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Optimal Phase-Insensitive Force Sensing with Non-Gaussian States

Piotr T. Grochowski* and Radim Filip†

  • Department of Optics, Palacký University, 17. listopadu 1192/12, 771 46 Olomouc, Czech Republic

  • *Contact author: piotr.grochowski@upol.cz
  • †Contact author: filip@optics.upol.cz
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Phys. Rev. Lett. 135, 230802 – Published 3 December, 2025

DOI: https://doi.org/10.1103/7pyw-tgjd

Abstract

Quantum metrology enables sensitivity to approach the limits set by fundamental physical laws. Even a single continuous mode offers enhanced precision, with the improvement scaling with its occupation number. Due to their high information capacity, continuous modes allow for the engineering of quantum non-Gaussian states, which not only improve metrological performance but can also be tailored to specific experimental platforms and conditions. Recent advancements in control over continuous platforms operating in the quantum regime have renewed interest in sensing weak forces, also coupling to massive macroscopic objects. In this work, we investigate a force-sensing scheme where a physical process completely randomizes the direction of the induced phase-space displacement, and the unknown force strength is inferred through excitation-number-resolving measurements. We find that N-spaced states, where only every Nth Fock state occupation is nonzero, approach the achievable sensing bound. Additionally, non-Gaussian states are shown to be more resilient against decoherence than their Gaussian counterparts with the same occupation number. While Fock states typically offer the best protection against decoherence, we uncover a transition in the metrological landscape—revealed through a tailored decoherence-aware Fisher-information-based reward functional—where experimental constraints favor a family of number-squeezed Schrödinger cat states. Specifically, by implementing quantum optimal control in a minimal spin-boson system, we identify these states as maximizing force sensitivity under lossy dynamics and finite system controllability. Our results provide a pathway for enhancing force sensing in a variety of continuous quantum systems, ranging from massive systems like mechanical oscillators to massless systems such as quantum light and microwave resonators.

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References (110)

