental assistance and fruitful discussions. This work was supported by the Air Force Office of Scientific Research under award no. FA8655-20-1-7009, Swiss National Science Foundation under grant agreement no. 204927, and European Research Council (ERC) under the EU H2020 research and innovation programme, grant agreement no. 835329 (ExCOM-cCEO). This work was further supported by United States Air Force Research Laboratory Award no. FA8750-21-2-0500, as well as United States National Science Foundation’s International Collaboration Supplements in Quantum Information Science and Engineering Research for RAISE-TAQS Award no. 18-39164. A.S. acknowledges support from the European Space Technology Centre with ESA Contract No. 4000135357/21/NL/GLC/my. Samples were fabricated in the Center of MicroNanoTechnology (CMi) at EPFL and Birck Nanotechnology Center at Purdue University. AUTHOR CONTRIBUTIONS T.B. designed the device. R.N.W, A.A. and H.T. fabricated the device. T.B., A.S. and W.K. conducted the experiments. T.B. and W.K. wrote the paper with input from all authors. T.J.K. and S.A.B. supervised the project. [1] K.-L. Lee, B. Sedighi, R. S. Tucker, H. Chow, and P. Vetter, J. Opt. Commun. Netw. 4, A59 (2012). [2] Q. Cheng, M. Bahadori, M. Glick, S. Rumley, and K. Bergman, Optica 5, 1354 (2018). [3] J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, Phys. Rev. Lett. 78, 3221 (1997). [4] L.-M. Duan, M. D. Lukin, J. 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