- Physical Review Journals
All JournalsPhysics Magazine

Physical Review A

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

OH molecule as a quantum probe to jointly estimate electric and magnetic fields

Luca Previdi1,2, Francesco Albarelli3,4,5, and Matteo G. A. Paris2

PDF

Phys. Rev. A 113, 062612 – Published 11 June, 2026

DOI: https://doi.org/10.1103/gpmd-ck38

Abstract

The hydroxyl radical (the OH molecule) carries both electric and magnetic dipole moments and, as a diatomic molecule, admits a comparatively simple and accurate model. This makes it a natural quantum probe for the joint estimation of electric and magnetic fields. Here we study simultaneous estimation of both fields using the tools of multiparameter quantum estimation theory, explicitly accounting for the performance loss caused by measurement incompatibility. We analyze and optimize both stationary and dynamical estimation strategies. In the stationary regime we consider ground and thermal states of the Stark-Zeeman Hamiltonian and identify optimal operating points. For thermal probes we find a nontrivial multiparameter effect: Increasing the temperature can reduce the overall estimation error by weakening parameter correlations. In the dynamical regime we study both pure and thermal initial states, illustrating nontrivial manifestations of incompatibility for mixed probes. Finally, we show that an optimal sequential control protocol can overcome limitations due to noncommutativity, and we assess its robustness in the multiparameter setting.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (90)

