Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content

Thank you for visiting nature.com. 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.

  • Article
  • Published:

Multi-qubit gates and Schrödinger cat states in an optical clock

Nature volume 634, pages 315–320 (2024)Cite this article

Subjects

Abstract

Many-particle entanglement is a key resource for achieving the fundamental precision limits of a quantum sensor1. Optical atomic clocks2, the current state of the art in frequency precision, are a rapidly emerging area of focus for entanglement-enhanced metrology3,4,5,6. Augmenting tweezer-based clocks featuring microscopic control and detection7,8,9,10 with the high-fidelity entangling gates developed for atom-array information processing11,12 offers a promising route towards making use of highly entangled quantum states for improved optical clocks. Here we develop and use a family of multi-qubit Rydberg gates to generate Schrödinger cat states of the Greenberger–Horne–Zeilinger (GHZ) type with up to nine optical clock qubits in a programmable atom array. In an atom-laser comparison at sufficiently short dark times, we demonstrate a fractional frequency instability below the standard quantum limit (SQL) using GHZ states of up to four qubits. However, because of their reduced dynamic range, GHZ states of a single size fail to improve the achievable clock precision at the optimal dark time compared with unentangled atoms13. Towards overcoming this hurdle, we simultaneously prepare a cascade of varying-size GHZ states to perform unambiguous phase estimation over an extended interval14,15,16,17. These results demonstrate key building blocks for approaching Heisenberg-limited scaling of optical atomic clock precision.

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

Access options

Buy this article

  • Purchase on SpringerLink
  • Instant access to full article PDF
Buy now

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

Fig. 1: Global multi-qubit gates in a 88Sr atom array.
Fig. 2: Preparing N-particle GHZ states with multi-qubit gates.
Fig. 3: Atom-laser-frequency comparisons with GHZ states.
Fig. 4: Preparing cascaded GHZ states for multi-ensemble metrology.

Similar content being viewed by others

Data availability

The data that support the findings of this study are available from the corresponding author on reasonable request. Source data for Figs. 1–4 are provided with this paper.

References

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

    Article  ADS  CAS  Google Scholar 

  3. Colombo, S., Pedrozo-Peñafiel, E. & Vuletić, V. Entanglement-enhanced optical atomic clocks. Appl. Phys. Lett. 121, 210502 (2022).

    Article  CAS  Google Scholar 

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

    Article  ADS  PubMed  Google Scholar 

  5. Robinson, J. M. et al. Direct comparison of two spin-squeezed optical clock ensembles at the 10−17 level. Nat. Phys. 20, 208–213 (2024).

    Article  CAS  Google Scholar 

  6. Eckner, W. J. et al. Realizing spin squeezing with Rydberg interactions in an optical clock. Nature 621, 734–739 (2023).

    Article  ADS  CAS  PubMed  Google Scholar 

  7. Norcia, M. A. et al. Seconds-scale coherence on an optical clock transition in a tweezer array. Science 366, 93–97 (2019).

    Article  ADS  CAS  PubMed  Google Scholar 

  8. Madjarov, I. S. et al. An atomic-array optical clock with single-atom readout. Phys. Rev. X 9, 041052 (2019).

    CAS  Google Scholar 

  9. Young, A. W. et al. Half-minute-scale atomic coherence and high relative stability in a tweezer clock. Nature 588, 408–413 (2020).

    Article  ADS  CAS  PubMed  Google Scholar 

  10. Shaw, A. L. et al. Multi-ensemble metrology by programming local rotations with atom movements. Nat. Phys. 20, 195–201 (2024).

    Article  CAS  Google Scholar 

  11. Evered, S. J. et al. High-fidelity parallel entangling gates on a neutral-atom quantum computer. Nature 622, 268–272 (2023).

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  12. Ma, S. et al. High-fidelity gates and mid-circuit erasure conversion in an atomic qubit. Nature 622, 279–284 (2023).

    Article  ADS  CAS  PubMed  Google Scholar 

  13. Huelga, S. F. et al. Improvement of frequency standards with quantum entanglement. Phys. Rev. Lett. 79, 3865 (1997).

    Article  ADS  CAS  Google Scholar 

  14. Higgins, B. et al. Demonstrating Heisenberg-limited unambiguous phase estimation without adaptive measurements. New J. Phys. 11, 073023 (2009).

    Article  ADS  Google Scholar 

  15. Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009).

    Article  ADS  Google Scholar 

  16. Kessler, E. M. et al. Heisenberg-limited atom clocks based on entangled qubits. Phys. Rev. Lett. 112, 190403 (2014).

    Article  ADS  CAS  PubMed  Google Scholar 

  17. Komar, P. et al. A quantum network of clocks. Nat. Phys. 10, 582–587 (2014).

    Article  CAS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  19. Schirhagl, R., Chang, K., Loretz, M. & Degen, C. L. Nitrogen-vacancy centers in diamond: nanoscale sensors for physics and biology. Annu. Rev. Phys. Chem. 65, 83–105 (2014).

    Article  ADS  CAS  PubMed  Google Scholar 

  20. Bongs, K. et al. Taking atom interferometric quantum sensors from the laboratory to real-world applications. Nat. Rev. Phys. 1, 731–739 (2019).

    Article  Google Scholar 

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

    Article  ADS  CAS  PubMed  Google Scholar 

  22. Backes, K. M. et al. A quantum enhanced search for dark matter axions. Nature 590, 238–242 (2021).

    Article  ADS  CAS  PubMed  Google Scholar 

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

    Article  ADS  CAS  PubMed  Google Scholar 

  24. Bluvstein, D. et al. A quantum processor based on coherent transport of entangled atom arrays. Nature 604, 451–456 (2022).

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  25. Graham, T. M. et al. Multi-qubit entanglement and algorithms on a neutral-atom quantum computer. Nature 604, 457–462 (2022).

    Article  ADS  CAS  PubMed  Google Scholar 

  26. Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024).

    Article  ADS  CAS  PubMed  Google Scholar 

  27. Jandura, S. & Pupillo, G. Time-optimal two- and three-qubit gates for Rydberg atoms. Quantum 6, 712 (2022).

    Article  Google Scholar 

  28. Levine, H. et al. Parallel implementation of high-fidelity multiqubit gates with neutral atoms. Phys. Rev. Lett. 123, 170503 (2019).

    Article  ADS  CAS  PubMed  Google Scholar 

  29. Bloom, B. et al. An optical lattice clock with accuracy and stability at the 10−18 level. Nature 506, 71–75 (2014).

    Article  ADS  CAS  PubMed  Google Scholar 

  30. Ushijima, I., Takamoto, M., Das, M., Ohkubo, T. & Katori, H. Cryogenic optical lattice clocks. Nat. Photon. 9, 185–189 (2015).

    Article  ADS  CAS  Google Scholar 

  31. McGrew, W. F. et al. Atomic clock performance enabling geodesy below the centimetre level. Nature 564, 87–90 (2018).

    Article  ADS  CAS  PubMed  Google Scholar 

  32. Brewer, S. M. et al. 27Al+ quantum-logic clock with a systematic uncertainty below 10−18. Phys. Rev. Lett. 123, 033201 (2019).

    Article  ADS  CAS  PubMed  Google Scholar 

  33. Oelker, E. et al. Demonstration of 4.8 × 10−17 stability at 1 s for two independent optical clocks. Nat. Photon. 13, 714–719 (2019).

    Article  ADS  CAS  Google Scholar 

  34. Bothwell, T. et al. Resolving the gravitational redshift across a millimetre-scale atomic sample. Nature 602, 420–424 (2022).

    Article  ADS  CAS  PubMed  Google Scholar 

  35. Zheng, X. et al. Differential clock comparisons with a multiplexed optical lattice clock. Nature 602, 425–430 (2022).

    Article  ADS  CAS  PubMed  Google Scholar 

  36. Schine, N., Young, A. W., Eckner, W. J., Martin, M. J. & Kaufman, A. M. Long-lived Bell states in an array of optical clock qubits. Nat. Phys. 18, 1067–1073 (2022).

    Article  CAS  Google Scholar 

  37. Scholl, P. et al. Erasure-cooling, control, and hyper-entanglement of motion in optical tweezers. Preprint at https://arxiv.org/abs/2311.15580 (2023).

