- Physical Review Journals
All JournalsPhysics Magazine

PRX Quantum

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Loss Mechanisms in High-Coherence Multimode Mechanical Resonators Coupled to Superconducting Circuits

Raquel Garcia-Belles1,2,*,§, Alexander Anferov1,2,§, Lukas F. Deeg3,4, Loris Colicchio1,2, Arianne Brooks1,2, Tom Schatteburg1,2, Maxwell Drimmer1,2, Ines C. Rodrigues1,2, Rodrigo Benevides1,2,‡ et al.

Marco Liffredo5, Jyotish Patidar6, Oleksandr Pshyk6, Matteo Fadel1,2, Luis Guillermo Villanueva5, Sebastian Siol6, Gerhard Kirchmair3,4, and Yiwen Chu1,2,†

  • *Contact author: rgarcia@phys.ethz.ch
  • †Contact author: yiwen.chu@phys.ethz.ch
  • ‡Present address: Institute of Physics, University of Sao Paulo, Sao Paulo 05508-090, Brazil.
  • §These authors contributed equally to this work.
PDF

PRX Quantum 7, 033013 – Published 17 July, 2026

DOI: https://doi.org/10.1103/vszq-z1sh

Abstract

Circuit quantum acoustodynamics (cQAD) devices have a wide range of applications in quantum science, all of which depend crucially on the quantum coherence of the mechanical subsystem. In this context, high-overtone bulk acoustic-wave resonators (HBARs) are particularly promising, since they have shown very high quality factors with negligible dephasing. However, the introduction of piezoelectric films, which are necessary for coupling to a superconducting circuit, can lead to additional loss channels, such as surface scattering and two-level systems. Here, we study the acoustic dissipation of HBAR resonators in cQAD systems and find that the defect density of the piezoelectric material and its interface with the bulk are limiting factors for the coherence. We measure acoustic modes with phonon lifetimes up to 400  μs and lifetime-limited coherence times approaching 1 ms in the quantum regime. When coupled to a superconducting qubit, this leads to a hybrid system with a large quantum coherence cooperativity of CT2=1.1×105. These results represent a new milestone for the performance of cQAD devices and offer concrete paths forward for further improvements.

View figure in article

Physics Subject Headings (PhySH)

Popular Summary

Hybrid quantum devices that couple mechanical resonators to superconducting circuits are promising platforms for quantum technologies such as processors, sensors, and quantum transducers. High-overtone bulk acoustic resonators (HBARs) are particularly attractive because they support long-lived mechanical modes that can interact with superconducting qubits. Nevertheless, as is the case with most quantum devices, they are still limited by the ratio of interaction strengths to decoherence rates. In this work, we investigate the origin of decoherence in HBAR devices with an aluminum nitride piezoelectric layer that couples acoustic modes to superconducting circuits. Using spectroscopy and qubit-based quantum protocols, we probe mechanical modes at the single-quantum level. We find that the dominant losses depend strongly on the growth method of the piezoelectric layer, arising either from two-level-system defects or from surface scattering. Materials analysis supports this interpretation and suggests clear routes for reducing losses through improved fabrication. Our devices achieve a coherence cooperativity of about 10, indicating that many coherent operations are possible and highlighting the potential of circuit quantum acousto-dynamic systems for future quantum information applications.

Article Text

Supplemental Material

References (89)

