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).

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