  1. M. S. Safronova, D. Budker, D. DeMille, D. F. J. Kimball, A. Derevianko, and C. W. Clark, Search for new physics with atoms and molecules, Rev. Mod. Phys. 90, 025008 (2018).
  2. A. Chou, K. Irwin, R. H. Maruyama et al., Quantum sensors for high energy physics, arXiv:2311.01930.
  3. M. Tse, H. Yu, N. Kijbunchoo et al., Quantum-enhanced Advanced LIGO detectors in the era of gravitational-wave astronomy, Phys. Rev. Lett. 123, 231107 (2019).
  4. F. Acernese, M. Agathos et al. (Virgo Collaboration), Increasing the astrophysical reach of the Advanced Virgo detector via the application of squeezed vacuum states of light, Phys. Rev. Lett. 123, 231108 (2019).
  5. J.-P. Karr, D. Marchand, and E. Voutier, The proton size, Nat. Rev. Phys. 2, 601 (2020).
  6. J. Ye, H. J. Kimble, and H. Katori, Quantum state engineering and precision metrology using state-insensitive light traps, Science 320, 1734 (2008).
  7. M. de Angelis, A. Bertoldi, L. Cacciapuoti, A. Giorgini, G. Lamporesi, M. Prevedelli, G. Saccorotti, F. Sorrentino, and G. M. Tino, Precision gravimetry with atomic sensors, Meas. Sci. Technol. 20, 022001 (2008).
  8. N. Aslam, H. Zhou, E. K. Urbach, M. J. Turner, R. L. Walsworth, M. D. Lukin, and H. Park, Quantum sensors for biomedical applications, Nat. Rev. Phys. 5, 157 (2023).
  9. M. G. A. Paris, Quantum estimation for quantum technology, Int. J. Quantum. Inform. 07, 125 (2009).
  10. C. L. Degen, F. Reinhard, and P. Cappellaro, Quantum sensing, Rev. Mod. Phys. 89, 035002 (2017).
  11. S. Pirandola, B. R. Bardhan, T. Gehring, C. Weedbrook, and S. Lloyd, Advances in photonic quantum sensing, Nat. Photonics 12, 724 (2018).
  12. K. Duivenvoorden, B. M. Terhal, and D. Weigand, Single-mode displacement sensor, Phys. Rev. A 95, 012305 (2017).
  13. D. Braun, G. Adesso, F. Benatti, R. Floreanini, U. Marzolino, M. W. Mitchell, and S. Pirandola, Quantum-enhanced measurements without entanglement, Rev. Mod. Phys. 90, 035006 (2018).
  14. M. Fadel, N. Roux, and M. Gessner, Quantum metrology with a continuous-variable system, Rep. Prog. Phys. 88, 106001 (2025).
  15. C. Rouvière, D. Barral, A. Grateau, I. Karuseichyk, G. Sorelli, M. Walschaers, and N. Treps, Ultra-sensitive separation estimation of optical sources, Optica 11, 166 (2024).
  16. X. Pan, T. Krisnanda, A. Duina, K. Park, P. Song, C. Y. Fontaine, A. Copetudo, R. Filip, and Y. Y. Gao, Realization of versatile and effective quantum metrology using a single bosonic mode, PRX Quantum 6, 010304 (2025).
  17. A. Facon, E.-K. Dietsche, D. Grosso, S. Haroche, J.-M. Raimond, M. Brune, and S. Gleyzes, A sensitive electrometer based on a Rydberg atom in a Schrödinger-cat state, Nature (London) 535, 262 (2016).
  18. S. C. Burd, R. Srinivas, J. J. Bollinger, A. C. Wilson, D. J. Wineland, D. Leibfried, D. H. Slichter, and D. T. C. Allcock, Quantum amplification of mechanical oscillator motion, Science 364, 1163 (2019).
  19. K. A. Gilmore, M. Affolter, R. J. Lewis-Swan, D. Barberena, E. Jordan, A. M. Rey, and J. J. Bollinger, Quantum-enhanced sensing of displacements and electric fields with two-dimensional trapped-ion crystals, Science 373, 673 (2021).
  20. A. Delakouras, D. Rodríguez, and J. Cerrillo, Production of Fock mixtures in trapped ions for motional metrology, Quantum Sci. Technol. 9, 015006 (2023).
  21. A. A. Rakhubovsky, D. W. Moore, and R. Filip, Quantum non-Gaussian optomechanics and electromechanics, Prog. Quantum Electron. 93, 100495 (2024).
  22. K. J. Satzinger, Y. P. Zhong, H.-S. Chang et al., Quantum control of surface acoustic-wave phonons, Nature (London) 563, 661 (2018).
  23. D. Mason, J. Chen, M. Rossi, Y. Tsaturyan, and A. Schliesser, Continuous force and displacement measurement below the standard quantum limit, Nat. Phys. 15, 745 (2019).
  24. E. A. Wollack, A. Y. Cleland, R. G. Gruenke, Z. Wang, P. Arrangoiz-Arriola, and A. H. Safavi-Naeini, Quantum state preparation and tomography of entangled mechanical resonators, Nature (London) 604, 463 (2022).
  25. U. von Lüpke, Y. Yang, M. Bild, L. Michaud, M. Fadel, and Y. Chu, Parity measurement in the strong dispersive regime of circuit quantum acoustodynamics, Nat. Phys. 18, 794 (2022).
  26. M. Bild, M. Fadel, Y. Yang, U. von Lüpke, P. Martin, A. Bruno, and Y. Chu, Schrödinger cat states of a 16-microgram mechanical oscillator, Science 380, 274 (2023).
  27. A. Youssefi, S. Kono, M. Chegnizadeh, and T. J. Kippenberg, A squeezed mechanical oscillator with millisecond quantum decoherence, Nat. Phys. 19, 1697 (2023).
  28. J. Millen, T. S. Monteiro, R. Pettit, and A. N. Vamivakas, Optomechanics with levitated particles, Rep. Prog. Phys. 83, 026401 (2020).
  29. C. Gonzalez-Ballestero, M. Aspelmeyer, L. Novotny, R. Quidant, and O. Romero-Isart, Levitodynamics: Levitation and control of microscopic objects in vacuum, Science 374, eabg3027 (2021).
  30. L. Lachman and R. Filip, Quantum non-Gaussianity of light and atoms, Prog. Quantum Electron. 83, 100395 (2022).
  31. M. Walschaers, Non-Gaussian quantum states and where to find them, PRX Quantum 2, 030204 (2021).
  32. M. Frigerio, M. G. A. Paris, C. E. Lopetegui, and M. Walschaers, Joint estimation of position and momentum with arbitrarily high precision using non-Gaussian states, arXiv:2504.01910.
  33. B. Vlastakis, G. Kirchmair, Z. Leghtas, S. E. Nigg, L. Frunzio, S. M. Girvin, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, Deterministically encoding quantum information using 100-photon Schrödinger cat states, Science 342, 607 (2013).
  34. Y. Chu, P. Kharel, T. Yoon, L. Frunzio, P. T. Rakich, and R. J. Schoelkopf, Creation and control of multi-phonon Fock states in a bulk acoustic-wave resonator, Nature (London) 563, 666 (2018).
  35. F. Wolf, C. Shi, J. C. Heip, M. Gessner, L. Pezzè, A. Smerzi, M. Schulte, K. Hammerer, and P. O. Schmidt, Motional Fock states for quantum-enhanced amplitude and phase measurements with trapped ions, Nat. Commun. 10, 2929 (2019).
  36. L. Podhora, L. Lachman, T. Pham, A. Lešundák, O. Číp, L. Slodička, and R. Filip, Quantum non-gaussianity of multiphonon states of a single atom, Phys. Rev. Lett. 129, 013602 (2022).
  37. M. Hofheinz, E. M. Weig, M. Ansmann, R. C. Bialczak, E. Lucero, M. Neeley, A. D. O’Connell, H. Wang, J. M. Martinis, and A. N. Cleland, Generation of Fock states in a superconducting quantum circuit, Nature (London) 454, 310 (2008).
  38. A. Eickbusch, V. Sivak, A. Z. Ding, S. S. Elder, S. R. Jha, J. Venkatraman, B. Royer, S. M. Girvin, R. J. Schoelkopf, and M. H. Devoret, Fast universal control of an oscillator with weak dispersive coupling to a qubit, Nat. Phys. 18, 1464 (2022).
  39. X. Deng, S. Li, Z.-J. Chen, Z. Ni, Y. Cai, J. Mai, L. Zhang, P. Zheng, H. Yu, C.-L. Zou, S. Liu, F. Yan, Y. Xu, and D. Yu, Quantum-enhanced metrology with large Fock states, Nat. Phys. 20, 1874 (2024).
  40. Q. R. Rahman, I. Kladarić, M.-E. Kern, L. Lachman, Y. Chu, R. Filip, and M. Fadel, Genuine quantum non-gaussianity and metrological sensitivity of Fock states prepared in a mechanical resonator, Phys. Rev. Lett. 134, 180801 (2025).
  41. K. C. McCormick, J. Keller, S. C. Burd, D. J. Wineland, A. C. Wilson, and D. Leibfried, Quantum-enhanced sensing of a single-ion mechanical oscillator, Nature (London) 572, 86 (2019).
  42. W. Wang, Y. Wu, Y. Ma, W. Cai, L. Hu, X. Mu, Y. Xu, Z.-J. Chen, H. Wang, Y. P. Song, H. Yuan, C.-L. Zou, L.-M. Duan, and L. Sun, Heisenberg-limited single-mode quantum metrology in a superconducting circuit, Nat. Commun. 10, 4382 (2019).
  43. A. Kovalenko, L. Lachman, T. Pham, K. Singh, O. Číp, J. Fiurášek, L. Slodička, and R. Filip, Quantum non-Gaussian coherences of an oscillating atom, Phys. Rev. Res. 7, 033075 (2025).
  44. C. Hempel, B. P. Lanyon, P. Jurcevic, R. Gerritsma, R. Blatt, and C. F. Roos, Entanglement-enhanced detection of single-photon scattering events, Nat. Photonics 7, 630 (2013).
  45. H. Y. Lo, D. Kienzler, L. De Clercq, M. Marinelli, V. Negnevitsky, B. C. Keitch, and J. P. Home, Spin-motion entanglement and state diagnosis with squeezed oscillator wavepackets, Nature (London) 521, 336 (2015).
  46. C. H. Valahu, M. P. Stafford, Z. Huang, V. G. Matsos, M. J. Millican, T. Chalermpusitarak, N. C. Menicucci, J. Combes, B. Q. Baragiola, and T. R. Tan, Quantum-enhanced multiparameter sensing in a single mode, Sci. Adv. 11, eadw9757 (2025).
  47. W. Górecki, A. Riccardi, and L. Maccone, Quantum metrology of noisy spreading channels, Phys. Rev. Lett. 129, 240503 (2022).
  48. R. Ballantini, P. Bernard, E. Chiaveri, A. Chincarini, G. Gemme, R. Losito, R. Parodi, and E. Picasso, A detector of high frequency gravitational waves based on coupled microwave cavities, Classical Quantum Gravity 20, 3505 (2003).