  1. R. Linehan, T. Trickle, C. R. Conner, S. Ghosh, T. Lin, M. Sholapurkar, and A. N. Cleland, Listening for new physics with quantum acoustics, Phys. Rev. D 112, 115005 (2025).
  2. D. Mason, J. Chen, M. Rossi, Y. Tsaturyan, and A. Schliesser, Continuous force and displacement measurement below the standard quantum limit, Nat. Phys. 15, 745 (2019).
  3. I. Pikovski, M. R. Vanner, M. Aspelmeyer, M. S. Kim, and Časlav Brukner, Probing planck-scale physics with quantum optics, Nat. Phys. 8, 393 (2012).
  4. Börn Schrinski, Y. Yang, U. von Lüpke, M. Bild, Y. Chu, K. Hornberger, S. Nimmrichter, and M. Fadel, Macroscopic quantum test with bulk acoustic wave resonators, Phys. Rev. Lett. 130, 133604 (2023).
  5. C. T. Hann, C.-L. Zou, Y. Zhang, Y. Chu, R. J. Schoelkopf, S M. Girvin, and L. Jiang, Hardware-efficient quantum random access memory with hybrid quantum acoustic systems, Phys. Rev. Lett. 123, 250501 (2019).
  6. T. C. van Thiel, M. J. Weaver, F. Berto, P. Duivestein, M. Lemang, K. L. Schuurman, M. Žemlička, F. Hijazi, A. C. Bernasconi, C. Ferrer, E. Cataldo, E. Lachman, M. Field, Y. Mohan, F. K. de Vries, C. C. Bultink, J. C. van Oven, J. Y. Mutus, R. Stockill, and S. Gröblacher, Optical readout of a superconducting qubit using a piezo-optomechanical transducer, Nat. Phys. 21, 401 (2025).
  7. A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, Quantum ground state and single-phonon control of a mechanical resonator, Nature (London) 464, 697 (2010).
  8. K. J. Satzinger, Y. P. Zhong, H.-S. Chang, G. A. Peairs, A. Bienfait, M.-H. Chou, A. Y. Cleland, C. R. Conner, É Dumur, J. Grebel, I. Gutierrez, B. H. November, R. G. Povey, S. J. Whiteley, D. D. Awschalom, D. I. Schuster, and A. N. Cleland, Quantum control of surface acoustic-wave phonons, Nature (London) 563, 661 (2018).
  9. Y. Yang, I. Kladarić, M. Drimmer, U. von Lüpke, D. Lenterman, J. Bus, S. Marti, M. Fadel, and Y. Chu, A mechanical qubit, Science 386, 783 (2024).
  10. M.-H. Chou, H. Qiao, H. Yan, G. Andersson, C. R. Conner, J. Grebel, Y. J. Joshi, J. M. Miller, R. G. Povey, X. Wu, and A. N. Cleland, Deterministic multi-phonon entanglement between two mechanical resonators on separate substrates, Nat. Commun. 16, 1450 (2025).
  11. Alk𝚤m B. Bozkurt, O. Golami, Y. Yu, H. Tian, and M. Mirhosseini, A mechanical quantum memory for microwave photons, Nat. Phys. 21, 1469 (2025).
  12. E. A. Wollack, A. Y. Cleland, R. G. Gruenke, Z. Wang, P. Arrangoiz-Arriola, and A. H. Safavi-Naeini, Quantum state preparation and tomography of entangled mechanical resonators, Nature (London) 604, 463 (2022).
  13. A. A. Clerk, K. W. Lehnert, P. Bertet, J. R. Petta, and Y. Nakamura, Hybrid quantum systems with circuit quantum electrodynamics, Nat. Phys. 16, 257 (2020).
  14. P. Arrangoiz-Arriola, E. A. Wollack, Z. Wang, M. Pechal, W. Jiang, T. P. McKenna, J. D. Witmer, R. B.;l Van Laer, and A. H. Safavi-Naeini, Resolving the energy levels of a nanomechanical oscillator, Nature (London) 571, 537 (2019).
  15. N. R. A. Lee, Y. Guo, A. Y. Cleland, E. A. Wollack, R. G. Gruenke, T. Makihara, Z. Wang, T. Rajabzadeh, W. Jiang, F. M. Mayor, P. Arrangoiz-Arriola, C. J. Sarabalis, and A. H. Safavi-Naeini, Strong dispersive coupling between a mechanical resonator and a fluxonium superconducting qubit, PRX Quantum 4, 040342 (2023).
  16. J. M. Kitzman, J. R. Lane, C. Undershute, P. M. Harrington, N. R. Beysengulov, C. A. Mikolas, K. W. Murch, and J. Pollanen, Phononic bath engineering of a superconducting qubit, Nat. Commun. 14, 3910 (2023).
