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Regularized counterdiabatic driving for the quantum Rabi model

Julián Ferreiro-Vélez1,2,*, Pablo García-Azorín2, Francisco Andrés Cárdenas-López3, and Xi Chen4,5,†

  • *Contact author: julian.ferreiro@tecnalia.com
  • †Contact author: xi.chen@csic.es
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Phys. Rev. Research 8, 033244 – Published 28 August, 2026

DOI: https://doi.org/10.1103/7wnd-t4xy

Abstract

Counterdiabatic (CD) driving provides a powerful route to fast and robust state preparation by suppressing diabatic excitations during finite-time evolution. Yet, deriving analytical CD protocols for complex systems remains challenging, motivating the development of variational approaches. These methods typically rely on minimizing trace-based functionals to construct approximate control Hamiltonians. However, in unbounded systems, such functionals can become ill-defined because of the unbounded bosonic Hilbert space, leading to divergent cost functions and unphysical variational coefficients. Here, we introduce a variational optimization framework equipped with physically motivated renormalization schemes that regularize the trace-based metric by restricting it to relevant displaced and low-energy subspaces. As a paradigmatic example, we apply our method to the quantum Rabi model beyond the dispersive approximation and identify two distinct CD contributions that couple the atomic degree of freedom to the position and momentum quadratures of the field. These terms suppress diabatic excitations across coupling regimes ranging from strong to deep-strong light-matter interaction. We further formulate a fidelity-based quantum optimal-control strategy that bypasses the limitations of trace-based variational methods. Finally, we show that the resulting CD terms can be implemented via Floquet engineering through parametric modulation of the native Hamiltonian. Our results demonstrate that CD driving can be consistently extended to continuous-variable systems with unbounded Hilbert spaces, providing a controlled and scalable framework for quantum control in strongly interacting light-matter platforms.

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References (84)

  1. I. I. Rabi, On the process of space quantization, Phys. Rev. 49, 324 (1936).
  2. E. Jaynes and F. Cummings, Comparison of quantum and semiclassical radiation theories with application to the beam maser, Proc. IEEE 51, 89 (1963).
  3. D. C. McKay, R. Naik, P. Reinhold, L. S. Bishop, and D. I. Schuster, High-contrast qubit interactions using multimode cavity QED, Phys. Rev. Lett. 114, 080501 (2015).
  4. C. Wang, Y. Y. Gao, P. Reinhold, R. W. Heeres, N. Ofek, K. Chou, C. Axline, M. Reagor, J. Blumoff, K. Sliwa, et al., A Schrödinger cat living in two boxes, Science 352, 1087 (2016).
  5. F. Yoshihara, T. Fuse, S. Ashhab, K. Kakuyanagi, S. Saito, and K. Semba, Superconducting qubit-oscillator circuit beyond the ultrastrong-coupling regime, Nat. Phys. 13, 44 (2017).
  6. X. Zhao, Q. Bin, W. Hou, Y. Li, Y. Li, Y. Lin, X.-Y. Lü, and J. Du, Experimental observation of parity-symmetry-protected phenomena in the quantum Rabi model with a trapped ion, Phys. Rev. Lett. 134, 193604 (2025).
  7. S. Ashhab, Superradiance transition in a system with a single qubit and a single oscillator, Phys. Rev. A 87, 013826 (2013).
  8. M.-J. Hwang, R. Puebla, and M. B. Plenio, Quantum phase transition and universal dynamics in the Rabi model, Phys. Rev. Lett. 115, 180404 (2015).
  9. T. Ye, Y.-Z. Wang, X.-Y. Chen, Q.-H. Chen, and H.-Q. Lin, Superradiant phase transitions in the quantum Rabi model: Overcoming the no-go theorem through anisotropy, Phys. Rev. A 111, 043716 (2025).
  10. M.-L. Cai, Z.-D. Liu, W.-D. Zhao, Y.-K. Wu, Q.-X. Mei, Y. Jiang, L. He, X. Zhang, Z.-C. Zhou, and L.-M. Duan, Observation of a quantum phase transition in the quantum Rabi model with a single trapped ion, Nat. Commun. 12, 1126 (2021).
