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# Quantum-enhanced sensing of a single-ion mechanical oscillator

## Abstract

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.

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## Data availability

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

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## Acknowledgements

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

### Contributions

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.

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## 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.

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McCormick, K.C., Keller, J., Burd, S.C. et al. Quantum-enhanced sensing of a single-ion mechanical oscillator. Nature 572, 86–90 (2019). https://doi.org/10.1038/s41586-019-1421-y

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