  17. Y. Chu, P. Kharel, W. H. Renninger, L. D. Burkhart, L. Frunzio, P. T. Rakich, and R. J. Schoelkopf, Quantum acoustics with superconducting qubits, Science 358, 199 (2017).
  18. U. von Lüpke, Y. Yang, M. Bild, L. Michaud, M. Fadel, and Y. Chu, Parity measurement in the strong dispersive regime of circuit quantum acoustodynamics, Nat. Phys. 18, 794 (2022).
  19. M. Kervinen, A. Välimaa, J. E. Ramírez-Muñoz, and M. A. Sillanpää, Sideband control of a multimode quantum bulk acoustic system, Phys. Rev. Appl. 14, 054023 (2020).
  20. V. J. Gokhale, B. P. Downey, D. S. Katzer, N. Nepal, A. C. Lang, R. M. Stroud, and D. J. Meyer, Epitaxial bulk acoustic wave resonators as highly coherent multi-phonon sources for quantum acoustodynamics, Nat. Commun. 11, 2314 (2020).
  21. G. S. MacCabe, H. Ren, J. Luo, J. D. Cohen, H. Zhou, A. Sipahigil, M. Mirhosseini, and O. Painter, Nano-acoustic resonator with ultralong phonon lifetime, Science 370, 840 (2020).
  22. A. Vorobiev, J. Berge, S. Gevorgian, M. Löffler, and E. Olsson, Effect of interface roughness on acoustic loss in tunable thin film bulk acoustic wave resonators, J. Appl. Phys. 110, 024116 (2011).
  23. Y. Luo, H. H. Diamandi, and H. Li et al., Lifetime-limited gigahertz-frequency mechanical oscillators with millisecond coherence times, arXiv:2504.07523.
  24. S. Galliou, M. Goryachev, R. Bourquin et al., Extremely low loss phonon-trapping cryogenic acoustic cavities for future physical experiments, Sci. Rep. 3, 2132 (2013).
  25. Y. Tsaturyan, A. Barg, E. S. Polzik, and A. Schliesser, Ultracoherent nanomechanical resonators via soft clamping and dissipation dilution, Nat. Nanotechnol. 12, 776 (2017).
  26. A. H. Ghadimi, S. A. Fedorov, N. J. Engelsen, M. J. Bereyhi, R. Schilling, D. J. Wilson, and T. J. Kippenberg, Elastic strain engineering for ultralow mechanical dissipation, Science 360, 764 (2018).
  27. A. Y. Cleland, E. A. Wollack, and A. H. Safavi-Naeini, Studying phonon coherence with a quantum sensor, Nat. Commun. 15, 4979 (2024).
  28. R. O. Behunin, F. Intravaia, and P. T. Rakich, Dimensional transformation of defect-induced noise, dissipation, and nonlinearity, Phys. Rev. B 93, 224110 (2016).
  29. M. P. Maksymowych, M. Yuksel, O. A. Hitchcock, N. R. Lee, F. M. Mayor, W. Jiang, M. L. Roukes, and A. H. Safavi-Naeini, Spectral diffusion of nanomechanical resonators due to single quantum defects, Phys. Rev. Appl. 24, 044066 (2025).
  30. R. G. Gruenke-Freudenstein, E. Szakiel, G. P. Multani, T. Makihara, A. G. Hayden, A. Khalatpour, E. A. Wollack, A. Akoto-Yeboah, S. Salmani-Rezaie, and A. H. Safavi-Naeini, Surface and bulk two-level-system losses in lithium niobate acoustic resonators, Phys. Rev. Appl. 23, 064055 (2025).
  31. Y. Chu, P. Kharel, T. Yoon, L. Frunzio, P. T. Rakich, and R. J. Schoelkopf, Creation and control of multi-phonon Fock states in a bulk acoustic-wave resonator, Nature (London) 563, 666 (2018).
  32. P. Kharel, Y. Chu, M. Power, W. H. Renninger, R. J. Schoelkopf, and P. T. Rakich, Ultra-high-Qphononic resonators on-chip at cryogenic temperatures, APL Photon. 3, 066101 (2018).
  33. See Supplemental Material at http://link.aps.org/supplemental/10.1103/vszq-z1sh for theory derivations and additional measurements, which includes Refs. [34–51].
  34. M. Goryachev, D. L. Creedon, S. Galliou, and M. E. Tobar, Observation of Rayleigh phonon scattering through excitation of extremely high overtones in low-loss cryogenic acoustic cavities for hybrid quantum systems, Phys. Rev. Lett. 111, 085502 (2013).