  11. H. Walther, B. T. Varcoe, B.-G. Englert, and T. Becker, Cavity quantum electrodynamics, Rep. Prog. Phys. 69, 1325 (2006).
  12. D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, Quantum dynamics of single trapped ions, Rev. Mod. Phys. 75, 281 (2003).
  13. A. Blais, A. L. Grimsmo, S. M. Girvin, and A. Wallraff, Circuit quantum electrodynamics, Rev. Mod. Phys. 93, 025005 (2021).
  14. J. Bourassa, J. M. Gambetta, A. A. Abdumalikov, O. Astafiev, Y. Nakamura, and A. Blais, Ultrastrong coupling regime of cavity QED with phase-biased flux qubits, Phys. Rev. A 80, 032109 (2009).
  15. P. Forn-Díaz, J. Lisenfeld, D. Marcos, J. J. García-Ripoll, E. Solano, C. J. P. M. Harmans, and J. E. Mooij, Observation of the Bloch-Siegert shift in a qubit-oscillator system in the ultrastrong coupling regime, Phys. Rev. Lett. 105, 237001 (2010).
  16. I. Chiorescu, P. Bertet, K. Semba, Y. Nakamura, C. Harmans, and J. Mooij, Coherent dynamics of a flux qubit coupled to a harmonic oscillator, Nature (London) 431, 159 (2004).
  17. T. Niemczyk, F. Deppe, H. Huebl, E. Menzel, F. Hocke, M. Schwarz, J. Garcia-Ripoll, D. Zueco, T. Hümmer, E. Solano, et al., Circuit quantum electrodynamics in the ultrastrong-coupling regime, Nat. Phys. 6, 772 (2010).
  18. K. Akbari, F. Nori, and S. Hughes, Floquet engineering the quantum Rabi model in the ultrastrong coupling regime, Phys. Rev. Lett. 134, 063602 (2025).
  19. P. Forn-Díaz, L. Lamata, E. Rico, J. Kono, and E. Solano, Ultrastrong coupling regimes of light-matter interaction, Rev. Mod. Phys. 91, 025005 (2019).
  20. A. Frisk Kockum, A. Miranowicz, S. De Liberato, S. Savasta, and F. Nori, Ultrastrong coupling between light and matter, Nat. Rev. Phys. 1, 19 (2019).
  21. J. Casanova, G. Romero, I. Lizuain, J. J. García-Ripoll, and E. Solano, Deep strong coupling regime of the Jaynes-Cummings model, Phys. Rev. Lett. 105, 263603 (2010).
  22. J. Koch, G. R. Hunanyan, T. Ockenfels, E. Rico, E. Solano, and M. Weitz, Quantum Rabi dynamics of trapped atoms far in the deep strong coupling regime, Nat. Commun. 14, 954 (2023).
  23. D. Lv, S. An, Z. Liu, J.-N. Zhang, J. S. Pedernales, L. Lamata, E. Solano, and K. Kim, Quantum simulation of the quantum Rabi model in a trapped ion, Phys. Rev. X 8, 021027 (2018).
  24. W. Qin, A. F. Kockum, C. S. Muñoz, A. Miranowicz, and F. Nori, Quantum amplification and simulation of strong and ultrastrong coupling of light and matter, Phys. Rep. 1078, 1 (2024).
  25. P. Domokos, J. M. Raimond, M. Brune, and S. Haroche, Simple cavity-QED two-bit universal quantum logic gate: The principle and expected performances, Phys. Rev. A 52, 3554 (1995).
  26. A. Rauschenbeutel, G. Nogues, S. Osnaghi, P. Bertet, M. Brune, J. M. Raimond, and S. Haroche, Coherent operation of a tunable quantum phase gate in cavity QED, Phys. Rev. Lett. 83, 5166 (1999).
  27. R. Di Candia, F. Minganti, K. Petrovnin, G. S. Paraoanu, and S. Felicetti, Critical parametric quantum sensing, npj Quantum Inf. 9, 23 (2023).