  1. C. Ticknor and J. L. Bohn, Influence of magnetic fields on cold collisions of polar molecules, Phys. Rev. A 71, 022709 (2005).
  2. A. V. Avdeenkov and J. L. Bohn, Linking ultracold polar molecules, Phys. Rev. Lett. 90, 043006 (2003).
  3. B. C. Sawyer, B. K. Stuhl, D. Wang, M. Yeo, and J. Ye, Molecular beam collisions with a magnetically trapped target, Phys. Rev. Lett. 101, 203203 (2008).
  4. T. V. Tscherbul, Z. Pavlovic, H. R. Sadeghpour, R. Côté, and A. Dalgarno, Collisions of trapped molecules with slow beams, Phys. Rev. A 82, 022704 (2010).
  5. E. R. Hudson, H. J. Lewandowski, B. C. Sawyer, and J. Ye, Cold molecule spectroscopy for constraining the evolution of the fine structure constant, Phys. Rev. Lett. 96, 143004 (2006).
  6. M. G. Kozlov, Λ-doublet spectra of diatomic radicals and their dependence on fundamental constants, Phys. Rev. A 80, 022118 (2009).
  7. B. L. Lev, E. R. Meyer, E. R. Hudson, B. C. Sawyer, J. L. Bohn, and J. Ye, OH hyperfine ground state: From precision measurement to molecular qubits, Phys. Rev. A 74, 061402(R) (2006).
  8. M. Lara, B. L. Lev, and J. L. Bohn, Loss of molecules in magneto-electrostatic traps due to nonadiabatic transitions, Phys. Rev. A 78, 033433 (2008).
  9. M. Bhattacharya, Z. Howard, and M. Kleinert, Ground-state OH molecule in combined electric and magnetic fields: Analytic solution of the effective Hamiltonian, Phys. Rev. A 88, 012503 (2013).
  10. C. W. Helstrom, Quantum detection and estimation theory, J. Stat. Phys. 1, 231 (1969).
  11. A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory, 2nd ed. (Edizioni della Normale, Pisa, 2011).
  12. M. G. A. Paris, Quantum estimation for quantum technology, Int. J. Quantum Inf. 07, 125 (2009).
  13. V. Giovannetti, S. Lloyd, and L. Maccone, Advances in quantum metrology, Nat. Photonics 5, 222 (2011).
  14. C. L. Degen, F. Reinhard, and P. Cappellaro, Quantum sensing, Rev. Mod. Phys. 89, 035002 (2017).
  15. C. Vaneph, T. Tufarelli, and M. G. Genoni, Quantum estimation of a two-phase spin rotation, Quantum Meas. Quantum Metrol. 1, 12 (2013).
  16. T. Baumgratz and A. Datta, Quantum enhanced estimation of a multidimensional field, Phys. Rev. Lett. 116, 030801 (2016).
  17. H. Yuan, Sequential feedback scheme outperforms the parallel scheme for Hamiltonian parameter estimation, Phys. Rev. Lett. 117, 160801 (2016).
  18. Z. Hou, Z. Zhang, G.-Y. Xiang, C.-F. Li, G.-C. Guo, H. Chen, L. Liu, and H. Yuan, Minimal tradeoff and ultimate precision limit of multiparameter quantum magnetometry under the parallel scheme, Phys. Rev. Lett. 125, 020501 (2020).
  19. W. Górecki and R. Demkowicz-Dobrzański, Multiparameter quantum metrology in the Heisenberg limit regime: Many-repetition scenario versus full optimization, Phys. Rev. A 106, 012424 (2022).
  20. R. Kaubruegger, A. Shankar, D. V. Vasilyev, and P. Zoller, Optimal and variational multiparameter quantum metrology and vector-field sensing, PRX Quantum 4, 020333 (2023).
  21. Y. Baamara, M. Gessner, and A. Sinatra, Quantum-enhanced multiparameter estimation and compressed sensing of a field, SciPost Phys. 14, 050 (2023).
  22. M. Hayashi and Y. Ouyang, Finding the optimal probe state for multiparameter quantum metrology using conic programming, npj Quantum Inf. 10, 111 (2024).
  23. J. Guo, F. Sun, Q. He, and M. Fadel, Assisted metrology and preparation of macroscopic superpositions with split spin-squeezed states, Phys. Rev. A 108, 053327 (2023).
  24. J. Kahn, Fast rate estimation of a unitary operation in SU(d), Phys. Rev. A 75, 022326 (2007).
  25. H. Imai and A. Fujiwara, Geometry of optimal estimation scheme for SU(D) channels, J. Phys. A 40, 4391 (2007).
  26. N. Kura and M. Ueda, Finite-error metrological bounds on multiparameter Hamiltonian estimation, Phys. Rev. A 97, 012101 (2018).
  27. T. Esat, D. Borodin, J. Oh, A. J. Heinrich, F. S. Tautz, Y. Bae, and R. Temirov, A quantum sensor for atomic-scale electric and magnetic fields, Nat. Nanotechnol. 19, 1466 (2024).
  28. For clarity, we briefly recall the spectroscopic term-symbol notation for diatomic molecules. The symbol Π denotes Λ=1, the projection of the electronic orbital angular momentum on the internuclear axis. The superscript 2 gives the spin multiplicity 2S+1 (here a doublet with S=1/2 for an open-shell radical with one unpaired electron). The subscript (1/2 or 3/2) denotes Ω, the projection of the total electronic angular momentum on the internuclear axis [60].
  29. F. Albarelli, M. Barbieri, M. Genoni, and I. Gianani, A perspective on multiparameter quantum metrology: From theoretical tools to applications in quantum imaging, Phys. Lett. A 384, 126311 (2020).
  30. J. Liu, H. Yuan, X.-M. Lu, and X. Wang, Quantum Fisher information matrix and multiparameter estimation, J. Phys. A 53, 023001 (2020).
