Intrinsic phononic dressed states in a nanomechanical system

Nature Physics (2026)Cite this article Nanoelectromechanical systems provide a platform for probing the quantum nature of mechanical motion in mesoscopic systems. Quantum effects are most pronounced when device vibrations are nonlinear, but it has been difficult to achieve vibrational nonlinearity at the single-phonon level. Here we report the observation of intrinsic mesoscopic vibrational states that are dressed by the interactions with a nonlinear quantum system. The nonlinearity results from the strong resonant coupling between an eigenmode of a nanoelectromechanical system resonator and a single, two-level system that is intrinsic to the device material. We control the two-level system in situ by varying the mechanical strain in the device, tuning it in and out of resonance with the nanoelectromechanical system mode. Varying the resonant drive or temperature allows a controlled ascent of the non-equidistant energy ladder of the hybridized system. Fluctuations of the two-level system on and off resonance with the mode induce switching between the dressed and bare states. These quantum effects directly emerge from the intrinsic material properties of mechanical systems without the need for complex, external quantum circuits. Our work offers insight into mesoscopic dynamics and provides the opportunity to harness nanomechanics for quantum measurements.This is a preview of subscription content, access via your institution Access Nature and 54 other Nature Portfolio journals Get Nature+, our best-value online-access subscription $32.99 / 30 days cancel any timeSubscribe to this journal Receive 12 print issues and online access $259.00 per yearonly $21.58 per issueBuy this articleUSD 39.95Prices may be subject to local taxes which are calculated during checkoutSource data are provided with this paper. The data used to generate the figures in the main text are provided in the paper and are also available via Zenodo at https://doi.org/10.5281/zenodo.15299527 (ref. 65). All other data generated or analysed during this study are available from the corresponding authors upon reasonable request.Bachtold, A., Moser, J. & Dykman, M. Mesoscopic physics of nanomechanical systems. Rev. Mod. Phys. 94, 045005 (2022).Article ADS MathSciNet Google Scholar Ekinci, K. L. & Roukes, M. L. Nanoelectromechanical systems. Rev. Sci. Instrum. 76, 061101 (2005).Schwab, K. C. & Roukes, M. L. Putting mechanics into quantum mechanics. Phys. Today 58, 36–42 (2005).Article Google Scholar MacCabe, G. S. et al. Nano-acoustic resonator with ultralong phonon lifetime. Science 370, 840–843 (2020).Article ADS Google Scholar Arrangoiz-Arriola, P. et al. Resolving the energy levels of a nanomechanical oscillator. Nature 571, 537–540 (2019).Article ADS Google Scholar Bozkurt, A. et al. A quantum electromechanical interface for long-lived phonons. Nat. Phys. 19, 1326–1332 (2023).Article Google Scholar Pistolesi, F., Cleland, A. & Bachtold, A. Proposal for a nanomechanical qubit. Phys. Rev. X 11, 031027 (2021).

Google Scholar Samanta, C. et al. Nonlinear nanomechanical resonators approaching the quantum ground state. Nat. Phys. 19, 1340–1344 (2023).Article Google Scholar O’Connell, A. D. et al. Quantum ground state and single-phonon control of a mechanical resonator. Nature 464, 697–703 (2010).Article ADS Google Scholar Wollack, E. A. et al. Quantum state preparation and tomography of entangled mechanical resonators. Nature 604, 463–467 (2022).Article ADS Google Scholar Satzinger, K. J. et al. Quantum control of surface acoustic-wave phonons. Nature 563, 661–665 (2018).Article ADS Google Scholar Yang, Y. et al. A mechanical qubit. Science 386, 783–788 (2024).Article ADS MathSciNet Google Scholar Bozkurt, A. B. et al. A mechanical quantum memory for microwave photons. Nat. Phys. 21, 1469–1474 (2025).Haroche, S. & Raimond, J.