  28. L. Garbe, O. Abah, S. Felicetti, and R. Puebla, Critical quantum metrology with fully-connected models: From Heisenberg to Kibble–Zurek scaling, Quantum Sci. Technol. 7, 035010 (2022).
  29. D. Z. Rossatto, C. J. Villas-Bôas, M. Sanz, and E. Solano, Spectral classification of coupling regimes in the quantum Rabi model, Phys. Rev. A 96, 013849 (2017).
  30. E. Torrontegui, S. Ibáñez, S. Martínez-Garaot, M. Modugno, A. del Campo, D. Guéry-Odelin, A. Ruschhaupt, X. Chen, and J. G. Muga, in Chapter 2 - Shortcuts to Adiabaticity, Advances in Atomic, Molecular, and Optical Physics, Advances In Atomic, Molecular, and Optical Physics, edited by E. Arimondo, P. R. Berman, and C. C. Lin (Academic Press, 2013), Vol. 62, pp. 117–169.
  31. D. Guéry-Odelin, A. Ruschhaupt, A. Kiely, E. Torrontegui, S. Martínez-Garaot, and J. G. Muga, Shortcuts to adiabaticity: Concepts, methods, and applications, Rev. Mod. Phys. 91, 045001 (2019).
  32. M. Demirplak and S. A. Rice, Assisted adiabatic passage revisited, J. Phys. Chem. B 109, 6838 (2005).
  33. M. V. Berry, Transitionless quantum driving, J. Phys. A: Math. Theor. 42, 365303 (2009).
  34. X. Chen, I. Lizuain, A. Ruschhaupt, D. Guéry-Odelin, and J. G. Muga, Shortcut to adiabatic passage in two- and three-level atoms, Phys. Rev. Lett. 105, 123003 (2010).
  35. A. del Campo, Shortcuts to adiabaticity by counterdiabatic driving, Phys. Rev. Lett. 111, 100502 (2013).
  36. C. Jarzynski, Generating shortcuts to adiabaticity in quantum and classical dynamics, Phys. Rev. A 88, 040101(R) (2013).
  37. M. Kolodrubetz, D. Sels, P. Mehta, and A. Polkovnikov, Geometry and non-adiabatic response in quantum and classical systems, Phys. Rep. 697, 1 (2017).
  38. O. Abah, R. Puebla, and M. Paternostro, Quantum state engineering by shortcuts to adiabaticity in interacting spin-boson systems, Phys. Rev. Lett. 124, 180401 (2020).
  39. Y.-H. Chen, W. Qin, X. Wang, A. Miranowicz, and F. Nori, Shortcuts to adiabaticity for the quantum Rabi model: Efficient generation of giant entangled cat states via parametric amplification, Phys. Rev. Lett. 126, 023602 (2021).
  40. P. W. Claeys, M. Pandey, D. Sels, and A. Polkovnikov, Floquet-engineering counterdiabatic protocols in quantum many-body systems, Phys. Rev. Lett. 123, 090602 (2019).
  41. D. Sels and A. Polkovnikov, Minimizing irreversible losses in quantum systems by local counterdiabatic driving, Proc. Natl. Acad. Sci. USA 114, E3909 (2017).
  42. N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbrüggen, and S. J. Glaser, Optimal control of coupled spin dynamics: Design of NMR pulse sequences by gradient ascent algorithms, J. Magn. Reson. 172, 296 (2005).
  43. P. Doria, T. Calarco, and S. Montangero, Optimal control technique for many-body quantum dynamics, Phys. Rev. Lett. 106, 190501 (2011).
  44. T. Caneva, T. Calarco, and S. Montangero, Chopped random-basis quantum optimization, Phys. Rev. A 84, 022326 (2011).
  45. D. Braak, Integrability of the Rabi model, Phys. Rev. Lett. 107, 100401 (2011).
  46. S. Ashhab and F. Nori, Qubit-oscillator systems in the ultrastrong-coupling regime and their potential for preparing nonclassical states, Phys. Rev. A 81, 042311 (2010).