  49. D. Carney, G. Higgins, G. Marocco, and M. Wentzel, Superconducting levitated detector of gravitational waves, Phys. Rev. Lett. 134, 181402 (2025).
  50. J. D. Teufel, T. Donner, M. A. Castellanos-Beltran, J. W. Harlow, and K. W. Lehnert, Nanomechanical motion measured with an imprecision below that at the standard quantum limit, Nat. Nanotechnol. 4, 820 (2009).
  51. K. M. Backes, D. A. Palken, S. A. Kenany et al., A quantum enhanced search for dark matter axions, Nature (London) 590, 238 (2021).
  52. A. J. Brady, C. Gao, R. Harnik, Z. Liu, Z. Zhang, and Q. Zhuang, Entangled sensor-networks for dark-matter searches, PRX Quantum 3, 030333 (2022).
  53. G. Higgins, S. Kalia, and Z. Liu, Maglev for dark matter: Dark-photon and axion dark matter sensing with levitated superconductors, Phys. Rev. D 109, 055024 (2024).
  54. S. L. Braunstein and C. M. Caves, Statistical distance and the geometry of quantum states, Phys. Rev. Lett. 72, 3439 (1994).
  55. V. Giovannetti, S. Lloyd, and L. Maccone, Quantum-enhanced measurements: Beating the standard quantum limit, Science 306, 1330 (2004).
  56. L. Maccone and A. Riccardi, Squeezing metrology: A unified framework, Quantum 4, 292 (2020).
  57. A. L. Grimsmo, J. Combes, and B. Q. Baragiola, Quantum computing with rotation-symmetric bosonic codes, Phys. Rev. X 10, 011058 (2020).
  58. Y. Liu, S. Singh, K. C. Smith, E. Crane, J. M. Martyn, A. Eickbusch, A. Schuckert, R. D. Li, J. Sinanan-Singh, M. B. Soley, T. Tsunoda, I. L. Chuang, N. Wiebe, and S. M. Girvin, Hybrid oscillator-qubit quantum processors: Instruction set architectures, abstract machine models, and applications, arXiv:2407.10381.
  59. C. Oh, K. Park, R. Filip, H. Jeong, and P. Marek, Optical estimation of unitary Gaussian processes without phase reference using Fock states, New J. Phys. 22, 123039 (2020).
  60. F. A. M. de Oliveira, M. S. Kim, P. L. Knight, and V. Bužek, Properties of displaced number states, Phys. Rev. A 41, 2645 (1990).
  61. D. M. Meekhof, C. Monroe, B. E. King, W. M. Itano, and D. J. Wineland, Generation of nonclassical motional states of a trapped atom, Phys. Rev. Lett. 76, 1796 (1996).
  62. R. H. Hadfield, Single-photon detectors for optical quantum information applications, Nat. Photonics 3, 696 (2009).
  63. T. Weiss and O. Romero-Isart, Quantum motional state tomography with nonquadratic potentials and neural networks, Phys. Rev. Res. 1, 033157 (2019).
  64. P. T. Grochowski, H. Pichler, C. A. Regal, and O. Romero-Isart, Quantum control of continuous systems via nonharmonic potential modulation, Quantum 9, 1824 (2025).
  65. A. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (Edizioni della Normale, Pisa, 2011).
  66. R. Demkowicz-Dobrzański, M. Jarzyna, and J. Kołodyński, in Progress in Optics, edited by E. Wolf (Elsevier, New York, 2015), Vol. 60, pp. 345–435.
  67. Z. Hradil, J. Řeháček, L. Sánchez-Soto, and B.-G. Englert, Quantum Fisher information with coherence, Optica 6, 1437 (2019).
  68. A. Ritboon, L. Slodička, and R. Filip, Sequential phonon measurements of atomic motion, Quantum Sci. Technol. 7, 015023 (2022).
  69. W. H. Zurek, Sub-Planck structure in phase space and its relevance for quantum decoherence, Nature (London) 412, 712 (2001).
  70. D. Gottesman, A. Kitaev, and J. Preskill, Encoding a qubit in an oscillator, Phys. Rev. A 64, 012310 (2001).
  71. O. Băzăvan, S. Saner, D. J. Webb, E. M. Ainley, P. Drmota, D. P. Nadlinger, G. Araneda, D. M. Lucas, C. J. Ballance, and R. Srinivas, Squeezing, trisqueezing, and quadsqueezing in a spin-oscillator system, arXiv:2403.05471.
  72. J. Kołodyński and R. Demkowicz-Dobrzański, Phase estimation without a priori phase knowledge in the presence of loss, Phys. Rev. A 82, 053804 (2010).
  73. S. Knysh, V. N. Smelyanskiy, and G. A. Durkin, Scaling laws for precision in quantum interferometry and the bifurcation landscape of the optimal state, Phys. Rev. A 83, 021804 (2011).
  74. See Supplemental Material at http://link.aps.org/supplemental/10.1103/7pyw-tgjd for derivation of phase-randomizing channels and Fisher information, solution of the master equation, analysis of different families of states, expanded discussion on Fock versus Gaussian states, and numerical details.