  38. Fröwis, F. & Dür, W. Measures of macroscopicity for quantum spin systems. New J. Phys. 14, 093039 (2012).

    Article  ADS  Google Scholar 

  39. Tóth, G. & Apellaniz, I. Quantum metrology from a quantum information science perspective. J. Phys. A 47, 424006 (2014).

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

  41. Moses, S. A. et al. A race-track trapped-ion quantum processor. Phys. Rev. X 13, 041052 (2023).

    CAS  Google Scholar 

  42. Bao, Z. et al. Schrödinger cats growing up to 60 qubits and dancing in a cat scar enforced discrete time crystal. Preprint at https://arxiv.org/abs/2401.08284 (2024).

  43. Leibfried, D. et al. Toward Heisenberg-limited spectroscopy with multiparticle entangled states. Science 304, 1476–1478 (2004).

    Article  ADS  CAS  PubMed  Google Scholar 

  44. Nagata, T., Okamoto, R., O’Brien, J. L., Sasaki, K. & Takeuchi, S. Beating the standard quantum limit with four-entangled photons. Science 316, 726–729 (2007).

    Article  ADS  CAS  PubMed  Google Scholar 

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

    Article  ADS  CAS  PubMed  Google Scholar 

  46. Facon, A. et al. A sensitive electrometer based on a Rydberg atom in a Schrödinger-cat state. Nature 535, 262–265 (2016).

    Article  ADS  CAS  PubMed  Google Scholar 

  47. Lukin, M. D. et al. Dipole blockade and quantum information processing in mesoscopic atomic ensembles. Phys. Rev. Lett. 87, 037901 (2001).

    Article  ADS  CAS  PubMed  Google Scholar 

  48. Urban, E. et al. Observation of Rydberg blockade between two atoms. Nat. Phys. 5, 110–114 (2009).

    Article  CAS  Google Scholar 

  49. Dudin, Y., Li, L., Bariani, F. & Kuzmich, A. Observation of coherent many-body Rabi oscillations. Nat. Phys. 8, 790–794 (2012).

    Article  CAS  Google Scholar 

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

    Article  ADS  CAS  PubMed  Google Scholar 

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

    Article  ADS  MathSciNet  CAS  PubMed  Google Scholar 

  52. Monz, T. et al. 14-qubit entanglement: creation and coherence. Phys. Rev. Lett. 106, 130506 (2011).

    Article  ADS  PubMed  Google Scholar 

  53. 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  ADS  CAS  Google Scholar 

  54. Matei, D. G. et al. 1.5 μm lasers with sub-10 mHz linewidth. Phys. Rev. Lett. 118, 263202 (2017).

    Article  ADS  CAS  PubMed  Google Scholar 

  55. 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 

  56. Marciniak, C. D. et al. Optimal metrology with programmable quantum sensors. Nature 603, 604–609 (2022).

    Article  ADS  CAS  PubMed  Google Scholar 

  57. Nichol, B. et al. An elementary quantum network of entangled optical atomic clocks. Nature 609, 689–694 (2022).

    Article  ADS  CAS  PubMed  Google Scholar 

  58. Norcia, M. A. et al. Iterative assembly of 171Yb atom arrays in cavity-enhanced optical lattices. PRX Quantum 5, 030316 (2024).

    Article  Google Scholar 

  59. Gyger, F. et al. Continuous operation of large-scale atom arrays in optical lattices. Phys. Rev. Res. 6, 033104 (2024).

    Article  Google Scholar 

  60. Lis, J. W. et al. Midcircuit operations using the omg architecture in neutral atom arrays. Phys. Rev. X 13, 041035 (2023).

    CAS  Google Scholar 

  61. Finkelstein, R. et al. Universal quantum operations and ancilla-based readout for tweezer clocks. Nature https://doi.org/10.1038/s41586-024-08005-8 (2024).

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

    Article  ADS  CAS  PubMed  Google Scholar 

  63. 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  PubMed  Google Scholar 

  64. Young, A. W., Eckner, W. J., Schine, N., Childs, A. M. & Kaufman, A. M. Tweezer-programmable 2D quantum walks in a Hubbard-regime lattice. Science 377, 885–889 (2022).

    Article  ADS  MathSciNet  CAS  PubMed  Google Scholar 

  65. Dörscher, S. et al. Lattice-induced photon scattering in an optical lattice clock. Phys. Rev. A 97, 063419 (2018).

    Article  ADS  Google Scholar 

  66. Scholl, P. et al. Erasure conversion in a high-fidelity Rydberg quantum simulator. Nature 622, 273–278 (2023).

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  67. Madjarov, I. S. et al. High-fidelity entanglement and detection of alkaline-earth Rydberg atoms. Nat. Phys. 16, 857–861 (2020).

    Article  CAS  Google Scholar 

  68. Taichenachev, A. V. et al. Magnetic field-induced spectroscopy of forbidden optical transitions with application to lattice-based optical atomic clocks. Phys. Rev. Lett. 96, 083001 (2006).

    Article  ADS  CAS  PubMed  Google Scholar 

  69. Hein, M., Eisert, J. & Briegel, H. J. Multiparty entanglement in graph states. Phys. Rev. A 69, 062311 (2004).

    Article  ADS  MathSciNet  Google Scholar 

  70. Zeiher, J. et al. Microscopic characterization of scalable coherent Rydberg superatoms. Phys. Rev. X 5, 031015 (2015).

    Google Scholar 

  71. Bernien, H. et al. Probing many-body dynamics on a 51-atom quantum simulator. Nature 551, 579–584 (2017).

    Article  ADS  CAS  PubMed  Google Scholar 

  72. Khaneja, N., Reiss, T., Kehlet, C., Schulte-Herbrüggen, T. & Glaser, S. J. Optimal control of coupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithms. J. Magn. Reson. 172, 296–305 (2005).

    Article  ADS  CAS  PubMed  Google Scholar 

  73. Löw, R. et al. An experimental and theoretical guide to strongly interacting Rydberg gases. J. Phys. B. 45, 113001 (2012).

    Article  ADS  Google Scholar 

  74. Derevianko, A., Kómár, P., Topcu, T., Kroeze, R. M. & Lukin, M. D. Effects of molecular resonances on Rydberg blockade. Phys. Rev. A 92, 063419 (2015).

    Article  ADS  Google Scholar 

  75. Young, A. W. et al. An atomic boson sampler. Nature 629, 311–316 (2024).

    Article  ADS  CAS  PubMed  Google Scholar 

  76. Jandura, S., Thompson, J. D. & Pupillo, G. Optimizing Rydberg gates for logical-qubit performance. PRX Quantum 4, 020336 (2023).

    Article  ADS  Google Scholar 

  77. Demkowicz-Dobrzański, R., Jarzyna, M. & Kołodyński, J. in Progress in Optics (ed. Wolf, E.) 345–435 (Elsevier, 2015).

  78. Rosenband, T. & Leibrandt, D. R. Exponential scaling of clock stability with atom number. Preprint at https://arxiv.org/abs/1303.6357 (2013).

  79. Borregaard, J. & Sørensen, A. S. Efficient atomic clocks operated with several atomic ensembles. Phys. Rev. Lett. 111, 090802 (2013).

    Article  ADS  CAS  PubMed  Google Scholar 

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

    Article  ADS  Google Scholar 

  81. Jarzyna, M. & Demkowicz-Dobrzański, R. True precision limits in quantum metrology. New J. Phys. 17, 013010 (2015).

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  PubMed  Google Scholar 

  83. Zheng, X., Dolde, J. & Kolkowitz, S. Reducing the instability of an optical lattice clock using multiple atomic ensembles. Phys. Rev. X 14, 011006 (2024).

    CAS  Google Scholar 

Download references

Acknowledgements

We acknowledge earlier contributions to the experiment from M. A. Norcia and N. Schine, as well as fruitful discussions with R. Kaubruegger and P. Zoller. We also wish to thank S. Lannig and A. M. Rey for careful readings of the manuscript and helpful comments. Also, we thankfully acknowledge helpful technical discussions and contributions to the clock-laser system from A. Aeppli, M. N. Frankel, J. Hur, D. Kedar, S. Lannig, B. Lewis, M. Miklos, W. R. Milner, Y. M. Tso, W. Warfield, Z. Hu and Z. Yao. This material is based on work supported by the Army Research Office (W911NF-22-1-0104), the Air Force Office of Scientific Research (FA9550-19-1-0275), the National Science Foundation Quantum Leap Challenge Institutes (QLCI; OMA-2016244), the U.S. Department of Energy, Office of Science, the National Quantum Information Science Research Centers, Quantum Systems Accelerator and the National Institute of Standards and Technology. This research also received funding from the European Union’s Horizon 2020 programme under the Marie Skłodowska-Curie project 955479 (MoQS), the Horizon Europe programme HORIZON-CL4-2021-DIGITAL-EMERGING-01-30 through the project 101070144 (EuRyQa) and from the French National Research Agency under the Investments for the Future Program project ANR-21-ESRE-0032 (aQCess). We also acknowledge funding from Lockheed Martin. A.C. acknowledges support from the NSF Graduate Research Fellowship Program (grant no. DGE2040434). W.J.E. acknowledges support from the NDSEG Fellowship. N.D.O. acknowledges support from the Alexander von Humboldt Foundation.

Author information

Authors and Affiliations

  1. JILA, University of Colorado Boulder and National Institute of Standards and Technology, Boulder, CO, USA

    Alec Cao, William J. Eckner, Theodor Lukin Yelin, Aaron W. Young, Lingfeng Yan, Kyungtae Kim, Jun Ye, Nelson Darkwah Oppong & Adam M. Kaufman

  2. Department of Physics, University of Colorado Boulder, Boulder, CO, USA

    Alec Cao, William J. Eckner, Theodor Lukin Yelin, Aaron W. Young, Lingfeng Yan, Kyungtae Kim, Jun Ye, Nelson Darkwah Oppong & Adam M. Kaufman

  3. aQCess, University of Strasbourg and CNRS, CESQ and ISIS (UMR 7006), Strasbourg, France

    Sven Jandura & Guido Pupillo

Authors
  1. Alec Cao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. William J. Eckner
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Theodor Lukin Yelin
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Aaron W. Young
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Sven Jandura
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Lingfeng Yan
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Kyungtae Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

  8. Guido Pupillo
    View author publications

    You can also search for this author in PubMed Google Scholar

  9. Jun Ye
    View author publications

    You can also search for this author in PubMed Google Scholar

  10. Nelson Darkwah Oppong
    View author publications

    You can also search for this author in PubMed Google Scholar

  11. Adam M. Kaufman
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

A.C., W.J.E., T.L.Y., A.W.Y., N.D.O. and A.M.K. contributed to the experimental setup, performed the measurements and analysed the data. S.J. and G.P. conceptualized the multi-qubit gate design. L.Y. and K.K. contributed to the clock-laser system, under supervision from J.Y. A.M.K. supervised the work. All authors contributed to the manuscript.

Corresponding author

Correspondence to Adam M. Kaufman.

Ethics declarations

Competing interests

G.P. is co-founder and shareholder of QPerfect. The other authors declare no competing interests.

Peer review

Peer review information

Nature thanks Christian Marciniak and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

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 Characterizing clock and Rydberg operations.

a, Effective level diagram for clock qubits with Rydberg coupling. Wavy lines indicate Rydberg decay. We categorize the many possible Rydberg decay paths by whether the final state is dark (dark pink) or bright (orange) to our standard state-detection scheme; note that these final states include the ones that are explicitly shown, with the background colour indicating dark or bright. Jagged lines indicate Raman scattering paths (intermediate state not shown). Straight lines indicate coherent drives. b, Left, decay of Rydberg state over time to states dark (dark pink circles) and bright (orange squares) to the detection protocol. We fit an exponential 1/e decay time to dark (bright) states of \({\tau }_{{\rm{r}}}^{{\rm{d}}}=1/{\gamma }_{{\rm{r}}}^{{\rm{d}}}=51(3)\,{\rm{\mu }}s\) (\({\tau }_{{\rm{r}}}^{{\rm{b}}}=1/{\gamma }_{{\rm{r}}}^{{\rm{b}}}=86(3)\,{\rm{\mu }}s\)). Middle, single-atom Rydberg Rabi oscillations at Ωr = 2π × 3.7 MHz with a fitted 11(1)-μs Gaussian 1/e decay time. Right, single-atom Rydberg Ramsey oscillations at a 1-MHz detuning with a fitted 4.5(2)-μs Gaussian 1/e decay time. c, Left, population of |1⟩ (turquoise circles) and |0⟩ (green squares) over time owing to Raman scattering in the lattice. Fitting to a rate model (see Methods) yields scattering rates of Γ1→0 = 0.48(1) Hz, Γ1→2 = 0.26(2) Hz and Γ2→0 = 0.47(3) Hz in an approximately 920Er deep 2D lattice. Middle, clock Rabi oscillations at Ωc = 2π × 0.31 kHz yielding a fitted ground-state fraction of 0.96(1). Right, clock Ramsey oscillations at an 84-Hz detuning with a fitted 217(17)-ms Gaussian 1/e decay time. We note that the longer coherence time reported in Fig. 2d is obtained by a different method, in which the Ramsey fringe contrast is carefully measured at each dark time and out to substantially longer times. d, Rydberg π-pulse fidelity for single atoms (purple) and two-atom blockade (red). These data are SPAM-corrected (see Methods and ref. 67). Parabolic fits yield SPAM-corrected Rydberg π-pulse fidelities of 0.995(2) for single atoms and 0.986(3) for two-atom blockade. e, Clock π-pulse fidelity. A parabolic fit yields a raw clock π-pulse fidelity of 0.9962(7).

Extended Data Fig. 2 Release and recapture in optical lattices.

a, Schematic of the release and recapture process. The atoms expand from an initially well-localized state while the lattice is off and, when excited to the Rydberg state, the atoms will also undergo a centre-of-mass displacement owing to the momentum recoil of the UV photon. When the lattice is turned back on, the wavefunction will be projected both into higher-band Wannier orbitals as well as nearby sites, causing both loss and heating. b, Top, measured survival as a function of time that the lattice is turned off for the ground state (blue circles) and Rydberg state (purple). The solid lines are theoretical predictions for the recapture probability from an approximately 50Er lattice (see Methods). At short times, the Rydberg-state survival decreases quadratically and we fit a Gaussian 1/e decay time of 8.7(1) μs. Bottom, theoretically predicted increase in mean phonon number (see Methods) for recaptured atoms over the same duration. The heating is quadratic at short times but begins to taper off as the highest-energy atoms are lost. The lattice turn-off duration is <2 μs for all data shown in the main text.

Extended Data Fig. 3 Error modelling for GHZ-state fidelities.

a, Sensitivity of multi-qubit gate \(\widehat{{\mathcal{U}}}\) to Rydberg Rabi frequency and detuning deviations for various Nmax. Solid lines indicate infidelity for N = Nmax GHZ state; dashed lines indicate infidelity for N = 2 Bell state. b, Modelling of various error sources for N = 2 Bell state (blue) and N = 4 GHZ state (red, hatched). For the Bell state, we consider the CZ gate protocol shown in Fig. 1c; for the four-atom GHZ state, we consider the general Nmax = N scheme shown in Fig. 2a. The measurement-corrected Bell state (four-atom GHZ state) infidelity (see Extended Data Table 2) is shown as the blue dashed (red dotted) line. In both cases, our error model accounts for roughly a third of the observed infidelity. The presence of pulse discretization/rise time error for only the Bell state is because we use the exact time-optimal CZ gate implementation described in ref. 11 as opposed to a modulation optimized for our pulse model.

Extended Data Table 1 Atomic arrangement parameters for different GHZ ensemble sizes
Full size table
Extended Data Table 2 Summary of raw and measurement-corrected GHZ-state fidelities
Full size table

Supplementary information

Source data

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cao, A., Eckner, W.J., Lukin Yelin, T. et al. Multi-qubit gates and Schrödinger cat states in an optical clock. Nature 634, 315–320 (2024). https://doi.org/10.1038/s41586-024-07913-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41586-024-07913-z

Search

Advanced search

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