  47. C. Leroux, L. C. G. Govia, and A. A. Clerk, Simple variational ground state and pure-cat-state generation in the quantum Rabi model, Phys. Rev. A 96, 043834 (2017).
  48. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, Cambridge, 1997).
  49. J. Larson and T. Mavrogordatos, The JaynesCummings Model and its Descendants (Second Edition) (IOP Publishing, 2024).
  50. P. Forn-Díaz, G. Romero, C. J. P. M. Harmans, E. Solano, and J. E. Mooij, Broken selection rule in the quantum Rabi model, Sci. Rep. 6, 26720 (2016).
  51. Y. Wang, J. Zhang, C. Wu, J. Q. You, and G. Romero, Holonomic quantum computation in the ultrastrong-coupling regime of circuit QED, Phys. Rev. A 94, 012328 (2016).
  52. D. Ballester, G. Romero, J. J. García-Ripoll, F. Deppe, and E. Solano, Quantum simulation of the ultrastrong-coupling dynamics in circuit quantum electrodynamics, Phys. Rev. X 2, 021007 (2012).
  53. C. Leroux, L. C. G. Govia, and A. A. Clerk, Enhancing cavity quantum electrodynamics via antisqueezing: Synthetic ultrastrong coupling, Phys. Rev. Lett. 120, 093602 (2018).
  54. W. Qin, A. Miranowicz, P.-B. Li, X.-Y. Lü, J. Q. You, and F. Nori, Exponentially enhanced light-matter interaction, cooperativities, and steady-state entanglement using parametric amplification, Phys. Rev. Lett. 120, 093601 (2018).
  55. F. Petiziol, B. Dive, F. Mintert, and S. Wimberger, Fast adiabatic evolution by oscillating initial Hamiltonians, Phys. Rev. A 98, 043436 (2018).
  56. K. Takahashi and A. del Campo, Shortcuts to adiabaticity in Krylov space, Phys. Rev. X 14, 011032 (2024).
  57. Z. Yin, C. Li, J. Allcock, Y. Zheng, X. Gu, M. Dai, S. Zhang, and S. An, Shortcuts to adiabaticity for open systems in circuit quantum electrodynamics, Nat. Commun. 13, 188 (2022).
  58. F. Motzoi, J. M. Gambetta, P. Rebentrost, and F. K. Wilhelm, Simple pulses for elimination of leakage in weakly nonlinear qubits, Phys. Rev. Lett. 103, 110501 (2009).
  59. N. N. Hegade, K. Paul, Y. Ding, M. Sanz, F. Albarrán-Arriagada, E. Solano, and X. Chen, Shortcuts to adiabaticity in digitized adiabatic quantum computing, Phys. Rev. Appl. 15, 024038 (2021).
  60. J. Yao, L. Lin, and M. Bukov, Reinforcement learning for many-body ground-state preparation inspired by counterdiabatic driving, Phys. Rev. X 11, 031070 (2021).
  61. C. Mc Keever and M. Lubasch, Towards adiabatic quantum computing using compressed quantum circuits, PRX Quantum 5, 020362 (2024).
  62. J. Ferreiro-Vélez, I. Iriarte-Zendoia, Y. Ban, and X. Chen, Shortcuts for adiabatic and variational algorithms in molecular simulation, arXiv:2407.20957.
  63. T. Hatomura, Universal digitized counterdiabatic driving, arXiv:2601.15972.
  64. N. Ohga and T. Hatomura, Improving variational counterdiabatic driving with weighted actions and computer algebra, PRX Quantum 7, 020347 (2026).
  65. S. Morawetz and A. Polkovnikov, Efficient paths for local counterdiabatic driving, Phys. Rev. B 110, 024304 (2024).
  66. S. Ashhab, T. Fuse, F. Yoshihara, S. Kim, and K. Semba, Controlling qubit-oscillator systems using linear parameter sweeps, New J. Phys. 25, 093011 (2023).
  67. P. R. Hegde, G. Passarelli, A. Scocco, and P. Lucignano, Genetic optimization of quantum annealing, Phys. Rev. A 105, 012612 (2022).