-M. Exploring the Quantum: Atoms, Cavities, and Photons (Oxford Univ. Press, 2006).Simmonds, R. W. et al. Decoherence in Josephson phase qubits from junction resonators. Phys. Rev. Lett. 93, 077003 (2004).Article ADS Google Scholar Lisenfeld, J. et al. Observation of directly interacting coherent two-level systems in an amorphous material. Nat. Commun. 6, 6182 (2015).Article ADS Google Scholar Lisenfeld, J. et al. Electric field spectroscopy of material defects in transmon qubits. npj Quantum Inf. 5, 105 (2019).Article ADS Google Scholar Chen, M., Owens, J. C., Putterman, H., Schäfer, M. & Painter, O. Phonon engineering of atomic-scale defects in superconducting quantum circuits. Sci. Adv. 10, eado6240 (2024).Article Google Scholar Grabovskij, G. J., Peichl, T., Lisenfeld, J., Weiss, G. & Ustinov, A. V. Strain tuning of individual atomic tunneling systems detected by a superconducting qubit. Science 338, 232–234 (2012).Article ADS Google Scholar Sarabi, B., Ramanayaka, A. N., Burin, A. L., Wellstood, F. C. & Osborn, K. D. Projected dipole moments of individual two-level defects extracted using circuit quantum electrodynamics. Phys. Rev. Lett. 116, 167002 (2016).Article ADS Google Scholar Kristen, M. et al. Giant two-level systems in a granular superconductor. Phys. Rev. Lett. 132, 217002 (2024).Article ADS Google Scholar Pedurand, R. et al. Progress toward detection of individual TLS in nanomechanical resonators. J. Low Temp. Phys. 215, 440–451 (2024).Tavakoli, A. et al. Specific heat of thin phonon cavities at low temperature: very high values revealed by zeptojoule calorimetry. Phys. Rev. B 105, 224313 (2022).Article ADS Google Scholar Ramos, T., Sudhir, V., Stannigel, K., Zoller, P. & Kippenberg, T. J. Nonlinear quantum optomechanics via individual intrinsic two-level defects. Phys. Rev. Lett. 110, 193602 (2013).Article ADS Google Scholar Maksymowych, M. et al. Spectral diffusion of nanomechanical resonators due to single quantum defects. Phys. Rev. Appl. 24, 044066 (2025).Article ADS Google Scholar Kamppinen, T., Mäkinen, J. & Eltsov, V. Dimensional control of tunneling two-level systems in nanoelectromechanical resonators. Phys. Rev. B 105, 035409 (2022).Article ADS Google Scholar Cleland, A. Y., Wollack, E. A. & Safavi-Naeini, A. H. Studying phonon coherence with a quantum sensor. Nat. Commun. 15, 4979 (2024).Article ADS Google Scholar Müller, C., Cole, J. H. & Lisenfeld, J. Towards understanding two-level-systems in amorphous solids: insights from quantum circuits. Rep. Prog. Phys. 82, 124501 (2019).Article ADS Google Scholar Phillips, W. A. Tunneling states in amorphous solids. J. Low Temp. Phys. 7, 351–360 (1972).Article ADS Google Scholar Anderson, P. W., Halperin, B. I. & Varma, C. M. Anomalous low-temperature thermal properties of glasses and spin glasses. Philos. Mag. 25, 1–9 (1972).Article ADS Google Scholar Phillips, W. A. Two-level states in glasses. Rep. Prog. Phys. 50, 1657 (1987).Article ADS Google Scholar Black, J. & Halperin, B. Spectral diffusion, phonon echoes, and saturation recovery in glasses at low temperatures. Phys. Rev. B 16, 2879 (1977).Article ADS Google Scholar Mihailovich, R. & Parpia, J. Low temperature mechanical properties of boron-doped silicon. Phys. Rev. Lett. 68, 3052 (1992).Article ADS Google Scholar Esquinazi, P. Tunneling Systems in Amorphous and Crystalline Solids (Springer Science & Business Media, 2013).Gruenke, R. G. et al. Surface modification and coherence in lithium niobate saw resonators. Sci. Rep. 14, 6663 (2024).Article ADS Google Scholar Gruenke-Freudenstein, R. G. et al. Surface and bulk two-level-system losses in lithium niobate acoustic resonators. Phys. Rev. Appl. 23, 064055 (2025).Article ADS Google Scholar Arrangoiz-Arriola, P. et al. Coupling a superconducting quantum circuit to a phononic crystal defect cavity. Phys. Rev. X 8, 031007 (2018).