  31. R. Demkowicz-Dobrzański, W. Górecki, and M. Guţă, Multi-parameter estimation beyond quantum Fisher information, J. Phys. A 53, 363001 (2020).
  32. S. Razavian, M. G. A. Paris, and M. G. Genoni, On the quantumness of multiparameter estimation problems for qubit systems, Entropy 22, 1197 (2020).
  33. J. Yang, S. Pang, Y. Zhou, and A. N. Jordan, Optimal measurements for quantum multiparameter estimation with general states, Phys. Rev. A 100, 032104 (2019).
  34. K. Matsumoto, A new approach to the Cramér-Rao-type bound of the pure-state model, J. Phys. A 35, 3111 (2002).
  35. L. Pezzè, M. A. Ciampini, N. Spagnolo, P. C. Humphreys, A. Datta, I. A. Walmsley, M. Barbieri, F. Sciarrino, and A. Smerzi, Optimal measurements for simultaneous quantum estimation of multiple phases, Phys. Rev. Lett. 119, 130504 (2017).
  36. L. Pezzè and A. Smerzi, Advances in multiparameter quantum sensing and metrology, arXiv:2502.17396.
  37. H. Cramér, Mathematical Methods of Statistics, Princeton Mathematical Series Vol. 43 (Princeton University, Princetpn, NJ, 1999).
  38. E. L. Lehmann and G. Casella, Theory of Point Estimation, 2nd ed. (Springer, New York, 1998).
  39. C. W. Helstrom, Minimum mean-squared error of estimates in quantum statistics, Phys. Lett. A 25, 101 (1967).
  40. A. S. Holevo, Commutation superoperator of a state and its applications to the noncommutative statistics, Rep. Math. Phys. 12, 251 (1977).
  41. H. Nagaoka, A new approach to Cramér-Rao bounds for quantum state estimation, IEICE Tech. Rep. IT 89-42, 9 (1989).
  42. K. Yamagata, A. Fujiwara, and R. D. Gill, Quantum local asymptotic normality based on a new quantum likelihood ratio, Ann. Stat. 41, 2197 (2013).
  43. M. Hayashi and K. Matsumoto, Asymptotic performance of optimal state estimation in qubit system, J. Math. Phys. 49, 102101 (2008).
  44. Y. Yang, G. Chiribella, and M. Hayashi, Attaining the ultimate precision limit in quantum state estimation, Commun. Math. Phys. 368, 223 (2019).
  45. M. Hayashi and Y. Ouyang, Tight Cramér-Rao type bounds for multiparameter quantum metrology through conic programming, Quantum 7, 1094 (2023).
  46. L. O. Conlon, J. Suzuki, P. K. Lam, and S. M. Assad, Efficient computation of the Nagaoka-Hayashi bound for multiparameter estimation with separable measurements, npj Quantum Inf. 7, 110 (2021).
  47. F. Belliardo and V. Giovannetti, Incompatibility in quantum parameter estimation, New J. Phys. 23, 063055 (2021).
  48. F. Albarelli, J. F. Friel, and A. Datta, Evaluating the Holevo Cramér-Rao bound for multiparameter quantum metrology, Phys. Rev. Lett. 123, 200503 (2019).
  49. A. Carollo, B. Spagnolo, A. A. Dubkov, and D. Valenti, On quantumness in multi-parameter quantum estimation, J. Stat. Mech. (2019) 094010.
  50. M. Tsang, F. Albarelli, and A. Datta, Quantum semiparametric estimation, Phys. Rev. X 10, 031023 (2020).
  51. A. Candeloro, M. G. A. Paris, and M. G. Genoni, On the properties of the asymptotic incompatibility measure in multiparameter quantum estimation, J. Phys. A 54, 485301 (2021).
  52. A. Candeloro, Z. Pazhotan, and M. G. A. Paris, Dimension matters: Precision and incompatibility in multi-parameter quantum estimation models, Quantum Sci. Technol. 9, 045045 (2024).
  53. A. Uhlmann, Parallel transport and “quantum holonomy” along density operators, Rep. Math. Phys. 24, 229 (1986).
  54. J. Dittmann, The scalar curvature of the Bures metric on the space of density matrices, J. Geom. Phys. 31, 16 (1999).
  55. A. Carollo, B. Spagnolo, and D. Valenti, Uhlmann curvature in dissipative phase transitions, Sci. Rep. 8, 9852 (2018).
  56. F. Albarelli and R. Demkowicz-Dobrzański, Probe incompatibility in multiparameter noisy quantum metrology, Phys. Rev. X 12, 011039 (2022).
  57. S. Ragy, M. Jarzyna, and R. Demkowicz-Dobrzański, Compatibility in multiparameter quantum metrology, Phys. Rev. A 94, 052108 (2016).
  58. G. Mihailescu, A. Bayat, S. Campbell, and A. K. Mitchell, Multiparameter critical quantum metrology with impurity probes, Quantum Sci. Technol. 9, 035033 (2024).
  59. P. F. Bernath, Spectra of Atoms and Molecules, 4th ed. (Oxford University, New York, 2020).
  60. J. M. Brown and A. Carrington, Rotational Spectroscopy of Diatomic Molecules, 1st ed. (Cambridge University, New York, 2003).
  61. B. K. Stuhl, M. Yeo, B. C. Sawyer, M. T. Hummon, and J. Ye, Microwave state transfer and adiabatic dynamics of magnetically trapped polar molecules, Phys. Rev. A 85, 033427 (2012).
  62. F. Troiani and Matteo G. A. Paris, Universal quantum magnetometry with spin states at equilibrium, Phys. Rev. Lett. 120, 260503 (2018).
  63. L. P. García-Pintos, K. Bharti, J. Bringewatt, H. Dehghani, A. Ehrenberg, N. Yunger Halpern, and A. V. Gorshkov, Estimation of Hamiltonian parameters from thermal states, Phys. Rev. Lett. 133, 040802 (2024).
  64. P. Abiuso, P. Sekatski, J. Calsamiglia, and M. Perarnau-Llobet, Fundamental limits of metrology at thermal equilibrium, Phys. Rev. Lett. 134, 010801 (2025).
  65. D. Qiu and L. Li, Minimum-error discrimination of quantum states: Bounds and comparisons, Phys. Rev. A 81, 042329 (2010).
  66. R. P. Feynman, Forces in molecules, Phys. Rev. 56, 340 (1939).
  67. Z. Jiang, Quantum Fisher information for states in exponential form, Phys. Rev. A 89, 032128 (2014).
  68. As discussed in Sec. 3, this low-temperature expansion should be interpreted as an expansion within the effective Hamiltonian. The finite-temperature numerical results shown in Fig. 4, including the curves at T=10mK, are above the quoted 5-mK validity scale.
  69. G. Mihailescu, U. Alushi, R. D. Candia, S. Felicetti, and K. Gietka, Critical quantum sensing: A tutorial on parameter estimation near quantum phase transitions, arXiv:2510.02035.
  70. L. R. B. Picard, A. J. Park, G. E. Patenotte, S. Gebretsadkan, D. Wellnitz, A. M. Rey, and K.-K. Ni, Entanglement and iSWAP gate between molecular qubits, Nature (London) 637, 821 (2025).
  71. C. M. Holland, Y. Lu, and L. W. Cheuk, On-demand entanglement of molecules in a reconfigurable optical tweezer array, Science 382, 1143 (2023).
  72. Q. He, W. Zhang, D. Meschede, and A. Schenzle, Entangled atoms and fields: Cavity QED, in Springer Handbook of Atomic, Molecular, and Optical Physics, edited by G. W. F. Drake (Springer, New York, 2023), pp. 1207–1230.
  73. H. Yuan and C.-H. F. Fung, Optimal feedback scheme and universal time scaling for Hamiltonian parameter estimation, Phys. Rev. Lett. 115, 110401 (2015).
  74. M. Zhang, H.-M. Yu, H. Yuan, X. Wang, R. Demkowicz-Dobrzański, and J. Liu, QuanEstimation: An open-source toolkit for quantum parameter estimation, Phys. Rev. Res. 4, 043057 (2022).
  75. H.-M. Yu and J. Liu, Quanestimation.jl: An open-source Julia framework for quantum parameter estimation, Fundam. Res. (2025), doi: 10.1016/j.fmre.2025.02.020.
  76. J. He and M. G. A. Paris, Scrambling for precision: Optimizing multiparameter qubit estimation in the face of sloppiness and incompatibility, J. Phys. A 58, 325301 (2025).
  77. J. He, G. Fazio, and M. G. A. Paris, Weight-dependent and weight-independent measures of quantum incompatibility in multiparameter estimation, arXiv:2510.18864.
  78. Z. Hou, R.-J. Wang, J.-F. Tang, H. Yuan, G.-Y. Xiang, C.-F. Li, and G.-C. Guo, Control-enhanced sequential scheme for general quantum parameter estimation at the Heisenberg limit, Phys. Rev. Lett. 123, 040501 (2019).
  79. S. Li, D.-J. Pan, Y.-K. Zhu, J.-L. Zhou, W.-C. Liao, W.-X. Zhang, Z.-T. Liang, Q.-X. Lv, H. Yu, Z.-Y. Xue, H. Yan, and S.-L. Zhu, Ultrahigh-precision Hamiltonian parameter estimation in a superconducting circuit, Phys. Rev. Lett. 132, 250204 (2024).
  80. S. Pang and T. A. Brun, Quantum metrology for a general Hamiltonian parameter, Phys. Rev. A 90, 022117 (2014).
  81. See also Ref. [84] for a similar approach for multiple parameters.
  82. This is known as a “null measurement”: the outcome corresponding to the projection on |ψ0〉〈ψ0| is observed with overwhelming probability, and any deviation from this outcome carries a lot of information on the value of the parameter. It is optimal in local quantum estimation of pure states, however a rigorous convergence analysis presents some subtleties [86], and it is also particularly susceptible to noise in the implementation [87, 88].
  83. Z. Hu, S. Wang, L. Qiao, T. Isogawa, C. Li, Y. Yang, G. Wang, H. Yuan, and P. Cappellaro, Control incompatibility in multiparameter quantum metrology, arXiv:2411.18896.
  84. Q. Wei and S. Pang, Efficient adaptive control strategy for multi-parameter quantum metrology in two-dimensional systems, arXiv:2510.14811.
  85. L. Previdi, F. Albarelli, and M. G. A. Paris, Code for: “OH molecule as a quantum probe to jointly estimate electric and magnetic fields,” 2025, https://github.com/lucappp02/OH_molecule_estimation.
  86. F. Girotti, A. Godley, and M. Guţă, Optimal estimation of pure states with displaced-null measurements, J. Phys. A 57, 245304 (2024).
  87. S. Kurdziałek and R. Demkowicz-Dobrzański, Measurement noise susceptibility in quantum estimation, Phys. Rev. Lett. 130, 160802 (2023).
  88. F. Albarelli, I. Gianani, M. G. Genoni, and M. Barbieri, Fisher-information susceptibility for multiparameter quantum estimation, Phys. Rev. A 110, 032436 (2024).
  89. J. Liu, X.-X. Jing, and X. Wang, Quantum metrology with unitary parametrization processes, Sci. Rep. 5, 8565 (2015).
  90. R. M. Wilcox, Exponential operators and parameter differentiation in quantum physics, J. Math. Phys. 8, 962 (1967).

Outline

Information

Phys. Rev. A 113, 062612– Published 11 June, 2026
  • Received 28 May 2025
  • Revised 13 January 2026
  • Accepted 26 May 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