  68. T. Hatomura, Shortcuts to adiabaticity in the infinite-range Ising model by mean-field counter-diabatic driving, J. Phys. Soc. Jpn. 86, 094002 (2017).
  69. J. R. Finžgar, S. Notarnicola, M. Cain, M. D. Lukin, and D. Sels, Counterdiabatic driving with performance guarantees, Phys. Rev. Lett. 135, 180602 (2025).
  70. S. Morawetz and A. Polkovnikov, Universal counterdiabatic driving in Krylov space, PRX Quantum 6, 040320 (2025).
  71. S. Alipour, A. Chenu, A. T. Rezakhani, and A. del Campo, Shortcuts to adiabaticity in driven open quantum systems: Balanced gain and loss and non-Markovian evolution, Quantum 4, 336 (2020).
  72. M. Jerger, F. Motzoi, Y. Gao, C. Dickel, L. Buchmann, F. A. Cárdenas-López, A. Bengtsson, G. Tancredi, C. W. Warren, J. Bylander, D. DiVincenzo, R. Barends, and P. A. Bushev, Dispersive qubit readout with intrinsic resonator reset, PRX Quantum (2026).
  73. M. Bukov, L. D’Alessio, and A. Polkovnikov, Universal high-frequency behavior of periodically driven systems: From dynamical stabilization to Floquet engineering, Adv. Phys. 64, 139 (2015).
  74. N. Goldman and J. Dalibard, Periodically driven quantum systems: Effective Hamiltonians and engineered gauge fields, Phys. Rev. X 4, 031027 (2014).
  75. T. Villazon, P. W. Claeys, A. Polkovnikov, and A. Chandran, Shortcuts to dynamic polarization, Phys. Rev. B 103, 075118 (2021).
  76. H. Ribeiro, A. Baksic, and A. A. Clerk, Systematic Magnus-based approach for suppressing leakage and nonadiabatic errors in quantum dynamics, Phys. Rev. X 7, 011021 (2017).
  77. F. Petiziol, B. Dive, S. Carretta, R. Mannella, F. Mintert, and S. Wimberger, Accelerating adiabatic protocols for entangling two qubits in circuit QED, Phys. Rev. A 99, 042315 (2019).
  78. E. Boyers, M. Pandey, D. K. Campbell, A. Polkovnikov, D. Sels, and A. O. Sushkov, Floquet-engineered quantum state manipulation in a noisy qubit, Phys. Rev. A 100, 012341 (2019).
  79. P. Forn-Díaz, J. J. García-Ripoll, B. Peropadre, J. L. Orgiazzi, M. A. Yurtalan, R. Belyansky, C. M. Wilson, and A. Lupascu, Ultrastrong coupling of a single artificial atom to an electromagnetic continuum in the nonperturbative regime, Nat. Phys. 13, 39 (2017).
  80. S. A. Caldwell et al., Parametrically activated entangling gates using transmon qubits, Phys. Rev. Appl. 10, 034050 (2018).
  81. X. Y. Jin, Z. Parrott, K. Cicak, S. Kotler, F. Lecocq, J. Teufel, J. Aumentado, E. Kapit, and R. W. Simmonds, Superconducting architecture demonstrating fast, tunable high-fidelity CZ gates with parametric control of ZZ coupling, Phys. Rev. Appl. 24, 064026 (2025).
  82. J. A. Valery, S. Chowdhury, G. Jones, and N. Didier, Dynamical sweet spot engineering via two-tone flux modulation of superconducting qubits, PRX Quantum 3, 020337 (2022).
  83. Y.-H. Shi et al., Quantum simulation of topological zero modes on a 41-qubit superconducting processor, Phys. Rev. Lett. 131, 080401 (2023).
  84. M. Xia, C. Lledó, M. Capocci, J. Repicky, B. D’Anjou, I. Mondragon-Shem, R. Kaufman, J. Koch, A. Blais, and M. Hatridge, Exceeding the parametric drive strength threshold in nonlinear circuits, arXiv:2506.03456.

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Phys. Rev. Research 8, 033244– Published 28 August, 2026
  • Received 27 May 2026
  • Accepted 27 July 2026
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