Google Scholar Wollack, E. A. et al. Loss channels affecting lithium niobate phononic crystal resonators at cryogenic temperature. Appl. Phys. Lett. 118, 123501 (2021).Teufel, J. D. et al. Circuit cavity electromechanics in the strong-coupling regime. Nature 471, 204–208 (2011).Article ADS Google Scholar Cleland, A. N. Foundations of Nanomechanics: From Solid-State Theory to Device Applications (Springer Science & Business Media, 2013).Emser, A. L., Metzger, C., Rose, B. C. & Lehnert, K. W. Thin-film quartz for high-coherence piezoelectric phononic crystal resonators. Phys. Rev. Appl. 22, 064032 (2024).Article ADS Google Scholar Jaynes, E. T. & Cummings, F. W. Comparison of quantum and semiclassical radiation theories with application to the beam maser. Proc. IEEE 51, 89–109 (1963).Article ADS Google Scholar Bilmes, A., Volosheniuk, S., Ustinov, A. V. & Lisenfeld, J. Probing defect densities at the edges and inside Josephson junctions of superconducting qubits. npj Quantum Inf. 8, 24 (2022).Article ADS Google Scholar Meißner, S. M., Seiler, A., Lisenfeld, J., Ustinov, A. V. & Weiss, G. Probing individual tunneling fluctuators with coherently controlled tunneling systems. Phys. Rev. B 97, 180505 (2018).Article ADS Google Scholar Sarabi, B., Ramanayaka, A. N., Burin, A. L., Wellstood, F. C. & Osborn, K. D. Cavity quantum electrodynamics using a near-resonance two-level system: emergence of the Glauber state. Appl. Phys. Lett. 106, 172601 (2015).Dykman, M. I. & Ivanov, M. A. Spectral distribution of resonant localized impurity vibrations at finite temperatures. Fiz. Tverd. Tela 18, 720–729 (1976).

Google Scholar Blais, A., Grimsmo, A. L., Girvin, S. M. & Wallraff, A. Circuit quantum electrodynamics. Rev. Mod. Phys. 93, 025005 (2021).Article ADS MathSciNet Google Scholar Johansson, J. R., Nation, P. D. & Nori, F. QuTiP: an open-source Python framework for the dynamics of open quantum systems. Comput. Phys. Commun. 183, 1760–1772 (2012).Article ADS Google Scholar Gardiner, C. W. & Collett, M. J. Input and output in damped quantum systems: quantum stochastic differential equations and the master equation. Phys. Rev. A 31, 3761 (1985).Article ADS MathSciNet Google Scholar Brune, M. et al. Quantum Rabi oscillation: a direct test of field quantization in a cavity. Phys. Rev. Lett. 76, 1800 (1996).Article ADS Google Scholar Schuster, D. et al. Resolving photon number states in a superconducting circuit. Nature 445, 515–518 (2007).Article ADS Google Scholar Bishop, L. S. et al. Nonlinear response of the vacuum Rabi resonance. Nat. Phys. 5, 105–109 (2009).Article Google Scholar Fink, J. et al. Climbing the Jaynes–Cummings ladder and observing its nonlinearity in a cavity QED system. Nature 454, 315–318 (2008).Article ADS Google Scholar Ginossar, E., Bishop, L. & Girvin, S. in Fluctuating Nonlinear Oscillators: From Nanomechanics to Quantum Superconducting Circuits 198 (Oxford Univ. Press, 2012).Peano, V. & Thorwart, M. Quasienergy description of the driven Jaynes-Cummings model. Phys. Rev. B 82, 155129 (2010).Article ADS Google Scholar Bonsen, T. et al. Probing the Jaynes-Cummings ladder with spin circuit quantum electrodynamics. Phys. Rev. Lett. 130, 137001 (2023).Article ADS Google Scholar Fink, J. et al. Quantum-to-classical transition in cavity quantum electrodynamics. Phys. Rev. Lett. 105, 163601 (2010).Article ADS Google Scholar Rau, I., Johansson, G. & Shnirman, A. Cavity quantum electrodynamics in superconducting circuits: susceptibility at elevated temperatures. Phys. Rev. B 70, 054521 (2004).Article ADS Google Scholar Burnett, J. et al. Evidence for interacting two-level systems from the 1/f noise of a superconducting resonator. Nat. Commun. 5, 4119 (2014).Article ADS Google Scholar Béjanin, J. et al. Fluctuation spectroscopy of two-level systems in superconducting resonators. Phys. Rev. Appl. 18, 034009 (2022).Article ADS Google Scholar De Graaf, S., Mahashabde, S., Kubatkin, S., Tzalenchuk, A. Y. & Danilov, A. Quantifying dynamics and interactions of individual spurious low-energy fluctuators in superconducting circuits. Phys. Rev. B 103, 174103 (2021).Article ADS Google Scholar Faoro, L. & Ioffe, L. B. Interacting tunneling model for two-level systems in amorphous materials and its predictions for their dephasing and noise in superconducting microresonators. Phys. Rev. B 91, 014201 (2015).Article ADS Google Scholar Bowman, A. W. & Azzalini, A.

Applied Smoothing Techniques for Data Analysis: The Kernel Approach with S-Plus Illustrations Vol. 18 (OUP Oxford, 1997).Scott, D. W. On optimal and data-based histograms. Biometrika 66, 605–610 (1979).Article ADS MathSciNet Google Scholar Yuksel, M. et al. Dataset for intrinsic phononic dressed states in a nanomechanical system. Zenodo https://doi.org/10.5281/zenodo.15299527 (2025).Download referencesWe are grateful for support for this work from Gordon and Betty Moore Foundation grant number GBFM12214 (https://doi.org/10.37807/GBMF12214; M.L.R., A.H.S.-N. and M.I.D.) and a Moore Inventor Fellowship (A.H.S.-N.); the Wellcome Leap—Delta Tissue Program (M.L.R.); National Science Foundation award number 2016555 (M.L.R.) and CAREER award number ECCS-1941826 (A.H.S.-N.); Defense Advanced Research Projects Agency (DARPA) under cooperative agreement number HR0011-23-2-0004 (M.I.D.); Amazon Web Services, Inc. (A.H.S.-N.); and US Department of Energy, grant number DE-AC02-76SF00515 (A.H.S.-N.). M.P.M. is grateful for support from an NSERC graduate fellowship. We thank the Stanford Nanofabrication Facility (supported by National Science Foundation awards ECCS-2026822 and ECCS-1542152), Caltech’s Kavli Nanoscience Institute and Stanford’s Q-NEXT Center. We also acknowledge valuable discussions with W. Fon, S. Habermehl, U. Hatipoglu and A. Nunn, and fabrication assistance from W. Jiang, K. K. S. Multani and A. Y. Cleland.These authors contributed equally: M. Yuksel, M. P. Maksymowych.Department of Applied Physics, California Institute of Technology, Pasadena, CA, USAM. Yuksel & M. L. RoukesDepartment of Applied Physics and Ginzton Laboratory, Stanford University, Stanford, CA, USAM. P. Maksymowych, O. A. Hitchcock, F. M. Mayor, N. R. Lee & A. H. Safavi-NaeiniDepartment of Physics, Stanford University, Stanford, CA, USAO. A. HitchcockDepartment of Physics and Astronomy, Michigan State University, East Lansing, MI, USAM. I. DykmanDepartment of Physics and the Kavli Nanoscience Institute, California Institute of Technology, Pasadena, CA, USAM. L. RoukesDepartment of Bioengineering, California Institute of Technology, Pasadena, CA, USAM. L. RoukesSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarM.Y. performed the experiments at Caltech, analysed the data and conducted the simulations. M.P.M. performed the experiments at Stanford and analysed the data. M.Y. and M.P.M. developed the theoretical framework with assistance from M.I.D., A.H.S.-N. and M.L.R. M.P.M., O.A.H., F.M.M. and N.R.L. designed and fabricated the devices at Stanford. M.Y., M.P.M. and M.I.D. wrote the paper with input from all authors. M.I.D., A.H.S.-N. and M.L.R. supervised the project.Correspondence to M. Yuksel or M. L. Roukes.A.H.S.-N. is an Amazon Scholar. The other authors declare no competing interests.Nature Physics thanks the anonymous reviewers for their contribution to the peer review of this work. Peer reviewer reports are available.Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.Schematic of the horizontal dilution refrigerator measurement setup at Caltech used to acquire the data presented in the main text.The plot displays the mean squared error (MSE) as a function of the estimated phonon number n. After iterative minimization of the fitting parameters, we sweep n around the optimized neff value to quantify the uncertainty in our estimation. This sweep identifies the regions n neff where the MSE increases by 5% above its minimum value (indicated by the shaded area). Notably, the error rises more steeply for n neff as expected due to the \(\sqrt{n}\) scaling of the energy levels. As neff increases further, achieving an accurate determination becomes increasingly challenging and computationally expensive.Source dataSupplementary Figs. 1–9, Table 1, Sections 1–4, Discussion and Data.Evolution of the TLS,C1 crossing as the input power is increased from –150 dBm to –135 dBm in 1-dB increments.Source data comprising raw experimental and simulation data.Source data comprising raw experimental and simulation data.Source data comprising raw experimental and simulation data.Source data comprising raw experimental and simulation data.Source data comprising raw experimental and simulation data.Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.Reprints and permissionsYuksel, M., Maksymowych, M.P., Hitchcock, O.A. et al. Intrinsic phononic dressed states in a nanomechanical system. Nat. Phys. (2026). https://doi.org/10.1038/s41567-026-03225-3Download citationReceived: 04 September 2025Accepted: 23 February 2026Published: 14 April 2026Version of record: 14 April 2026DOI: https://doi.org/10.1038/s41567-026-03225-3Anyone you share the following link with will be able to read this content:Sorry, a shareable link is not currently available for this article. Provided by the Springer Nature SharedIt content-sharing initiative