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 amplification of boson-mediated interactions


Strong and precisely controlled interactions between quantum objects are essential for quantum information processing1,2, simulation3 and sensing4,5, and for the formation of exotic quantum matter6. A well-established paradigm for coupling otherwise weakly interacting quantum objects is to use auxiliary bosonic quantum excitations to mediate the interactions. Important examples include photon-mediated interactions between atoms7, superconducting qubits8, and colour centres in diamond9, and phonon-mediated interactions between trapped ions10,11,12 and between optical and microwave photons13. Boson-mediated interactions can, in principle, be amplified through parametric driving of the boson channel; the drive need not couple directly to the interacting quantum objects. This technique has been proposed for a variety of quantum platforms14,15,16,17,18,19,20,21,22,23,24, but has not, so far, been realized in the laboratory. Here we experimentally demonstrate the amplification of a boson-mediated interaction between two trapped-ion qubits by parametric modulation of the trapping potential21. The amplification provides up to a 3.25-fold increase in the interaction strength, validated by measuring the speed-up of two-qubit entangling gates. This amplification technique can be used in any quantum platform where parametric modulation of the boson channel is possible, enabling exploration of new parameter regimes and enhanced quantum information processing.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Fig. 1: Phase-space illustration of boson-mediated interactions.
Fig. 2: Bell-state fidelities and speed-up.
Fig. 3: Phase dependence of amplification.

Data availability

Source data are provided with this paper. All other data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.

Code availability

The simulation and analysis codes are available from the corresponding authors upon reasonable request.


  1. 1.

    Bruzewicz, C. D., Chiaverini, J., McConnell, R. & Sage, J. M. Trapped-ion quantum computing: progress and challenges. Appl. Phys. Rev. 6, 021314 (2019).

    ADS  Article  Google Scholar 

  2. 2.

    Blais, A., Girvin, S. M. & Oliver, W. D. Quantum information processing and quantum optics with circuit quantum electrodynamics. Nat. Phys. 16, 247–256 (2020).

    Article  Google Scholar 

  3. 3.

    Georgescu, I. M., Ashhab, S. & Nori, F. Quantum simulation. Rev. Mod. Phys. 86, 153–185 (2014).

    ADS  Article  Google Scholar 

  4. 4.

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

    ADS  MathSciNet  Article  Google Scholar 

  5. 5.

    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).

    ADS  MathSciNet  Article  Google Scholar 

  6. 6.

    Bloch, I., Dalibard, J. & Zwerger, W. Many-body physics with ultracold gases. Rev. Mod. Phys. 80, 885–964 (2008).

    ADS  Article  Google Scholar 

  7. 7.

    Gerry, C. C. & Knight, P. L. Introductory Quantum Optics (Cambridge Univ. Press, 2005).

    Google Scholar 

  8. 8.

    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).

    ADS  Article  Google Scholar 

  9. 9.

    Evans, R. E. et al. Photon-mediated interactions between quantum emitters in a diamond nanocavity. Science 362, 662–665 (2018).

    ADS  Article  Google Scholar 

  10. 10.

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

    ADS  Article  Google Scholar 

  11. 11.

    Sørensen, A. & Mølmer, K. Quantum computation with ions in thermal motion. Phys. Rev. Lett. 82, 1971–1974 (1999).

    ADS  Article  Google Scholar 

  12. 12.

    Milburn, G. J., Schneider, S. & James, D. F. V. Ion trap quantum computing with warm ions. Fortschr. Phys. 48, 801–810 (2000).

    Article  Google Scholar 

  13. 13.

    Higginbotham, A. P. et al. Harnessing electro-optic correlations in an efficient mechanical converter. Nat. Phys. 14, 1038–1042 (2018).

    Article  Google Scholar 

  14. 14.

    Lü, X.-Y. et al. Squeezed optomechanics with phase-matched amplification and dissipation. Phys. Rev. Lett. 114, 093602 (2015).

    ADS  Article  Google Scholar 

  15. 15.

    Lemonde, M.-A., Didier, N. & Clerk, A. A. Enhanced nonlinear interactions in quantum optomechanics via mechanical amplification. Nat. Commun. 7, 11338 (2016).

    ADS  Article  Google Scholar 

  16. 16.

    Zeytinoğlu, S., İmamoğlu, A. & Huber, S. Engineering matter interactions using squeezed vacuum. Phys. Rev. X 7, 021041 (2017).

    Google Scholar 

  17. 17.

    Qin, W. et al. Exponentially enhanced light-matter interaction, cooperativities, and steady-state entanglement using parametric amplification. Phys. Rev. Lett. 120, 093601 (2018).

    ADS  Article  Google Scholar 

  18. 18.

    Chen, Y.-H., Qin, W. & Nori, F. Fast and high-fidelity generation of steady-state entanglement using pulse modulation and parametric amplification. Phys. Rev. A 100, 012339 (2019).

    ADS  Article  Google Scholar 

  19. 19.

    Leroux, C., Govia, L. C. G. & Clerk, A. A. Enhancing cavity quantum electrodynamics via antisqueezing: synthetic ultrastrong coupling. Phys. Rev. Lett. 120, 093602 (2018).

    ADS  Article  Google Scholar 

  20. 20.

    Arenz, C., Bondar, D. I., Burgarth, D., Cormick, C. & Rabitz, H. Amplification of quadratic Hamiltonians. Quantum 4, 271 (2020).

    Article  Google Scholar 

  21. 21.

    Ge, W. et al. Trapped ion quantum information processing with squeezed phonons. Phys. Rev. Lett. 122, 030501 (2019).

    ADS  Article  Google Scholar 

  22. 22.

    Ge, W. et al. Stroboscopic approach to trapped-ion quantum information processing with squeezed phonons. Phys. Rev. A 100, 043417 (2019).

    ADS  Article  Google Scholar 

  23. 23.

    Groszkowski, P., Lau, H.-K., Leroux, C., Govia, L. C. G. & Clerk, A. A. Heisenberg-limited spin squeezing via bosonic parametric driving. Phys. Rev. Lett. 125, 203601 (2020).

    ADS  Article  Google Scholar 

  24. 24.

    Li, P.-B., Zhou, Y., Gao, W.-B. & Nori, F. Enhancing spin-phonon and spin-spin interactions using linear resources in a hybrid quantum system. Phys. Rev. Lett. 125, 153602 (2020).

    ADS  Article  Google Scholar 

  25. 25.

    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).

    ADS  Article  Google Scholar 

  26. 26.

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

    ADS  Article  Google Scholar 

  27. 27.

    McKay, D. C., Sheldon, S., Smolin, J. A., Chow, J. M. & Gambetta, J. M. Three-qubit randomized benchmarking. Phys. Rev. Lett. 122, 200502 (2019).

    ADS  Article  Google Scholar 

  28. 28.

    Meyer, V. et al. Experimental demonstration of entanglement-enhanced rotation angle estimation using trapped ions. Phys. Rev. Lett. 86, 5870–5873 (2001).

    ADS  Article  Google Scholar 

  29. 29.

    Cox, K. C., Greve, G. P., Weiner, J. M. & Thompson, J. K. Deterministic squeezed states with collective measurements and feedback. Phys. Rev. Lett. 116, 093602 (2016).

    ADS  Article  Google Scholar 

  30. 30.

    Hosten, O., Engelsen, N. J., Krishnakumar, R. & Kasevich, M. A. Measurement noise 100 times lower than the quantum-projection limit using entangled atoms. Nature 529, 505–508 (2016).

    ADS  MATH  Article  Google Scholar 

  31. 31.

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

    ADS  MathSciNet  MATH  Article  Google Scholar 

  32. 32.

    Mottl, R. et al. Roton-type mode softening in a quantum gas with cavity-mediated long-range interactions. Science 336, 1570–1573 (2012).

    ADS  Article  Google Scholar 

  33. 33.

    Léonard, J., Morales, A., Zupancic, P., Esslinger, T. & Donner, T. Supersolid formation in a quantum gas breaking a continuous translational symmetry. Nature 543, 87–90 (2017).

    ADS  Article  Google Scholar 

  34. 34.

    Ozeri, R. et al. Errors in trapped-ion quantum gates due to spontaneous photon scattering. Phys. Rev. A 75, 042329 (2007).

    ADS  Article  Google Scholar 

  35. 35.

    Sackett, C. A. et al. Experimental entanglement of four particles. Nature 404, 256–259 (2000).

    ADS  Article  Google Scholar 

  36. 36.

    Leibfried, D. et al. Experimental demonstration of a robust, high-fidelity geometric two ion-qubit phase gate. Nature 422, 412–415 (2003).

    ADS  Article  Google Scholar 

  37. 37.

    Tavis, M. & Cummings, F. W. Exact solution for an n-molecule—radiation-field Hamiltonian. Phys. Rev. 170, 379–384 (1968).

    ADS  Article  Google Scholar 

  38. 38.

    Mølmer, K. & Sørensen, A. Multiparticle entanglement of hot trapped ions. Phys. Rev. Lett. 82, 1835–1838 (1999).

    ADS  Article  Google Scholar 

  39. 39.

    Bogoljubov, N. N. On a new method in the theory of superconductivity. Nuovo Cimento 7, 794–805 (1958).

    MathSciNet  Article  Google Scholar 

  40. 40.

    Burd, S. C. et al. Quantum amplification of mechanical oscillator motion. Science 364, 1163–1165 (2019).

    ADS  Article  Google Scholar 

  41. 41.

    Seidelin, S. et al. Microfabricated surface-electrode ion trap for scalable quantum information processing. Phys. Rev. Lett. 96, 253003 (2006).

    ADS  Article  Google Scholar 

  42. 42.

    Srinivas, R. et al. Trapped-ion spin-motion coupling with microwaves and a near-motional oscillating magnetic field gradient. Phys. Rev. Lett. 122, 163201 (2019).

    ADS  Article  Google Scholar 

  43. 43.

    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).

    ADS  Article  Google Scholar 

  44. 44.

    Ospelkaus, C. et al. Trapped-ion quantum logic gates based on oscillating magnetic fields. Phys. Rev. Lett. 101, 090502 (2008).

    ADS  Article  Google Scholar 

  45. 45.

    Ospelkaus, C. et al. Microwave quantum logic gates for trapped ions. Nature 476, 181–184 (2011).

    ADS  Article  Google Scholar 

  46. 46.

    Keith, A. C., Baldwin, C. H., Glancy, S. & Knill, E. Joint quantum-state and measurement tomography with incomplete measurements. Phys. Rev. A 98, 042318 (2018).

    ADS  Article  Google Scholar 

  47. 47.

    Uys, H. et al. Decoherence due to elastic Rayleigh scattering. Phys. Rev. Lett. 105, 200401 (2010).

    ADS  Article  Google Scholar 

  48. 48.

    Sørensen, A. & Mølmer, K. Entanglement and quantum computation with ions in thermal motion. Phys. Rev. A 62, 022311 (2000).

    ADS  Article  Google Scholar 

  49. 49.

    Haddadfarshi, F. & Mintert, F. High fidelity quantum gates of trapped ions in the presence of motional heating. New J. Phys. 18, 123007 (2016).

    ADS  Article  Google Scholar 

  50. 50.

    Webb, A. E. et al. Resilient entangling gates for trapped ions. Phys. Rev. Lett. 121, 180501 (2018).

    ADS  Article  Google Scholar 

  51. 51.

    Shapira, Y., Shaniv, R., Manovitz, T., Akerman, N. & Ozeri, R. Robust entanglement gates for trapped-ion qubits. Phys. Rev. Lett. 121, 180502 (2018).

    ADS  Article  Google Scholar 

  52. 52.

    Zarantonello, G. et al. Robust and resource-efficient microwave near-field entangling 9Be+ gate. Phys. Rev. Lett. 123, 260503 (2019).

    ADS  Article  Google Scholar 

  53. 53.

    Sutherland, R. T. et al. Laser-free trapped-ion entangling gates with simultaneous insensitivity to qubit and motional decoherence. Phys. Rev. A 101, 042334 (2020).

    ADS  Article  Google Scholar 

  54. 54.

    Wineland, D. J. et al. Experimental issues in coherent quantum-state manipulation of trapped atomic ions. J. Res. Natl Inst. Stand. Technol. 103, 259–328 (1998).

    Article  Google Scholar 

  55. 55.

    Heinzen, D. J. & Wineland, D. J. Quantum-limited cooling and detection of radio-frequency oscillations by laser-cooled ions. Phys. Rev. A 42, 2977–2994 (1990).

    ADS  Article  Google Scholar 

  56. 56.

    Walls, D. F. & Milburn, G. J. Quantum Optics (Springer-Verlag, 1994).

    MATH  Book  Google Scholar 

  57. 57.

    Tellinghuisen, J. Statistical error propagation. J. Phys. Chem. A 105, 3917–3921 (2001).

    Article  Google Scholar 

  58. 58.

    Turchette, Q. A. et al. Decoherence and decay of motional quantum states of a trapped atom coupled to engineered reservoirs. Phys. Rev. A 62, 053807 (2000).

    ADS  Article  Google Scholar 

  59. 59.

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

    ADS  Article  Google Scholar 

  60. 60.

    Gardiner, C. W., Parkins, A. S. & Zoller, P. Wave-function quantum stochastic differential equations and quantum-jump simulation methods. Phys. Rev. A 46, 4363–4381 (1992).

    ADS  MathSciNet  Article  Google Scholar 

  61. 61.

    Sepiol, M. A High-Fidelity Microwave Driven Two-Qubit Quantum Logic Gate in 43Ca+. PhD thesis, Univ. Oxford (2016).

Download references


We thank R. W. Simmonds, J. Schmidt and L. J. Stephenson for a careful reading of the manuscript. These experiments were performed using the ARTIQ control system. At the time the work was performed, S.C.B., R.S., H.M.K. and D.T.C.A. were Associates in the Professional Research Experience Program (PREP) operated jointly by NIST and the University of Colorado. This work was supported by the NIST Quantum Information Program.

Author information




S.C.B. carried out the experiments with assistance from D.H.S., R.S., H.M.K. and D.T.C.A., based on protocols developed by W.G. and J.J.B. D.T.C.A., D.H.S., R.S., S.C.B. and H.M.K. built and maintained the apparatus. S.C.B., H.M.K., W.G. and D.H.S. analysed the data and performed simulations. S.C.B. wrote the manuscript with input from all authors. D.H.S. supervised the work, with support from J.J.B., D.T.C.A., D.L., A.C.W. and D.J.W.

Corresponding authors

Correspondence to S. C. Burd or D. H. Slichter.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature Physics thanks the anonymous reviewers for their contribution to the peer review of this work.

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

Extended data

Extended Data Fig. 1 Data and fitting to determine \({\tilde{t}}_{I,est}\).

Fidelity as a function of \(\delta ^{\prime} /2\pi\) for various values of tI for a calibrated value of the parametric coupling strength of g/2π = 49.7(6) kHz. Data points are fidelities obtained using the method described in ref. 46. Vertical error bars indicate 68% confidence intervals for the fidelity. Horizontal error bars indicate 68% confidence intervals for \(\delta ^{\prime} /2\pi\) calculated from error propagation of the measured uncertainties in δ and g within a given experiment. Red curves are slices of the 2D quadratic fitting function at the corresponding interaction times.

Source data

Source data

Source Data Fig. 2

Data for all plots.

Source Data Fig. 3

Data for all plots.

Source Data Extended Data Fig. 1

Data for all plots.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Burd, S.C., Srinivas, R., Knaack, H.M. et al. Quantum amplification of boson-mediated interactions. Nat. Phys. 17, 898–902 (2021).

Download citation


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