- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Beyond average Hamiltonian theory for quantum sensing

Jner Tzern Oon1,2,*,†, Sebastian C. Carrasco3,*, Connor A. Hart2, George A. Witt1, Vladimir S. Malinovsky3, and Ronald Walsworth1,2,4

  • *These authors contributed equally to this work.
  • †Contact author: jjoon@umd.edu
PDF

Phys. Rev. Research 8, 013222 – Published 26 February, 2026

DOI: https://doi.org/10.1103/w2sf-zxdk

Abstract

The Magnus expansion underpins average Hamiltonian theory (AHT), a central tool in quantum control and sensing that is used to generate efficient approximations of system dynamics. Here, we demonstrate that the conditions for Magnus expansion convergence are violated under typical operating regimes for sensing using solid-state spin systems. We also establish that certain symmetries, such as those found in rapid echo pulse sequences, cancel all higher-order Magnus terms outside standard convergence frameworks. We introduce an exact method to compute the sensor response beyond the AHT regime, and apply it to both broadband and narrow-band magnetometry protocols. Our results clarify AHT limits and enable opportunities in pulse sequence design through symmetry-driven convergence, with implications for entanglement-enhanced sensing and quantum Fisher information estimation in control protocols.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (60)

  1. W. Magnus, On the exponential solution of differential equations for a linear operator, Commun. Pure Appl. Math. 7, 649 (1954).
  2. J. A. Oteo and J. Ros, The Magnus expansion for classical Hamiltonian systems, J. Phys. A: Math. Gen. 24, 5751 (1991).
  3. A. Connes and D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem I: The Hopf algebra structure of graphs and the main theorem, Commun. Math. Phys. 210, 249 (2000).
  4. D. W. Robinson, Multiple Coulomb excitations of deformed nuclei, Helv. Phys. Acta (Switzerland) 36, 140 (1963).
  5. J. Eichler, Magnus approximation for K-shell ionization by heavy-ion impact, Phys. Rev. A 15, 1856 (1977).
  6. S. Blanes, F. Casas, J. Oteo, and J. Ros, The Magnus expansion and some of its applications, Phys. Rep. 470, 151 (2009).
  7. F. Casas, Sufficient conditions for the convergence of the Magnus expansion, J. Phys. A: Math. Theor. 40, 15001 (2007).
  8. A. Brinkmann, Introduction to average Hamiltonian theory. I. Basics, Concepts Magn. Reson. 45A, e21414 (2016).
  9. W. Evans, On some applications of the Magnus expansion in nuclear magnetic resonance, Ann. Phys. (NY) 48, 72 (1968).
  10. U. Haeberlen and J. S. Waugh, Coherent averaging effects in magnetic resonance, Phys. Rev. 175, 453 (1968).
  11. M. Edén and M. H. Levitt, Pulse sequence symmetries in the nuclear magnetic resonance of spinning solids: Application to heteronuclear decoupling, J. Chem. Phys. 111, 1511 (1999).
  12. M. Goldman, P. J. Grandinetti, A. Llor, Z. Olejniczak, J. R. Sachleben, and J. W. Zwanziger, Theoretical aspects of higher-order truncations in solid-state nuclear magnetic resonance, J. Chem. Phys. 97, 8947 (1992).
  13. E. S. Mananga and T. Charpentier, Introduction of the Floquet-Magnus expansion in solid-state nuclear magnetic resonance spectroscopy, J. Chem. Phys. 135, 044109 (2011).
  14. D. Candoli, I. K. Nikolov, L. Z. Brito, S. Carr, S. Sanna, and V. F. Mitrović, Pulsee: A software for the quantum simulation of an extensive set of magnetic resonance observables, Comput. Phys. Commun. 284, 108598 (2023).
  15. S. C. Carrasco, S. Lourette, I. Sola, and V. S. Malinovsky, Dynamically enhanced two-photon spectroscopy, Phys. Rev. Lett. 134, 163601 (2025).
  16. O. Sahin, E. de Leon Sanchez, S. Conti, A. Akkiraju, P. Reshetikhin, E. Druga, A. Aggarwal, B. Gilbert, S. Bhave, and A. Ajoy, High field magnetometry with hyperpolarized nuclear spins, Nat. Commun. 13, 5486 (2022).
  17. P. Balasubramanian, C. Osterkamp, Y. Chen, X. Chen, T. Teraji, E. Wu, B. Naydenov, and F. Jelezko, DC magnetometry with engineered nitrogen-vacancy spin ensembles in diamond, Nano Lett. 19, 6681 (2019).
  18. S. Choi, N. Y. Yao, and M. D. Lukin, Dynamical engineering of interactions in qudit ensembles, Phys. Rev. Lett. 119, 183603 (2017).
  19. J. Choi, H. Zhou, H. S. Knowles, R. Landig, S. Choi, and M. D. Lukin, Robust dynamic Hamiltonian engineering of many-body spin systems, Phys. Rev. X 10, 031002 (2020).
  20. M. F. O'Keeffe, L. Horesh, J. F. Barry, D. A. Braje, and I. L. Chuang, Hamiltonian engineering with constrained optimization for quantum sensing and control, New J. Phys. 21, 023015 (2019).
  21. H. Zhou, J. Choi, S. Choi, R. Landig, A. M. Douglas, J. Isoya, F. Jelezko, S. Onoda, H. Sumiya, P. Cappellaro, H. S. Knowles, H. Park, and M. D. Lukin, Quantum metrology with strongly interacting spin systems, Phys. Rev. X 10, 031003 (2020).
  22. N. Arunkumar, K. S. Olsson, J. T. Oon, C. A. Hart, D. B. Bucher, D. R. Glenn, M. D. Lukin, H. Park, D. Ham, and R. L. Walsworth, Quantum logic enhanced sensing in solid-state spin ensembles, Phys. Rev. Lett. 131, 100801 (2023).
  23. K. Khodjasteh and D. A. Lidar, Performance of deterministic dynamical decoupling schemes: Concatenated and periodic pulse sequences, Phys. Rev. A 75, 062310 (2007).
  24. G. S. Uhrig, Keeping a quantum bit alive by optimized π-pulse sequences, Phys. Rev. Lett. 98, 100504 (2007).
  25. H. Zhou, L. S. Martin, M. Tyler, O. Makarova, N. Leitao, H. Park, and M. D. Lukin, Robust higher-order Hamiltonian engineering for quantum sensing with strongly interacting systems, Phys. Rev. Lett. 131, 220803 (2023).
  26. O. Hosten, R. Krishnakumar, N. J. Engelsen, and M. A. Kasevich, Quantum phase magnification, Science 352, 1552 (2016).
  27. S. Colombo, E. Pedrozo-Peñafiel, A. F. Adiyatullin, Z. Li, E. Mendez, C. Shu, and V. Vuletić, Time-reversal-based quantum metrology with many-body entangled states, Nat. Phys. 18, 925 (2022).
  28. M. Kitagawa and M. Ueda, Squeezed spin states, Phys. Rev. A 47, 5138 (1993).
  29. S. L. Braunstein and C. M. Caves, Statistical distance and the geometry of quantum states, Phys. Rev. Lett. 72, 3439 (1994).
  30. Multiplicative factors of the reduced Planck constant ℏ are dropped in the exponent for simplicity. All Hamiltonian operators have nonangular frequency units in this work.
  31. C. L. Degen, F. Reinhard, and P. Cappellaro, Quantum sensing, Rev. Mod. Phys. 89, 035002 (2017).
  32. J. S. Waugh, L. M. Huber, and U. Haeberlen, Approach to high-resolution NMR in solids, Phys. Rev. Lett. 20, 180 (1968).
  33. Alternatively, Ref.  [34] calculates α by considering the largest difference between the eigenvalues of H¯(1), giving the same result for a two-level system.
  34. H. Zhou, H. Gao, N. T. Leitao, O. Makarova, I. Cong, A. M. Douglas, L. S. Martin, and M. D. Lukin, Robust Hamiltonian engineering for interacting qudit systems, Phys. Rev. X 14, 031017 (2024).
  35. S. Blanes, F. Casas, J. A. Oteo, and J. Ros, Magnus and Fer expansions for matrix differential equations: The convergence problem, J. Phys. A: Math. Gen. 31, 259 (1998).
  36. P. C. Moan and E. Ew, Efficient approximation of Sturm-Liouville problems using Lie-group methods, Technical Report No. 1998/NA11, University of Cambridge, Cambridge, United Kingdom, 1999.
  37. Alternately, one could prevent crossing this threshold by increasing the number of Floquet cycles nc to minimize t, while maintaining the total interferometric time, nct.
  38. G. Kucsko, S. Choi, J. Choi, P. C. Maurer, H. Zhou, R. Landig, H. Sumiya, S. Onoda, J. Isoya, F. Jelezko, E. Demler, N. Y. Yao, and M. D. Lukin, Critical thermalization of a disordered dipolar spin system in diamond, Phys. Rev. Lett. 121, 023601 (2018).
  39. N. J. Higham, Functions of Matrices: Theory and Computation (Society for Industrial and Applied Mathematics, Philadelphia, 2008).
  40. See Supplemental Material at http://link.aps.org/supplemental/10.1103/w2sf-zxdk for additional details.
  41. L. S. Martin, H. Zhou, N. T. Leitao, N. Maskara, O. Makarova, H. Gao, Q.-Z. Zhu, M. Park, M. Tyler, H. Park, S. Choi, and M. D. Lukin, Controlling local thermalization dynamics in a Floquet-engineered dipolar ensemble, Phys. Rev. Lett. 130, 210403 (2023).
  42. M. Lei, R. Fukumori, C.-J. Wu, E. Barnes, S. E. Economou, J. Choi, and A. Faraon, Quantum thermalization and Floquet engineering in a spin ensemble with a clock transition, Nat. Phys. 21, 1196 (2025).
  43. S. C. Carrasco, M. H. Goerz, Z. Li, S. Colombo, V. Vuletić, and V. S. Malinovsky, Extreme spin squeezing via optimized one-axis twisting and rotations, Phys. Rev. Appl. 17, 064050 (2022).
  44. S. C. Carrasco, M. H. Goerz, S. A. Malinovskaya, V. Vuletić, W. P. Schleich, and V. S. Malinovsky, Dicke state generation and extreme spin squeezing via rapid adiabatic passage, Phys. Rev. Lett. 132, 153603 (2024).
  45. S. Boixo, S. T. Flammia, C. M. Caves, and J. M. Geremia, Generalized limits for single-parameter quantum estimation, Phys. Rev. Lett. 98, 090401 (2007).
  46. M. Tyler, H. Zhou, L. S. Martin, N. Leitao, and M. D. Lukin, Higher-order methods for Hamiltonian engineering pulse sequence design, Phys. Rev. A 108, 062602 (2023).
  47. X.-D. Chen, C.-L. Zou, F.-W. Sun, and G.-C. Guo, Optical manipulation of the charge state of nitrogen-vacancy center in diamond, Appl. Phys. Lett. 103, 013112 (2013).
  48. P. Ji and M. V. G. Dutt, Charge state dynamics of the nitrogen vacancy center in diamond under 1064-nm laser excitation, Phys. Rev. B 94, 024101 (2016).
  49. S. Dhomkar, H. Jayakumar, P. R. Zangara, and C. A. Meriles, Charge dynamics in near-surface, variable-density ensembles of nitrogen-vacancy centers in diamond, Nano Lett. 18, 4046 (2018).
  50. I. C. Barbosa, J. Gutsche, and A. Widera, Impact of charge conversion on NV-center relaxometry, Phys. Rev. B 108, 075411 (2023).
  51. Ł. Cywiński, R. M. Lutchyn, C. P. Nave, and S. Das Sarma, How to enhance dephasing time in superconducting qubits, Phys. Rev. B 77, 174509 (2008).
  52. G. S. Uhrig, Exact results on dynamical decoupling by π pulses in quantum information processes, New J. Phys. 10, 083024 (2008).
  53. M. J. Biercuk, H. Uys, A. P. VanDevender, N. Shiga, W. M. Itano, and J. J. Bollinger, Optimized dynamical decoupling in a model quantum memory, Nature (London) 458, 996 (2009).
  54. F. Poggiali, P. Cappellaro, and N. Fabbri, Optimal control for one-qubit quantum sensing, Phys. Rev. X 8, 021059 (2018).
  55. P. Peng, X. Huang, C. Yin, L. Joseph, C. Ramanathan, and P. Cappellaro, Deep reinforcement learning for quantum Hamiltonian engineering, Phys. Rev. Appl. 18, 024033 (2022).
  56. H. J. Mamin, M. H. Sherwood, M. Kim, C. T. Rettner, K. Ohno, D. D. Awschalom, and D. Rugar, Multipulse double-quantum magnetometry with near-surface nitrogen-vacancy centers, Phys. Rev. Lett. 113, 030803 (2014).
  57. C. A. Hart, J. M. Schloss, M. J. Turner, P. J. Scheidegger, E. Bauch, and R. L. Walsworth, N−V-diamond magnetic microscopy using a double quantum 4-Ramsey protocol, Phys. Rev. Appl. 15, 044020 (2021).
  58. R. Jozsa, Fidelity for mixed quantum states, J. Mod. Opt. 41, 2315 (1994).
  59. A. Arnal, F. Casas, and C. Chiralt, A general formula for the Magnus expansion in terms of iterated integrals of right-nested commutators, J. Phys. Commun. 2, 035024 (2018).
  60. T. Macrì, A. Smerzi, and L. Pezzè, Loschmidt echo for quantum metrology, Phys. Rev. A 94, 010102(R) (2016).

Outline

Information

Phys. Rev. Research 8, 013222– Published 26 February, 2026
  • Received 10 October 2024
  • Revised 12 September 2025
  • Accepted 15 January 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation