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Optimal metrology with programmable quantum sensors

Nature volume 603, pages 604–609 (2022)Cite this article

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Abstract

Quantum sensors are an established technology that has created new opportunities for precision sensing across the breadth of science. Using entanglement for quantum enhancement will allow us to construct the next generation of sensors that can approach the fundamental limits of precision allowed by quantum physics. However, determining how state-of-the-art sensing platforms may be used to converge to these ultimate limits is an outstanding challenge. Here we merge concepts from the field of quantum information processing with metrology, and successfully implement experimentally a programmable quantum sensor operating close to the fundamental limits imposed by the laws of quantum mechanics. We achieve this by using low-depth, parametrized quantum circuits implementing optimal input states and measurement operators for a sensing task on a trapped-ion experiment. With 26 ions, we approach the fundamental sensing limit up to a factor of 1.45 ± 0.01, outperforming conventional spin-squeezing with a factor of 1.87 ± 0.03. Our approach reduces the number of averages to reach a given Allan deviation by a factor of 1.59 ± 0.06 compared with traditional methods not using entanglement-enabled protocols. We further perform on-device quantum-classical feedback optimization to ‘self-calibrate’ the programmable quantum sensor with comparable performance. This ability illustrates that this next generation of quantum sensor can be used without previous knowledge of the device or its noise environment.

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Fig. 1: Measurement and feedback concept for variational quantum Ramsey interferometry circuits.
Fig. 2: Generalized Ramsey sequence performance measurements.
Fig. 3: On-device hybrid quantum-classical optimization performance with 26 ions at \({\boldsymbol{\delta \phi }}{\boldsymbol{\approx }}{\bf{0.74}}\) (minimum BMSE versus δϕ).
Fig. 4: Frequency measurement using 12 ions with a standard and variationally optimized Ramsey sequence.

Data availability

All data obtained in the study are available from the corresponding author upon request. Source data are provided with this paper.

References

  1. Taylor, M. A. & Bowen, W. P. Quantum metrology and its application in biology. Phys. Rep. 615, 1–59 (2016).

    Article  MathSciNet  CAS  ADS  Google Scholar 

  2. Wu, Y., Jelezko, F., Plenio, M. B. & Weil, T. Diamond quantum devices in biology. Angew. Chem. Int. Edn 55, 6586–6598 (2016).

    Article  CAS  Google Scholar 

  3. Rej, E., Gaebel, T., Boele, T., Waddington, D. E. & Reilly, D. J. Hyperpolarized nanodiamond with long spin-relaxation times. Nat. Commun. 6, 8459 (2015).

    Article  CAS  PubMed  ADS  Google Scholar 

  4. Frasco, M. F. & Chaniotakis, N. Semiconductor quantum dots in chemical sensors and biosensors. Sensors 9, 7266–7286 (2009).

    Article  CAS  PubMed  PubMed Central  ADS  Google Scholar 

  5. Chen, Y.-J. et al. Single-source multiaxis cold-atom interferometer in a centimeter-scale cell. Phys. Rev. Appl. 12, 014019 (2019).

    Article  CAS  ADS  Google Scholar 

  6. Ahn, J. et al. Ultrasensitive torque detection with an optically levitated nanorotor. Nat. Nanotechnol. 15, 89–93 (2020).

    Article  CAS  PubMed  ADS  Google Scholar 

  7. Moser, J. et al. Ultrasensitive force detection with a nanotube mechanical resonator. Nat. Nanotechnol. 8, 493–496 (2013).

    Article  CAS  PubMed  ADS  Google Scholar 

  8. Chaste, J. et al. A nanomechanical mass sensor with yoctogram resolution. Nat. Nanotechnol. 7, 301–304 (2012).

    Article  CAS  PubMed  ADS  Google Scholar 

  9. Ludlow, A. D., Boyd, M. M., Ye, J., Peik, E. & Schmidt, P. O. Optical atomic clocks. Rev. Mod. Phys. 87, 637–701 (2015).

    Article  CAS  ADS  Google Scholar 

  10. Tse, M. et al. Quantum-enhanced advanced LIGO detectors in the era of gravitational-wave astronomy. Phys. Rev. Lett. 123, 231107 (2019).

    Article  CAS  PubMed  ADS  Google Scholar 

  11. Casacio, C. A. et al. Quantum-enhanced nonlinear microscopy. Nature 594, 201–206 (2021).

    Article  CAS  PubMed  ADS  Google Scholar 

  12. Pedrozo-Peñafiel, E. et al. Entanglement on an optical atomic-clock transition. Nature 588, 414–418 (2020).

    Article  PubMed  ADS  Google Scholar 

  13. Górecki, W., Demkowicz-Dobrzański, R., Wiseman, H. M. & Berry, D. W. π-corrected Heisenberg limit. Phys. Rev. Lett. 124, 030501 (2020).

    Article  MathSciNet  PubMed  ADS  Google Scholar 

  14. Kaubruegger, R., Vasilyev, D. V., Schulte, M., Hammerer, K. & Zoller, P. Quantum Variational optimization of Ramsey interferometry and atomic clocks. Phys. Rev. X 11, 041045 (2021).

    CAS  Google Scholar 

  15. Preskill, J. Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018).

    Article  Google Scholar 

  16. Omran, A. et al. Generation and manipulation of Schrödinger cat states in Rydberg atom arrays. Science 365, 570–574 (2019).

    Article  MathSciNet  CAS  PubMed  ADS  Google Scholar 

  17. Pogorelov, I. et al. Compact ion-trap quantum computing demonstrator. PRX Quantum 2, 020343 (2021).

    Article  ADS  Google Scholar 

  18. Scholl, P. et al. Quantum simulation of 2D antiferromagnets with hundreds of Rydberg atoms. Nature 595, 233–238 (2021).

    Article  CAS  PubMed  ADS  Google Scholar 

  19. Ebadi, S. et al. Quantum phases of matter on a 256-atom programmable quantum simulator. Nature 595, 227–232 (2021).

    Article  CAS  PubMed  ADS  Google Scholar 

  20. Semeghini, G. et al. Probing topological spin liquids on a programmable quantum simulator. Science 374, 1242–1247 (2021).

    Article  CAS  PubMed  ADS  Google Scholar 

  21. Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nature Commun. 5, 4213 (2014).

    Article  CAS  ADS  Google Scholar 

  22. Kandala, A. et al. Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. Nature 549, 242–246 (2017).

    Article  CAS  PubMed  ADS  Google Scholar 

  23. Kokail, C. et al. Self-verifying variational quantum simulation of lattice models. Nature 569, 355–360 (2019).

    Article  CAS  PubMed  ADS  Google Scholar 

  24. Cerezo, M. et al. Variational quantum algorithms. Nat. Rev. Phys. 3, 625–644 (2021).

    Article  Google Scholar 

  25. Davis, E., Bentsen, G. & Schleier-Smith, M. Approaching the Heisenberg limit without single-particle detection. Phys. Rev. Lett. 116, 053601 (2016).

    Article  PubMed  ADS  Google Scholar 

  26. Hosten, O., Krishnakumar, R., Engelsen, N. J. & Kasevich, M. A. Quantum phase magnification. Science 352, 1552–1555 (2016).

    Article  MathSciNet  CAS  PubMed  MATH  ADS  Google Scholar 

  27. Kaubruegger, R. et al. Variational spin-squeezing algorithms on programmable quantum sensors. Phys. Rev. Lett. 123, 260505 (2019).

    Article  CAS  PubMed  ADS  Google Scholar 

  28. Wineland, D. J., Bollinger, J. J., Itano, W. M., Moore, F. & Heinzen, D. Spin squeezing and reduced quantum noise in spectroscopy. Phys. Rev. A 46, R6797 (1992).

    Article  CAS  PubMed  ADS  Google Scholar 

  29. Bollinger, J. J., Itano, W. M., Wineland, D. J. & Heinzen, D. J. Optimal frequency measurements with maximally correlated states. Phys. Rev. A. 54, R4649 (1996).

    Article  CAS  PubMed  ADS  Google Scholar 

  30. 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  MathSciNet  ADS  Google Scholar 

  31. Degen, C. L., Reinhard, F. & Cappellaro, P. Quantum sensing. Rev. Mod. Phys. 89, 035002 (2017).

    Article  MathSciNet  ADS  Google Scholar 

  32. Leroux, I. D. et al. On-line estimation of local oscillator noise and optimisation of servo parameters in atomic clocks. Metrologia 54, 307–321 (2017).

    Article  CAS  ADS  Google Scholar 

  33. Macieszczak, K., Fraas, M. & Demkowicz-Dobrzański, R. Bayesian quantum frequency estimation in presence of collective dephasing. New J. Phys. 16, 113002 (2014).

    Article  MATH  ADS  Google Scholar 

  34. Kitagawa, M. & Ueda, M. Squeezed spin states. Phys. Rev. A. 47, 5138–5143 (1993).

    Article  CAS  PubMed  ADS  Google Scholar 

  35. Bohnet, J. G. et al. Quantum spin dynamics and entanglement generation with hundreds of trapped ions. Science 352, 1297–1301 (2016).

    Article  MathSciNet  CAS  PubMed  MATH  ADS  Google Scholar 

  36. Jones, J. A. et al. Magnetic field sensing beyond the standard quantum limit using 10-spin NOON states. Science 324, 1166–1168 (2009).

    Article  CAS  PubMed  ADS  Google Scholar 

  37. Bordé, C. J. Atomic clocks and inertial sensors. Metrologia 39, 435–463 (2002).

    Article  ADS  Google Scholar 

  38. Gilmore, K. A. et al. Quantum-enhanced sensing of displacements and electric fields with two-dimensional trapped-ion crystals. Science 373, 673–678 (2021).

    Article  CAS  PubMed  ADS  Google Scholar 

  39. Gilmore, K. A., Bohnet, J. G., Sawyer, B. C., Britton, J. W. & Bollinger, J. J. Amplitude sensing below the zero-point fluctuations with a two-dimensional trapped-ion mechanical oscillator. Phys. Rev. Lett. 118, 263602 (2017).

    Article  CAS  PubMed  ADS  Google Scholar 

  40. Demkowicz-Dobrzański, R., Górecki, W. & Guţă, M. Multi-parameter estimation beyond quantum fisher information. J. Phys. A. Math. Theor. 53, 363001 (2020).

    Article  MathSciNet  Google Scholar 

  41. André, A., Sørensen, A. & Lukin, M. Stability of atomic clocks based on entangled atoms. Phys. Rev. Lett. 92, 230801 (2004).

    Article  PubMed  ADS  Google Scholar 

  42. Demkowicz-Dobrzański, R., Jarzyna, M. & Kołodyński, J. Quantum Limits in Optical Interferometry Vol. 60 of Progress in Optics (Elsevier, 2015).

  43. Chabuda, K., Dziarmaga, J., Osborne, T. J. & Demkowicz-Dobrzałski, R. Tensor-network approach for quantum metrology in many-body quantum systems. Nat. Commun. 11, 250 (2020).

    Article  CAS  PubMed  PubMed Central  ADS  Google Scholar 

  44. Borregaard, J. & Sørensen, A. S. Near-Heisenberg-limited atomic clocks in the presence of decoherence. Phys. Rev. Lett. 111, 090801 (2013).

    Article  CAS  PubMed  ADS  Google Scholar 

  45. Trees, H. L. V. Detection, Estimation and Modulation (Wiley, 1968).

  46. Leroux, I. D. et al. On-line estimation of local oscillator noise and optimisation of servo parameters in atomic clocks. Metrologia 54, 307 (2017).

    Article  CAS  ADS  Google Scholar 

  47. Wineland, D. J., Bollinger, J. J., Itano, W. M. & Heinzen, D. Squeezed atomic states and projection noise in spectroscopy. Phys. Rev. A. 50, 67–88 (1994).

    Article  CAS  PubMed  ADS  Google Scholar 

Download references

Acknowledgements

We acknowledge funding from the EU H2020-FETFLAG-2018-03 under grant agreement no. 820495. We also acknowledge support by the Austrian Science Fund (FWF), through the SFB BeyondC (FWF Project No. F7109), and the IQI GmbH. P.S. acknowledges support from the Austrian Research Promotion Agency (FFG) contract 872766. P.S., T.M. and R.B. acknowledge funding by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), through US ARO grant no. W911NF-16-1-0070 and W911NF-20-1-0007, and the US Air Force Office of Scientific Research (AFOSR) via IOE grant no. FA9550-19-1-7044 LASCEM. R.K., D.V.V. and P.Z. are supported by the US Air Force Office of Scientific Research (AFOSR) through IOE grant no. FA9550-19-1-7044 LASCEM, D.V.V by a joint-project grant from the FWF (grant no. I04426, RSF/Russia 2019), R.v.B and P.Z. by the European Union’s Horizon 2020 research and innovation programme under grant agreement no. 817482 (PASQuanS) and R.v.B by the Austrian Research Promotion Agency (FFG) contract 884471 (ELQO). P.Z. acknowledges funding by the the European Union’s Horizon 2020 research and innovation programme under grant agreement no. 731473 (QuantERA through QTFLAG), and by the Simons Collaboration on Ultra-Quantum Matter, which is a grant from the Simons Foundation (651440). Innsbruck theory is a member of the NSF Quantum Leap Challenge Institute Q-Sense. The computational results presented here have been achieved (in part) using the LEO HPC infrastructure of the University of Innsbruck. All statements of fact, opinions or conclusions contained herein are those of the authors and should not be construed as representing the official views or policies of the funding agencies.

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  1. These authors contributed equally: Christian D. Marciniak, Thomas Feldker

Authors and Affiliations

  1. Institut für Experimentalphysik, Innsbruck, Austria

    Christian D. Marciniak, Thomas Feldker, Ivan Pogorelov, Philipp Schindler, Rainer Blatt & Thomas Monz

  2. Institute for Quantum Optics and Quantum Information, Innsbruck, Austria

    Raphael Kaubruegger, Denis V. Vasilyev, Rick van Bijnen, Peter Zoller & Rainer Blatt

  3. Center for Quantum Physics, Innsbruck, Austria

    Raphael Kaubruegger, Denis V. Vasilyev, Rick van Bijnen & Peter Zoller

  4. Alpine Quantum Technologies, Innsbruck, Austria

    Thomas Monz

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Contributions

Ch.D.M. was the lead writer of the manuscript with assistance from R.K., D.V.V., R.v.B. and P.Z., and input from all coauthors. Ch.D.M., T.F. and I.P. built the experiment. Ch.D.M. and T.F. performed measurements. R.K., D.V.V. and P.Z. conceived of the method and provided theory. R.K. and R.v.B. developed the optimizer routines and implementation. Ch.D.M. and R.K. analysed the data. P.S., R.B. and T.M. supervised the experiment.

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Correspondence to Thomas Monz.

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Marciniak, C.D., Feldker, T., Pogorelov, I. et al. Optimal metrology with programmable quantum sensors. Nature 603, 604–609 (2022). https://doi.org/10.1038/s41586-022-04435-4

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