  75. A. Shukla and B. C. Sanders, Superposing compass states for asymptotic isotropic sub-Planck phase-space sensitivity, Phys. Rev. A 108, 043719 (2023).
  76. B. E. Asenbeck, L. Lachman, A. Boyer, P. Giri, A. Urvoy, R. Filip, and J. Laurat, Hierarchical verification of non-Gaussian coherence in bosonic quantum states, Phys. Rev. Lett. 134, 233604 (2025).
  77. L. Lachman, B. E. Asenbeck, A. Boyer, P. Giri, A. Urvoy, J. Laurat, and R. Filip, Hierarchies of quantum non-Gaussian coherences for bosonic systems: A theoretical study, Phys. Rev. A 111, 063704 (2025).
  78. H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, New York, 2007).
  79. Y. Yang, I. Kladarić, M. Drimmer, U. von Lüpke, D. Lenterman, J. Bus, S. Marti, M. Fadel, and Y. Chu, A mechanical qubit, Science 386, 783 (2024).
  80. C. Gonzalez-Ballestero, P. Maurer, D. Windey, L. Novotny, R. Reimann, and O. Romero-Isart, Theory for cavity cooling of levitated nanoparticles via coherent scattering: Master equation approach, Phys. Rev. A 100, 013805 (2019).
  81. K. Fujii, in Advances in Quantum Mechanics, edited by P. Bracken (IntechOpen, London, 2013).
  82. J. W. Gardner, T. Gefen, S. A. Haine, J. J. Hope, J. Preskill, Y. Chen, and L. McCuller, Stochastic waveform estimation at the fundamental quantum limit, PRX Quantum 6, 030311 (2025).
  83. C. A. Rosiek, M. Rossi, A. Schliesser, and A. S. Sørensen, Quadrature Squeezing enhances Wigner negativity in a mechanical duffing oscillator, PRX Quantum 5, 030312 (2024).
  84. S. J. Glaser, U. Boscain, T. Calarco, C. P. Koch, W. Köckenberger, R. Kosloff, I. Kuprov, B. Luy, S. Schirmer, T. Schulte-Herbrüggen, D. Sugny, and F. K. Wilhelm, Training Schrödinger’s cat: Quantum optimal control, Eur. Phys. J. D 69, 279 (2015).
  85. C. P. Koch, U. Boscain, T. Calarco, G. Dirr, S. Filipp, S. J. Glaser, R. Kosloff, S. Montangero, T. Schulte-Herbrüggen, D. Sugny, and F. K. Wilhelm, Quantum optimal control in quantum technologies. Strategic report on current status, visions and goals for research in Europe, Eur. Phys. J. Quantum Technol. 9, 1 (2022).
  86. R. W. Heeres, P. Reinhold, N. Ofek, L. Frunzio, L. Jiang, M. H. Devoret, and R. J. Schoelkopf, Implementing a universal gate set on a logical qubit encoded in an oscillator, Nat. Commun. 8, 94 (2017).
  87. T. Krisnanda, C. Y. Fontaine, A. Copetudo, P. Song, K. X. Lee, N.-N. Huang, F. Valadares, T. C. Liew, and Y. Y. Gao, Demonstrating efficient and robust bosonic state reconstruction via optimized excitation counting, PRX Quantum 6, 010303 (2025).
  88. V. G. Matsos, C. H. Valahu, T. Navickas, A. D. Rao, M. J. Millican, X. C. Kolesnikow, M. J. Biercuk, and T. R. Tan, Robust and deterministic preparation of bosonic logical states in a trapped ion, Phys. Rev. Lett. 133, 050602 (2024).
  89. M. M. Müller, R. S. Said, F. Jelezko, T. Calarco, and S. Montangero, One decade of quantum optimal control in the chopped random basis, Rep. Prog. Phys. 85, 076001 (2022).
  90. J. Liu and H. Yuan, Control-enhanced multiparameter quantum estimation, Phys. Rev. A 96, 042114 (2017).
  91. J. Liu and H. Yuan, Quantum parameter estimation with optimal control, Phys. Rev. A 96, 012117 (2017).
  92. S. Pang and A. N. Jordan, Optimal adaptive control for quantum metrology with time-dependent Hamiltonians, Nat. Commun. 8, 14695 (2017).
  93. D. Basilewitsch, H. Yuan, and C. P. Koch, Optimally controlled quantum discrimination and estimation, Phys. Rev. Res. 2, 033396 (2020).
  94. C. Lin, Y. Ma, and D. Sels, Application of Pontryagin’s maximum principle to quantum metrology in dissipative systems, Phys. Rev. A 105, 042621 (2022).
  95. N. Beato, P. Patil, and M. Bukov, Towards a theory of phase transitions in quantum control landscapes, Phys. Rev. X 15, 041014 (2025).
  96. R. L. de Matos Filho and W. Vogel, Nonlinear coherent states, Phys. Rev. A 54, 4560 (1996).
  97. V. V. Dodonov, ‘Nonclassical’ states in quantum optics: A ‘squeezed’ review of the first 75 years, J. Opt. B 4, R1 (2002).
  98. I. Rojkov, M. Simoni, E. Zapusek, F. Reiter, and J. Home, Stabilization of cat-state manifolds using nonlinear reservoir engineering, arXiv:2407.18087.
  99. R. Rousseau, D. Ruiz, E. Albertinale et al., Enhancing dissipative cat qubit protection by squeezing, arXiv:2502.07892.

  100. We take γ=1/gT. The reported values read: g=2π·{2.2,1.4,0.29,0.002}  MHz and T={2.7,1,0.089,5000}  ms for [16, 40, 86, 88], respectively.

  101. S. Deffner, Optimal control of a qubit in an optical cavity, J. Phys. B 47, 145502 (2014).
  102. M. Roda-Llordes, A. Riera-Campeny, D. Candoli, P. T. Grochowski, and O. Romero-Isart, Macroscopic quantum superpositions via dynamics in a wide double-well potential, Phys. Rev. Lett. 132, 023601 (2024).
  103. S. Casulleras, P. T. Grochowski, and O. Romero-Isart, Optimization of static potentials for large delocalization and non-Gaussian quantum dynamics of levitated nanoparticles under decoherence, Phys. Rev. A 110, 033511 (2024).
  104. J. R. Johansson, P. D. Nation, and F. Nori, qutip: An open-source python framework for the dynamics of open quantum systems, Comput. Phys. Commun. 183, 1760 (2012).
  105. J. R. Johansson, P. D. Nation, and F. Nori, qutip 2: A python framework for the dynamics of open quantum systems, Comput. Phys. Commun. 184, 1234 (2013).
  106. N. Lambert, E. Giguère, P. Menczel et al., qutip 5: The quantum toolbox in python, arXiv:2412.04705.
  107. M. Rossignolo, T. Reisser, A. Marshall, P. Rembold, A. Pagano, P. J. Vetter, R. S. Said, M. M. Müller, F. Motzoi, T. Calarco, F. Jelezko, and S. Montangero, QuOCS: The quantum optimal control suite, Comput. Phys. Commun. 291, 108782 (2023).
  108. All data and simulation codes are available at 10.5281/zenodo.15582205.
  109. F. Crameri, G. E. Shephard, and P. J. Heron, The misuse of colour in science communication, Nat. Commun. 11, 5444 (2020).
  110. F. Crameri, Scientific colour maps, Zenodo, 10.5281/zenodo.8409685 (2023).

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Phys. Rev. Lett. 135, 230802– Published 3 December, 2025
  • Received 7 June 2025
  • Accepted 3 November 2025
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