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Anharmonic motion of a trapped ion under the influence of different drives in an ion-nanomechanical hybrid system

Adrien Poindron and Stefan Willitsch*

  • *Contact author: stefan.willitsch@unibas.ch

Phys. Rev. Research 8, 033028 – Published 7 July, 2026

DOI: https://doi.org/10.1103/mw5k-p3gg

Abstract

Ions in traps hybridized with nanomechanical oscillators represent promising platforms for studying interactions between microscopic and macroscopic systems and for manipulating the classical and quantum motion of trapped particles. In a previous report [Weegen et al., Phys. Rev. Lett. 133, 223201 (2024)], we presented a hybrid system combining laser-cooled trapped ions with an oscillating and electrically charged nanowire driving the ion motion. Here, we focus on the ramifications of trap-potential anharmonicites introduced by the nanooscillator and show that the driven classical motion of the ions in this hybrid system obeys Duffing-type dynamics. We combine electrical and mechanical drives to explore the details of the anharmonic ion motion and demonstrate how anharmonicity and damping parameters can be determined from analyzing the steady-state amplitude and phase of the motion under these conditions. The present results provide new insights into the dynamical properties of ion-nanomechanical hybrid systems and the anharmonic motion of trapped particles under the influence of different types of drives.

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References (64)

  1. In Proceedings of the International School of Physics "Enrico Fermi", edited by M. Knoop, I. Marzoli, and G. Morigi (IOS Press, Amsterdam-Washington DC, 2015), Vol. 189.
  2. E. Peik, Optical atomic clocks, in Photonic Quantum Technologies, edited by M. Benyoucef (John Wiley & Sons, Ltd., Weinheim, Germany, 2023), Chap. 14, pp. 333–348.
  3. M. C. Marshall, Daniel A. Rodriguez Castillo, W. J. Arthur-Dworschack, A. Aeppli, K. Kim, D. Lee, W. Warfield, J. Hinrichs, N. V. Nardelli, T. M. Fortier, J. Ye, D. R. Leibrandt, and D. B. Hume, High-stability single-ion clock with 5.5×10−19 systematic uncertainty, Phys. Rev. Lett. 135, 033201 (2025).
  4. H. N. Hausser, J. Keller, T. Nordmann, N. M. Bhatt, J. Kiethe, H. Liu, I. M. Richter, M. von Boehn, J. Rahm, S. Weyers, E. Benkler, B. Lipphardt, S. Dörscher, K. Stahl, J. Klose, C. Lisdat, M. Filzinger, N. Huntemann, E. Peik, and T. E. Mehlstäubler, In+115–Yb+172 Coulomb crystal clock with 2.5×10−18 systematic uncertainty, Phys. Rev. Lett. 134, 023201 (2025).
  5. M. Biercuk, J. Britton, H. Uys, A. Vandevender, and J. Bollinger, Yocto-newton force detection sensitivity using trapped ions, Nat. Nanotechnol. 5, 646 (2010).
  6. B. Deng, M. Göb, B. A. Stickler, M. Masuhr, K. Singer, and D. Wang, Amplifying a zeptonewton force with a single-ion nonlinear oscillator, Phys. Rev. Lett. 131, 153601 (2023).
  7. R. Blatt and D. Wineland, Entangled states of trapped atomic ions, Nature (London) 453, 1008 (2008).
  8. C. D. Bruzewicz, J. Chiaverini, R. McConnell, and J. M. Sage, Trapped-ion quantum computing: Progress and challenges, Appl. Phys. Rev. 6, 021314 (2019).
  9. S. Willitsch, Chemistry with controlled ions, Adv. Chem. Phys. 162, 307 (2017).
  10. M. Tomza, K. Jachymski, R. Gerritsma, A. Negretti, T. Calarco, Z. Idziaszek, and P. S. Julienne, Cold hybrid ion-atom systems, Rev. Mod. Phys. 91, 035001 (2019).
  11. P. O. Schmidt, T. Rosenband, C. Langer, W. M. Itano, J. C. Bergquist, and D. J. Wineland, Spectroscopy using quantum logic, Science 309, 749 (2005).
  12. M. Sinhal, Z. Meir, K. Najafian, G. Hegi, and S. Willitsch, Quantum-nondemolition state detection and spectroscopy of single trapped molecules, Science 367, 1213 (2020).
  13. M. Sinhal and S. Willitsch, Molecular-ion quantum technologies, in Photonic Quantum Technologies, edited by M. Benyoucef (John Wiley & Sons, Ltd., Weinheim, Germany, 2023), Chap. 13, pp. 305–332.
  14. D. Hunger, S. Camerer, M. Korppi, A. Jöckel, T. Hänsch, and P. Treutlein, Coupling ultracold atoms to mechanical oscillators, Compt. Rend. Phys. 12, 871 (2011).
  15. M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Cavity optomechanics, Rev. Mod. Phys. 86, 1391 (2014).
  16. N. Ghaemi, A. Nikoobin, and M. R. Ashory, A comprehensive categorization of micro/nanomechanical resonators and their practical applications from an engineering perspective: A review, Adv. Elect. Materials 8, 2200229 (2022).
  17. S. Singh, M. Bhattacharya, O. Dutta, and P. Meystre, Coupling nanomechanical cantilevers to dipolar molecules, Phys. Rev. Lett. 101, 263603 (2008).
  18. A. Sanz-Mora, S. Wüster, and J.-M. Rost, On-chip quantum tomography of mechanical nanoscale oscillators with guided Rydberg atoms, Phys. Rev. A 96, 013855 (2017).
  19. M. Weegen, M. Poggio, and S. Willitsch, Coupling trapped ions to a nanomechanical oscillator, Phys. Rev. Lett. 133, 223201 (2024).
  20. U. Delić, M. Reisenbauer, K. Dare, D. Grass, V. Vuletić, N. Kiesel, and M. Aspelmeyer, Cooling of a levitated nanoparticle to the motional quantum ground state, Science 367, 892 (2020).
  21. J. Millen, T. S. Monteiro, R. Pettit, and A. N. Vamivakas, Optomechanics with levitated particles, Rep. Prog. Phys. 83, 026401 (2020).
  22. F. Tebbenjohanns, M. L. Mattana, M. Rossi, M. Frimmer, and L. Novotny, Quantum control of a nanoparticle optically levitated in cryogenic free space, Nature (London) 595, 378 (2021).
  23. D. S. Bykov, L. Dania, F. Goschin, and T. E. Northup, Nanoparticle stored with an atomic ion in a linear Paul trap, Phys. Rev. Lett. 135, 213602 (2025).
  24. B. M. Zwickl, W. E. Shanks, A. M. Jayich, C. Yang, A. C. Bleszynski Jayich, J. D. Thompson, and J. G. E. Harris, High quality mechanical and optical properties of commercial silicon nitride membranes, Appl. Phys. Lett. 92, 103125 (2008).
  25. K. Hammerer, K. Stannigel, C. Genes, P. Zoller, P. Treutlein, S. Camerer, D. Hunger, and T. W. Hänsch, Optical lattices with micromechanical mirrors, Phys. Rev. A 82, 021803(R) (2010).
  26. 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).
  27. S. Camerer, M. Korppi, A. Jöckel, D. Hunger, T. W. Hänsch, and P. Treutlein, Realization of an optomechanical interface between ultracold atoms and a membrane, Phys. Rev. Lett. 107, 223001 (2011).
  28. N. Daniilidis and H. Häffner, Quantum interfaces between atomic and solid-state systems, Annu. Rev. Condens. Matter Phys. 4, 83 (2013).
  29. G. Kurizki, P. Bertet, Y. Kubo, K. Mølmer, D. Petrosyan, P. Rabl, and J. Schmiedmayer, Quantum technologies with hybrid systems, Proc. Natl. Acad. Sci. USA 112, 3866 (2015).
  30. P. N. Fountas, M. Poggio, and S. Willitsch, Classical and quantum dynamics of a trapped ion coupled to a charged nanowire, New J. Phys. 21, 013030 (2019).
  31. T. M. Karg, B. Gouraud, C. T. Ngai, G.-L. Schmid, K. Hammerer, and P. Treutlein, Light-mediated strong coupling between a mechanical oscillator and atomic spins 1 meter apart, Science 369, 174 (2020).
  32. A. Chowdhury, M. G. Clerc, S. Barbay, I. Robert-Philip, and R. Braive, Weak signal enhancement by nonlinear resonance control in a forced nano-electromechanical resonator, Nat. Commun. 11, 2400 (2020).
  33. M. Weegen, P. N. Fountas, M. Poggio, and S. Willitsch, Experimental implementation of an ion-nanowire hybrid system, Rev. Sci. Instrum. 96, 063203 (2025).
  34. Y. Wang, X. Zhang, Y. Feng, R. Shao, X. Xiong, X. Fang, Y. Deng, and W. Xu, Characterization of geometry deviation effects on ion trap mass analysis: A comparison study, Int. J. Mass Spectrom. 370, 125 (2014).
  35. A. Maitra, D. Leibfried, D. Ullmo, and H. Landa, Far-from-equilibrium noise-heating and laser-cooling dynamics in radio-frequency Paul traps, Phys. Rev. A 99, 043421 (2019).
  36. N. Akerman, S. Kotler, Y. Glickman, Y. Dallal, A. Keselman, and R. Ozeri, Single-ion nonlinear mechanical oscillator, Phys. Rev. A 82, 061402(R) (2010).
  37. G. H. Low, P. F. Herskind, and I. L. Chuang, Finite-geometry models of electric field noise from patch potentials in ion traps, Phys. Rev. A 84, 053425 (2011).
  38. J. P. Home, D. Hanneke, J. D. Jost, D. Leibfried, and D. J. Wineland, Normal modes of trapped ions in the presence of anharmonic trap potentials, New J. Phys. 13, 073026 (2011).
  39. M. Johanning, Isospaced linear ion strings, Appl. Phys. B 122, 71 (2016).
  40. T. Gudjons, P. Seibert, and G. Werth, Influence of anharmonicities of a Paul trap potential on the motion of stored ions, Appl. Phys. B 65, 57 (1997).
  41. S. S. Sharma and N. K. Sharma, The Schrödinger cat state of trapped ions in harmonic and anharmonic oscillator traps, J. Phys. B: At. Mol. Opt. Phys. 35, 1643 (2002).
  42. L. M. A. Nguyen, B. Bowers, and S. Mouradian, The effect of trap design on the scalability of trapped-ion quantum technologies, Entropy 27, 576 (2025).
  43. P. T. Grochowski, H. Pichler, C. A. Regal, and O. Romero-Isart, Quantum control of continuous systems via nonharmonic potential modulation, Quantum 9, 1824 (2025).
  44. A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations (John Wiley & Sons, Ltd., New York, 1995).
  45. M. Drewsen, A. Mortensen, R. Martinussen, P. Staanum, and J. L. Sørensen, Nondestructive identification of cold and extremely localized single molecular ions, Phys. Rev. Lett. 93, 243201 (2004).
  46. F. G. Major, V. N. Gheorghe, and G. Werth, Charged Particle Traps (Springer, Berlin, 2005).
  47. H. J. Metcalf and P. van der Straten, Laser Cooling and Trapping (Springer, New York, 1999).
  48. H. Duifhuis, Nonlinear tools, in Cochlear Mechanics: Introduction to a Time Domain Analysis of the Nonlinear Cochlea (Springer, New York, 2008), Chap. 9, pp. 211–235.
  49. S. Zaitsev, O. Shtempluck, E. Buks, and O. Gottlieb, Nonlinear damping in a micromechanical oscillator, Nonlinear Dyn. 67, 859 (2012).
  50. S. J. Elliott, M. G. Tehrani, and R. S. Langley, Nonlinear damping and quasi-linear modelling, Philos. Trans. R. Soc. A 373, 20140402 (2015).
  51. D. J. Berkeland, J. D. Miller, J. C. Bergquist, W. M. Itano, and D. J. Wineland, Minimization of ion micromotion in a Paul trap, J. Appl. Phys. 83, 5025 (1998).
  52. B. Wang, J. W. Zhang, Z. H. Lu, and L. J. Wang, Direct measurement of micromotion speed in a linear quadrupole trap, J. Appl. Phys. 108, 013108 (2010).
  53. R. Lifshitz and M. C. Cross, Nonlinear Dynamics of Nanomechanical and Micromechanical Resonators (John Wiley & Sons, Ltd., Weinheim, Germany, 2008), Chap. 1, pp. 1–52.
  54. T. Barois, S. Perisanu, P. Vincent, S. T. Purcell, and A. Ayari, Frequency modulated self-oscillation and phase inertia in a synchronized nanowire mechanical resonator, New J. Phys. 16, 083009 (2014).
  55. D. Younesian, E. Esmailzadeh, and R. Sedaghati, Existence of periodic solutions for the generalized form of Mathieu equation, Nonlinear Dyn. 39, 335 (2005).
  56. G. T. Abraham and A. Chatterjee, Approximate asymptotics for a nonlinear Mathieu equation using harmonic balance based averaging, Nonlinear Dyn. 31, 347 (2003).
  57. M. Sudakov, Effective potential and the ion axial beat motion near the boundary of the first stable region in a nonlinear ion trap, Int. J. Mass Spectrom. 206, 27 (2001).
  58. I. Filippov and M. Sudakov, Analysis of electric fields in mass spectrometry, Int. J. Mass Spectrom. 467, 116620 (2021).
  59. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Mechanics, 3rd ed. (Butterworth-Heinemann, Oxford, UK, 1976), Vol. 1.
  60. D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, Quantum dynamics of single trapped ions, Rev. Mod. Phys. 75, 281 (2003).
  61. https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.solve_ivp.html.
  62. J. Dormand and P. Prince, A family of embedded Runge-Kutta formulae, J. Comput. Appl. Math. 6, 19 (1980).
  63. J. Pedregosa-Gutierrez, C. Champenois, M. R. Kamsap, and M. Knoop, Ion transport in macroscopic rf linear traps, Int. J. Mass Spectrom. 381-382, 33 (2015).
  64. M. Feldman, Hilbert Transform Applications in Mechanical Vibration (Wiley, 2011).

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