  35. A. N. Cleland, Foundations of Nanomechanics: From Solid-State Theory to Device Applications (Springer Science & Business Media, Berlin, Heidelberg, 2013).
  36. J. M. Ziman, Electrons and Phonons: The Theory of Transport Phenomena in Solids, Oxford Classic Texts in the Physical Sciences (Oxford University Press, Oxford, 2001).
  37. A. A. Maznev, Boundary scattering of phonons: Specularity of a randomly rough surface in the small-perturbation limit, Phys. Rev. B 91, 134306 (2015).
  38. D. S. Stevens and H. F. Tiersten, An analysis of doubly rotated quartz resonators utilizing essentially thickness modes with transverse variation, J. Acoust. Soc. Am. 79, 1811 (1986).
  39. V. J. Gokhale, B. P. Downey, D. S. Katzer, and D. J. Meyer, Phonon diffraction limited performance of Fabry-Pérot cavities in piezoelectric epi–HBARs, in Proceedings of the 2021 IEEE 34th International Conference on Micro Electro Mechanical Systems (MEMS) (2021), pp. 206–209.
  40. W. H. Renninger, P. Kharel, R. O. Behunin, and P. T. Rakich, Bulk crystalline optomechanics, Nat. Phys. 14, 601 (2018).
  41. J. M. Martinis, K. B. Cooper, R. McDermott, M. Steffen, M. Ansmann, K. D. Osborn, K. Cicak, S. Oh, D. P. Pappas, R. W. Simmonds, and C. C. Yu, Decoherence in Josephson qubits from dielectric loss, Phys. Rev. Lett. 95, 210503 (2005).
  42. M. Trif, O. Dmytruk, Hélène Bouchiat, Rón Aguado, and P. Simon, Dynamic current susceptibility as a probe of Majorana bound states in nanowire-based Josephson junctions, Phys. Rev. B 97, 041415 (2018).
  43. H. Banderier, M. Drimmer, and Y. Chu, Unified simulation methods for quantum acoustic devices, Phys. Rev. Appl. 20, 024024 (2023).
  44. M. Reagor, W. Pfaff, C. Axline, R. W. Heeres, N. Ofek, K. Sliwa, E. Holland, C. Wang, J. Blumoff, K. Chou, M. J. Hatridge, L. Frunzio, M. H. Devoret, L. Jiang, and R. J. Schoelkopf, Quantum memory with millisecond coherence in circuit QED, Phys. Rev. B 94, 014506 (2016).
  45. V. J. Gokhale, B. P. Downey, D. S. Katzer, and D. J. Meyer, Temperature evolution of frequency and anharmonic phonon loss for multi-mode epitaxial HBARs, Appl. Phys. Lett. 117, 124003 (2020).
  46. M. Bild, M. Fadel, Y. Yang, U. von Lüpke, P. Martin, A. Bruno, and Y. Chu, Schrödinger cat states of a 16-microgram mechanical oscillator, Science 380, 274 (2023).
  47. H. Yu, C.-Y. Lee, W. Pang, H. Zhang, A. Brannon, J. Kitching, and E. S. Kim, HBAR-based 3.6 GHz oscillator with low power consumption and low phase noise, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 56, 400 (2009).
  48. T. Baron, G. Martin, E. Lebrasseur et al., rf oscillators stabilized by temperature compensated HBARs based on LiNbO3/Quartz combination, in Proceedings of the 2011 Joint Conference of the IEEE International Frequency Control and the European Frequency and Time Forum (FCS) (IEEE, San Francisco, CA, USA, 2011), pp. 1–4.
  49. K. Capannelli, B. Undseth, I. Fernández de Fuentes, E. Raymenants, F. K. Unseld, O. Pietx-Casas, S. G. J. Philips, M. T. Mądzik, S. V. Amitonov, L. Tryputen, G. Scappucci, and L. M. K. Vandersypen, Tracking spin qubit frequency variations over 912 days, npj Quantum Inf. 11, 192 (2025).
  50. R. Boudot, G. Martin, J.-M. Friedt, and E. Rubiola, Frequency flicker of 2.3 GHz AlN-sapphire high-overtone bulk acoustic resonators, J. Appl. Phys. 120, 224903 (2016).
  51. M. Carroll, S. Rosenblatt, P. Jurcevic, I. Lauer, and A. Kandala, Dynamics of superconducting qubit relaxation times, npj Quantum Inf. 8, 132 (2022).
  52. E. A. Wollack, A. Y. Cleland, P. Arrangoiz-Arriola, T. P. McKenna, R. G. Gruenke, R. N. Patel, W. Jiang, C. J. Sarabalis, and A. H. Safavi-Naeini, Loss channels affecting lithium niobate phononic crystal resonators at cryogenic temperature, Appl. Phys. Lett. 118, 123501 (2021).
  53. A. J. Välimaa, J. T. Santos, C. F. Ockeloen-Korppi, and M. A. Sillanpää, Electrode configuration and electrical dissipation of mechanical energy in quartz crystal resonators, J. Micromech. Microeng. 28, 095014 (2018).
  54. M.-A. Dubois and P. Muralt, Stress and piezoelectric properties of aluminum nitride thin films deposited onto metal electrodes by pulsed direct current reactive sputtering, J. Appl. Phys. 89, 6389 (2001).
  55. M. Lee, H. Son, H.-Y. Lee, J. Moon, H. Kim, J.-I. Park, Z. Liu, M. G. Hahm, M. Yang, and U. J. Kim, Nanovoid-driven highly crystalline aluminum nitride and its application in solar-blind UV photodetectors, J. Mater. Chem. C 8, 14431 (2020).
  56. J. Patidar, O. Pshyk, K. Thorwarth, L. Sommerhäuser, and S. Siol, Low temperature deposition of functional thin films on insulating substrates enabled by selective ion acceleration using synchronized floating potential HiPIMS, Nat. Commun. 16, 4719 (2025).
  57. Y. Zhang, Z. Wang, and J. D. N. Cheeke, Resonant spectrum method to characterize piezoelectric films in composite resonators, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 50, 321 (2003).
  58. W. P. Mason, Electromechanical Transducers and Wave Filters, 2nd ed. (D. Van Nostrand Company, New York, NY, 1948).
  59. S.-J. Lee, S. R. Jeon, S. H. Jung, Y.-J. Choi, H.-G. Oh, H.-Y. Lee, M.-K. Kwon, and S.-K. Hong, Realization of low dislocation density AIN on patterned sapphire substrate by hydride vapor-phase epitaxy for deep ultraviolet light-emitting diodes, Phys. Status Solidi (a) 220, 2200835 (2023).
  60. C. Wang, C. Axline, Y. Y. Gao, T. Brecht, Y. Chu, L. Frunzio, M. H. Devoret, and R. J. Schoelkopf, Surface participation and dielectric loss in superconducting qubits, Appl. Phys. Lett. 107, 162601 (2015).
  61. Z. K. Minev, Z. Leghtas, S. O. Mundhada, L. Christakis, I. M. Pop, and M. H. Devoret, Energy-participation quantization of Josephson circuits, npj Quantum Inf. 7, 131 (2021).
  62. V. B. Braginsky, V. P. Mitrofanov, and V. I. Panov, Systems with Small Dissipation (University of Chicago Press, Chicago, IL, 1986).
  63. T. S. Hutchison, Ultrasonic absorption in solids, Science 132, 643 (1960).
  64. A. S. Nowick and B. S. Berry, Boundary relaxation processes and internal friction at high temperatures, in Anelastic Relaxation in Crystalline Solids (Academic Press, New York, 1972), pp. 435–462.
  65. A. Akhiezer, On the absorption of sound in solids, J. Phys. (USSR) 1, 277 (1939).
  66. S. Ghaffari, S. A. Chandorkar, S. Wang et al., Quantum limit of quality factor in silicon micro and nano mechanical resonators, Sci. Rep. 3, 3244 (2013).
  67. W. A. Phillips, Two-level states in glasses, Rep. Prog. Phys. 50, 1657 (1987).
  68. S. Hunklinger and W. Arnold, Ultrasonic properties of glasses at low temperatures, in Physical Acoustics (Academic Press, New York, 1976), Vol. 12, pp. 155–215.
  69. A. L. Emser, C. Metzger, B. C. Rose, and K. W. Lehnert, Thin-film quartz for high-coherence piezoelectric phononic crystal resonators, Phys. Rev. Appl. 22, 064032 (2024).
  70. O. F. Wolff, H. Mantry, R. Raja et al., Structural control of two-level defect density revealed by high-throughput correlative measurements of josephson junctions, arXiv:2602.11469.
  71. A. Premkumar, C. Weiland, S. Hwang, B. Jäck, A. P. M. Place, I. Waluyo, A. Hunt, V. Bisogni, J. Pelliciari, A. Barbour, M. S. Miller, P. Russo, F. Camino, K. Kisslinger, X. Tong, M. S. Hybertsen, A. A. Houck, and I. Jarrige, Microscopic relaxation channels in materials for superconducting qubits, Commun. Mater. 2, 72 (2021).
  72. C. Undershute, J. M. Kitzman, C. A. Mikolas, and J. Pollanen, Decoherence of surface phonons in a quantum acoustic system, Phys. Rev. A 111, 012615 (2025).
  73. A. N. Bolgar, J. I. Zotova, D. D. Kirichenko, I. S. Besedin, A. V. Semenov, R. S. Shaikhaidarov, and O. V. Astafiev, Quantum regime of a two-dimensional phonon cavity, Phys. Rev. Lett. 120, 223603 (2018).
  74. B. A. Moores, L. R. Sletten, J. J. Viennot, and K W. Lehnert, Cavity quantum acoustic device in the multimode strong coupling regime, Phys. Rev. Lett. 120, 227701 (2018).
  75. A. Bienfait, K. J. Satzinger, Y. P. Zhong, H.-S. Chang, M.-H. Chou, C. R. Conner, É Dumur, J. Grebel, G. A. Peairs, R. G. Povey, and A. N. Cleland, Phonon-mediated quantum state transfer and remote qubit entanglement, Science 364, 368 (2019).
  76. W. Crump, A. Välimaa, and M. A. Sillanpää, Coupling high-overtone bulk acoustic wave resonators via superconducting qubits, Appl. Phys. Lett. 123, 134004 (2023).
  77. S. Galliou, S. Deléglise, M. Goryachev, L. Neuhaus, G. Cagnoli, S. Zerkani, V. Dolique, Jérémy Bon, X. Vacheret, P. Abbé, L. Pinard, C. Michel, T. Karassouloff, T. Briant, P.-F.çois Cohadon, A. Heidmann, M. E. Tobar, and R. Bourquin, A new method of probing mechanical losses of coatings at cryogenic temperatures, Rev. Sci. Instrum. 87, 123906 (2016).
  78. H. Zhang, W. Pang, H. Yu, and E. S. Kim, High-tone bulk acoustic resonators on sapphire, crystal quartz, fused silica, and silicon substrates, J. Appl. Phys. 99, 124911 (2006).
  79. H. Tian, J. Liu, B. Dong, J. C. Skehan, M. Zervas, T. J. Kippenberg, and S. A. Bhave, Hybrid integrated photonics using bulk acoustic resonators, Nat. Commun. 11, 3073 (2020).
  80. U. von Lüpke, Quantum control of a multimode bulk acoustic wave resonator, Doctoral Thesis, ETH Zurich, 2023.
  81. S. Balachandran, D. B. Smathers, J. Kim et al., A method for measuring interface roughness from cross-sectional micrographs, IEEE Trans. Appl. Supercond. 33, 1 (2023).
  82. O. Gargiulo, S. Oleschko, J. Prat-Camps, M. Zanner, and G. Kirchmair, Fast flux control of 3D transmon qubits using a magnetic hose, Appl. Phys. Lett. 118, 012601 (2021).
  83. N. E. Frattini, V. V. Sivak, A. Lingenfelter, S. Shankar, and M. H. Devoret, Optimizing the nonlinearity and dissipation of a snail parametric amplifier for dynamic range, Phys. Rev. Appl. 10, 054020 (2018).
  84. D. Rieger, S. Günzler, M. Spiecker, A. Nambisan, W. Wernsdorfer, and I. M. Pop, Fano interference in microwave resonator measurements, Phys. Rev. Appl. 20, 014059 (2023).
  85. S. Probst, F. B. Song, P. A. Bushev, A. V. Ustinov, and M. Weides, Efficient and robust analysis of complex scattering data under noise in microwave resonators, Rev. Sci. Instrum. 86, 024706 (2015).
  86. Steelelab-delft/stlab, https://github.com/steelelab-delft/stlab.
  87. D. M. Pozar, Microwave Engineering: Theory and Techniques (John Wiley & Sons, Ltd, Hoboken, NJ, 2021).
  88. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (John Wiley & Sons, Ltd, New York, 1991).
  89. R. Garcia-Belles, A. Anferov, L. Deeg et al., Loss mechanisms in high-coherence multimode mechanical resonators coupled to superconducting circuits [data set], https://zenodo.org/records/20429959.

Outline

Information

PRX Quantum 7, 033013– Published 17 July, 2026
  • Received 7 March 2026
  • Accepted 26 May 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation