Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Quantum-enhanced sensing of a single-ion mechanical oscillator


Special quantum states are used in metrology to achieve sensitivities below the limits established by classically behaving states1,2. In bosonic interferometers, squeezed states3, number states4,5 and ‘Schrödinger cat’ states5 have been implemented on various platforms and have demonstrated improved measurement precision over interferometers using coherent states6,7. Another metrologically useful state is an equal superposition of two eigenstates with maximally different energies; this state ideally reaches the full interferometric sensitivity allowed by quantum mechanics8,9. Here we demonstrate the enhanced sensitivity of these quantum states in the case of a harmonic oscillator. We extend an existing experimental technique10 to create number states of order up to n = 100 and to generate superpositions of a harmonic oscillator ground state and a number state of the form \(\frac{1}{\sqrt{2}}\left(\left|0\right\rangle +\left|n\right\rangle \right)\) with n up to 18 in the motion of a single trapped ion. Although experimental imperfections prevent us from reaching the ideal Heisenberg limit, we observe enhanced sensitivity to changes in the frequency of the mechanical oscillator. This sensitivity initially increases linearly with n and reaches a maximum at n = 12, where we observe a metrological enhancement of 6.4(4) decibels (the uncertainty is one standard deviation of the mean) compared to an ideal measurement on a coherent state with the same average occupation number. Such measurements should provide improved characterization of motional decoherence, which is an important source of error in quantum information processing with trapped ions11,12. It should also be possible to use the quantum advantage from number-state superpositions to achieve precision measurements in other harmonic oscillator systems.

This is a preview of subscription content, access via your institution

Relevant articles

Open Access articles citing this article.

Access options

Rent or buy this article

Prices vary by article type



Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Generating number states and number-state superpositions.
Fig. 2: Sideband flopping on number states.
Fig. 3: Interference and sensitivity of different number-state superpositions.
Fig. 4: Tracking of oscillator frequency using number-state interferometers.

Data availability

The datasets generated or analysed during the current study are available from the corresponding author on reasonable request.


  1. Pezzè, L., Smerzi, A., Oberthaler, M. K., Schmied, R. & Treutlein, P. Quantum metrology with nonclassical states of atomic ensembles. Rev. Mod. Phys. 90, 035005 (2018).

    Article  ADS  MathSciNet  Google Scholar 

  2. Braun, D. et al. Quantum-enhanced measurements without entanglement. Rev. Mod. Phys. 90, 035006 (2018).

    Article  ADS  MathSciNet  Google Scholar 

  3. Caves, C. M., Thorne, K. S., Drever, R. W. P., Sandberg, V. D. & Zimmermann, M. On the measurement of a weak classical force coupled to a quantum-mechanical oscillator. I. Issues of principle. Rev. Mod. Phys. 52, 341–392 (1980).

    Article  ADS  Google Scholar 

  4. Boto, A. N. et al. Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit. Phys. Rev. Lett. 85, 2733–2736 (2000).

    Article  ADS  CAS  Google Scholar 

  5. Haroche, S. & Raymond, J.-M. Exploring the Quantum (Oxford University Press, 2006).

  6. Schrödinger, E. Der stetige Übergang von der Mikro- zur Makromechanik. Naturwissenschaften 14, 664–666 (1926).

    Article  ADS  Google Scholar 

  7. Glauber, R. J. Nobel lecture: one hundred years of light quanta. Rev. Mod. Phys. 78, 1267–1278 (2006).

    Article  ADS  Google Scholar 

  8. Margolus, N. & Levitin, L. B. The maximum speed of dynamical evolution. Physica D 120, 188–195 (1998).

    Article  ADS  Google Scholar 

  9. Caves, C. M. & Shaji, A. Quantum-circuit guide to optical and atomic interferometry. Opt. Commun. 283, 695–712 (2010).

    Article  ADS  CAS  Google Scholar 

  10. Meekhof, D. M., Monroe, C., King, B. E., Itano, W. M. & Wineland, D. J. Generation of nonclassical motional states of a trapped atom. Phys. Rev. Lett. 77, 2346 (1996).

    Article  ADS  CAS  Google Scholar 

  11. Ballance, C. J., Harty, T. P., Linke, N. M., Sepiol, M. A. & Lucas, D. M. High-fidelity quantum logic gates using trapped-ion hyperfine qubits. Phys. Rev. Lett. 117, 060504 (2016).

    Article  ADS  CAS  Google Scholar 

  12. Gaebler, J. P. et al. High-fidelity universal gate set for 9Be+ ion qubits. Phys. Rev. Lett. 117, 060505 (2016).

    Article  ADS  CAS  Google Scholar 

  13. Blais, A., Huang, R.-S., Wallraff, A., Girvin, S. M. & Schoelkopf, R. J. Cavity quantum electrodynamics for superconducting electrical circuits: an architecture for quantum computation. Phys. Rev. A 69, 062320 (2004).

    Article  ADS  Google Scholar 

  14. Aspelmeyer, M., Kippenberg, T. & Marquardt, F. Cavity optomechanics. Rev. Mod. Phys. 86, 1391–1452 (2014).

    Article  ADS  Google Scholar 

  15. Grimm, R., Weidemüller, M. & Ovchinnikov, Y. B. Optical dipole traps for neutral atoms. 42 Adv. Atom. Mol. Opt. Phys. 95–170 (2000).

  16. Leibfried, D., Blatt, R., Monroe, C. & Wineland, D. J. Quantum dynamics of single trapped ions. Rev. Mod. Phys. 75, 281–324 (2003).

    Article  ADS  CAS  Google Scholar 

  17. Aasi, J. et al. Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light. Nat. Photon. 7, 613–619 (2013).

    Article  ADS  CAS  Google Scholar 

  18. Ladd, T. D. et al. Quantum computers. Nature 464, 45–53 (2010).

    Article  ADS  CAS  Google Scholar 

  19. Wineland, D. J., Bollinger, J. J. & Itano, W. M. Laser fluorescence mass spectroscopy. Phys. Rev. Lett. 50, 628–631 (1983).

    Article  ADS  CAS  Google Scholar 

  20. Sheridan, K. & Keller, M. Weighing of trapped ion crystals and its applications. New J. Phys. 13, 123002 (2011).

    Article  ADS  Google Scholar 

  21. Alonso, J. et al. Generation of large coherent states by bang–bang control of a trapped-ion oscillator. Nat. Commun. 7, 11243 (2016).

    Article  ADS  CAS  Google Scholar 

  22. Home, J. P., Hanneke, D., Jost, J. D., Leibfried, D. & Wineland, D. J. Normal modes of trapped ions in the presence of anharmonic trap potentials. New J. Phys. 13, 073026 (2011).

    Article  ADS  Google Scholar 

  23. Leibfried, D. et al. Trapped-ion quantum simulator: experimental application to nonlinear interferometers. Phys. Rev. Lett. 89, 247901 (2002).

    Article  ADS  CAS  Google Scholar 

  24. Cirac, J. I. & Zoller, P. Quantum computations with cold trapped ions. Phys. Rev. Lett. 74, 4091–4094 (1995).

    Article  ADS  CAS  Google Scholar 

  25. Brownnutt, M., Kumph, M., Rabl, P. & Blatt, R. Ion-trap measurements of electric-field noise near surfaces. Rev. Mod. Phys. 87, 1419–1482 (2015).

    Article  ADS  CAS  Google Scholar 

  26. Itano, W. M. et al. Quantum projection noise: population fluctuations in two-level systems. Phys. Rev. A 47, 3554–3570 (1993).

    Article  ADS  CAS  Google Scholar 

  27. Monroe, C., Meekhof, D. M., King, B. & Wineland, D. J. A “Schrödinger cat” superposition state of an atom. Science 272, 1131–1136 (1996).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  28. Wilson, A. C. et al. Tunable spin–spin interactions and entanglement of ions in separate potential wells. Nature 512, 57–60 (2014).

    Article  ADS  CAS  Google Scholar 

  29. Monroe, C. et al. Resolved-sideband Raman cooling of a bound atom to the 3D zero-point energy. Phys. Rev. Lett. 75, 4011–4014 (1995).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  30. Sanner, C., Huntemann, N., Lange, R., Tamm, C. & Peik, E. Autobalanced Ramsey spectroscopy. Phys. Rev. Lett. 120, 053602 (2018).

    Article  ADS  CAS  Google Scholar 

  31. Howe, D. A., Allan, D. U. & Barnes, J. A. Properties of signal sources and measurement methods. In Proc. 1981 Freq. Cont. Symp. 1–47 (IEEE, 1981).

  32. Kotler, S., Akerman, N., Glickman, Y., Keselman, A. & Ozeri, R. Single-ion quantum lock-in amplifier. Nature 473, 61–65 (2011).

    Article  ADS  CAS  Google Scholar 

  33. Ziesel, F. et al. Experimental creation and analysis of displaced number states. J. Phys. At. Mol. Opt. Phys. 46, 104008 (2013).

    Article  ADS  CAS  Google Scholar 

  34. Wolf, F. et al. Motional Fock states for quantum-enhanced amplitude and phase measurements with trapped ions. Preprint at (2018).

  35. Chu, Y. et al. Creation and control of multi-phonon Fock states in a bulk acoustic-wave resonator. Nature 563, 666–670 (2018).

    Article  ADS  CAS  Google Scholar 

  36. Wilson, A. C. et al. A 750 mW, continuous-wave, solid-state laser source at 313 nm for cooling and manipulating trapped 9Be+ ions. Appl. Phys. B 105, 741–748 (2011).

    Article  ADS  CAS  Google Scholar 

  37. Colombe, Y., Slichter, D. H., Wilson, A. C., Leibfried, D. & Wineland, D. J. Single-mode optical fiber for high-power, low-loss UV transmission. Opt. Express 22, 19783–19793 (2014).

    Article  ADS  Google Scholar 

Download references


We thank D. Allcock, D. Slichter and R. Srinivas for discussions and assistance with the experimental setup, and H. F. Leopardi and H. Knaack for comments on the manuscript. This work was supported by IARPA, ARO, ONR and the NIST Quantum Information Program. K.C.M. acknowledges support by an ARO QuaCGR fellowship through grant W911NF-14-1-0079. J.K. acknowledges support by the Alexander von Humboldt foundation.

Author information

Authors and Affiliations



K.C.M. and D.L. conceived the experiments, carried out the measurements, analysed the data and wrote the main part of the manuscript. K.C.M., J.K., S.C.B. and A.C.W. built and maintained the experimental setup. D.J.W., A.C.W. and D.L. developed parts of the experimental setup and supervised the work. All authors discussed the results and contributed to the manuscript.

Corresponding author

Correspondence to Katherine C. McCormick.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data figures and tables

Extended Data Fig. 1 Schematic illustrating the auto-balanced feedback loop.

The feedback is applied to the LO, a frequency source used as a reference to compare to the ion’s oscillation frequency. The LO controls the phases and frequencies of the BSB and RSB laser pulses (see Fig. 1a) during the mode-frequency tracking experiments. The difference between the populations measured after a pair of Ramsey experiments with long waiting times provides an error signal, plong, which is used to feed back on the LO frequency, ωLO. Similarly, a second pair of Ramsey experiments with short waiting times provides and error signal, pshort, which is used to feed back on an additional LO phase offset ϕLO between the first and second effective π/2-pulses (“π/2”). The long- and short-waiting-time Ramsey experiments are interleaved, with ϕLO and ωLO applied equally to both. For more details on auto-balanced Ramsey experiments, see ref. 30.

Extended Data Table 1 Duration of pulse sequences used to produce number states and number-state superpositions

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

McCormick, K.C., Keller, J., Burd, S.C. et al. Quantum-enhanced sensing of a single-ion mechanical oscillator. Nature 572, 86–90 (2019).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

This article is cited by


By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing