- 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

Effective Hamiltonian for an off-resonantly driven qubit-cavity system

Martin Jirlow*, Kunal Helambe, Axel M. Eriksson, Simone Gasparinetti, and Tahereh Abad†

  • *Contact author: martin.jirlow@chalmers.se
  • †Contact author: tahereh.abad@chalmers.se
PDF

Phys. Rev. A 114, 042605 – Published 6 October, 2026

DOI: https://doi.org/10.1103/y7xr-jq5w

Abstract

Accurate modeling of driven light-matter interactions is essential for quantum technologies, where natural and synthetic atoms are used to store and process quantum information, mediate interactions between bosonic modes, and enable nonlinear operations. In systems subject to multitone drives, however, the theoretical description becomes challenging and existing models cannot quantitatively reproduce the experimental data. Here, we derive an effective Hamiltonian that retains slowly rotating terms, providing a general framework for accurately describing driven dynamics across platforms. As a concrete application, we validate the theory in circuit QED, where it quantitatively reproduces experimentally measured ac Stark shifts and captures key interactions such as two-mode squeezing and beam splitting. Our results establish a broadly applicable tool to engineer driven interactions in quantum information processing platforms.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (58)

  1. A. Joshi, K. Noh, and Y. Y. Gao, Quantum information processing with bosonic qubits in circuit QED, Quantum Sci. Technol. 6, 033001 (2021).
  2. O. Milul, B. Guttel, U. Goldblatt, S. Hazanov, L. M. Joshi, D. Chausovsky, N. Kahn, E. Çiftyürek, F. Lafont, and S. Rosenblum, Superconducting cavity qubit with tens of milliseconds single-photon coherence time, PRX Quantum 4, 030336 (2023).
  3. A. Krasnok, P. Dhakal, A. Fedorov, P. Frigola, M. Kelly, and S. Kutsaev, Superconducting microwave cavities and qubits for quantum information systemss, Appl. Phys. Rev. 11, 011302 (2024).
  4. C. Flühmann, T. L. Nguyen, M. Marinelli, V. Negnevitsky, K. Mehta, and J. P. Home, Encoding a qubit in a trapped-ion mechanical oscillator, Nature (London) 566, 513 (2019).
  5. C. Flühmann and J. P. Home, Direct characteristic-function tomography of quantum states of the trapped-ion motional oscillator, Phys. Rev. Lett. 125, 043602 (2020).
  6. B. de Neeve, T. L. Nguyen, T. Behrle, and J. P. Home, Error correction of a logical grid state qubit by dissipative pumping, Nat. Phys. 18, 296 (2022).
  7. P. K. Shandilya, D. P. Lake, M. J. Mitchell, D. D. Sukachev, and P. E. Barclay, Optomechanical interface between telecom photons and spin quantum memory, Nat. Phys. 17, 1420 (2021).
  8. T. H. Haug, A. Frisk Kockum, and R. Van Laer, Heralding entangled optical photons from a microwave quantum processor, Phys. Rev. Appl. 22, 034068 (2024).
  9. M. Hofheinz, H. Wang, M. Ansmann, R. C. Bialczak, E. Lucero, M. Neeley, A. D. O'Connell, D. Sank, J. Wenner, J. M. Martinis, and A. N. Cleland, Synthesizing arbitrary quantum states in a superconducting resonator, Nature (London) 459, 546 (2009).
  10. M. Kudra, M. Kervinen, I. Strandberg, S. Ahmed, M. Scigliuzzo, A. Osman, D. P. Lozano, M. O. Tholén, R. Borgani, D. B. Haviland, G. Ferrini, J. Bylander, A. F. Kockum, F. Quijandría, P. Delsing, and S. Gasparinetti, Robust preparation of Wigner-negative states with optimized SNAP-displacement sequences, PRX Quantum 3, 030301 (2022).
  11. Z. Ni, S. Li, X. Deng, Y. Cai, L. Zhang, W. Wang, Z.-B. Yang, H. Yu, F. Yan, S. Liu, C.-L. Zou, L. Sun, S.-B. Zheng, Y. Xu, and D. Yu, Beating the break-even point with a discrete-variable-encoded logical qubit, Nature (London) 616, 56 (2023).
  12. R. W. Heeres, B. Vlastakis, E. Holland, S. Krastanov, V. V. Albert, L. Frunzio, L. Jiang, and R. J. Schoelkopf, Cavity state manipulation using photon-number selective phase gates, Phys. Rev. Lett. 115, 137002 (2015).
  13. J. M. Gertler, B. Baker, J. Li, S. Shirol, J. Koch, and C. Wang, Protecting a bosonic qubit with autonomous quantum error correction, Nature (London) 590, 243 (2021).
  14. P. Campagne-Ibarcq, E. Zalys-Geller, A. Narla, S. Shankar, P. Reinhold, L. Burkhart, C. Axline, W. Pfaff, L. Frunzio, R. J. Schoelkopf, and M. H. Devoret, Deterministic remote entanglement of superconducting circuits through microwave two-photon transitions, Phys. Rev. Lett. 120, 200501 (2018).
  15. M. Kudra, M. Jirlow, M. Kervinen, A. M. Eriksson, F. Quijandría, P. Delsing, T. Abad, and S. Gasparinetti, Experimental realization of deterministic and selective photon addition in a bosonic mode assisted by an ancillary qubit, Quantum Sci. Technol. 10, 045037 (2025).
  16. M. Mirrahimi, Z. Leghtas, V. V. Albert, S. Touzard, R. J. Schoelkopf, L. Jiang, and M. H. Devoret, Dynamically protected cat-qubits: A new paradigm for universal quantum computation, New J. Phys. 16, 045014 (2014).
  17. A. P. Saiko, R. Fedaruk, and S. A. Markevich, Kerr-like nonlinearities in an optomechanical system with an asymmetric anharmonic mechanical resonator, JETP Lett. 113, 487 (2021).
  18. P. Rabl, Photon blockade effect in optomechanical systems, Phys. Rev. Lett. 107, 063601 (2011).
  19. A. Nunnenkamp, K. Børkje, and S. M. Girvin, Single-photon optomechanics, Phys. Rev. Lett. 107, 063602 (2011).
  20. C. Flühmann, V. Negnevitsky, M. Marinelli, and J. P. Home, Sequential modular position and momentum measurements of a trapped ion mechanical oscillator, Phys. Rev. X 8, 021001 (2018).
  21. K. R. Motes, B. Q. Baragiola, A. Gilchrist, and N. C. Menicucci, Encoding qubits into oscillators with atomic ensembles and squeezed light, Phys. Rev. A 95, 053819 (2017).
  22. N. Ofek, A. Petrenko, R. Heeres, P. Reinhold, Z. Leghtas, B. Vlastakis, Y. Liu, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, Extending the lifetime of a quantum bit with error correction in superconducting circuits, Nature (London) 536, 441 (2016).
  23. L. Hu, Y. Ma, W. Cai, X. Mu, Y. Xu, W. Wang, Y. Wu, H. Wang, Y. P. Song, C.-L. Zou, S. M. Girvin, L.-M. Duan, and L. Sun, Quantum error correction and universal gate set operation on a binomial bosonic logical qubit, Nat. Phys. 15, 503 (2019).
  24. P. Campagne-Ibarcq, A. Eickbusch, S. Touzard, E. Zalys-Geller, N. E. Frattini, V. V. Sivak, P. Reinhold, S. Puri, S. Shankar, R. J. Schoelkopf, L. Frunzio, M. Mirrahimi, and M. H. Devoret, Quantum error correction of a qubit encoded in grid states of an oscillator, Nature (London) 584, 368 (2020).
  25. A. Grimm, N. E. Frattini, S. Puri, S. O. Mundhada, S. Touzard, M. Mirrahimi, S. M. Girvin, S. Shankar, and M. H. Devoret, Stabilization and operation of a Kerr-cat qubit, Nature (London) 584, 205 (2020).
  26. W. Cai, Y. Ma, W. Wang, C.-L. Zou, and L. Sun, Bosonic quantum error correction codes in superconducting quantum circuits, Fundam. Res. 1, 50 (2021).
  27. V. V. Sivak, A. Eickbusch, B. Royer, S. Singh, I. Tsioutsios, S. Ganjam, A. Miano, B. L. Brock, A. Z. Ding, L. Frunzio, S. M. Girvin, R. J. Schoelkopf, and M. H. Devoret, Real-time quantum error correction beyond break-even, Nature (London) 616, 50 (2023).
  28. D. Lachance-Quirion, M.-A. Lemonde, J. O. Simoneau, L. St-Jean, P. Lemieux, S. Turcotte, W. Wright, A. Lacroix, J. Fréchette-Viens, R. Shillito, F. Hopfmueller, M. Tremblay, N. E. Frattini, J. Camirand Lemyre, and P. St-Jean, Autonomous quantum error correction of Gottesman-Kitaev-Preskill states, Phys. Rev. Lett. 132, 150607 (2024).
  29. M. Jirlow, martin-jw/late-RWA-Hamiltonian (Version 1.0.0) [Computer software], Zenodo, 2026, https://doi.org/10.5281/zenodo.22304140.
  30. See Supplemental Material at http://link.aps.org/supplemental/10.1103/y7xr-jq5w for system parameters, derivations, and additional simulation results, which includes Refs. [31, 32].
  31. M. Malekakhlagh, A. Petrescu, and H. E. Türeci, Lifetime renormalization of weakly anharmonic superconducting qubits. I. Role of number nonconserving terms, Phys. Rev. B 101, 134509 (2020).
  32. H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, London, 2002).
  33. B. J. Chapman, S. J. De Graaf, S. H. Xue, Y. Zhang, J. Teoh, J. C. Curtis, T. Tsunoda, A. Eickbusch, A. P. Read, A. Koottandavida, S. O. Mundhada, L. Frunzio, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, High-on-off-ratio beam-splitter interaction for gates on bosonically encoded qubits, PRX Quantum 4, 020355 (2023).
  34. S. Rosenblum, Y. Y. Gao, P. Reinhold, C. Wang, C. J. Axline, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, A CNOT gate between multiphoton qubits encoded in two cavities, Nat. Commun. 9, 652 (2018).
  35. X. You, Y. Lu, T. Kim, D. M. Kürkçüolu, S. Zhu, D. Van Zanten, T. Roy, Y. Lu, S. Chakram, A. Grassellino, A. Romanenko, J. Koch, and S. Zorzetti, Crosstalk-robust quantum control in multimode bosonic systems, Phys. Rev. Appl. 22, 044072 (2024).
  36. Y. Y. Gao, B. J. Lester, K. S. Chou, L. Frunzio, M. H. Devoret, L. Jiang, S. M. Girvin, and R. J. Schoelkopf, Entanglement of bosonic modes through an engineered exchange interaction, Nature (London) 566, 509 (2019).
  37. J. Landgraf, C. Flühmann, T. Fösel, F. Marquardt, and R. J. Schoelkopf, Fast quantum control of cavities using an improved protocol without coherent errors, Phys. Rev. Lett. 133, 260802 (2024).
  38. S. Li, Z. Ni, L. Zhang, Y. Cai, J. Mai, S. Wen, P. Zheng, X. Deng, S. Liu, et al., Autonomous stabilization of Fock states in an oscillator against multiphoton losses, Phys. Rev. Lett. 132, 203602 (2024).
  39. S. Shirol, S. van Geldern, H. Xi, and C. Wang, Passive quantum error correction of photon loss at breakeven Phys. Rev. X 16, 021042 (2026).
  40. C.-H. Wang, K. Noh, J. Lebreuilly, S. M. Girvin, and L. Jiang, Photon-number-dependent Hamiltonian engineering for cavities, Phys. Rev. Appl. 15, 044026 (2021).
  41. V. Maurya, H. Zhang, D. Kowsari, A. Kuo, D. M. Hartsell, C. Miyamoto, J. Liu, S. Shanto, E. Vlachos, A. Zarassi, K. W. Murch, and E. M. Levenson-Falk, On-demand driven dissipation for cavity reset and cooling, PRX Quantum 5, 020321 (2024).
  42. Y. Zhang, J. C. Curtis, C. S. Wang, R. J. Schoelkopf, and S. M. Girvin, Drive-induced nonlinearities of cavity modes coupled to a transmon ancilla, Phys. Rev. A 105, 022423 (2022).
  43. J. Huang, T. J. DiNapoli, G. Rockwood, M. Yuan, P. Narasimhan, E. Gupta, M. Bal, F. Crisa, S. Garattoni, Y. Lu, L. Jiang, and S. Chakram, Fast sideband control of a multimode cavity memory with weak dispersive coupling to a transmon, Phys. Rev. X 16, 011058 (2026).
  44. J. Koch, V. Manucharyan, M. H. Devoret, and L. I. Glazman, Charging effects in the inductively shunted Josephson junction, Phys. Rev. Lett. 103, 217004 (2009).
  45. V. E. Manucharyan, J. Koch, L. I. Glazman, and M. H. Devoret, Fluxonium: Single Cooper-pair circuit free of charge offsets, Science 326, 113 (2009).
  46. L. B. Nguyen, Y.-H. Lin, A. Somoroff, R. Mencia, N. Grabon, and V. E. Manucharyan, High-coherence fluxonium qubit, Phys. Rev. X 9, 041041 (2019).
  47. K. Nie, J. N. Bradford, S. Mandal, A. Bista, W. Pfaff, and A. Kou, Fluxonium as a control qubit for bosonic quantum information, PRX Quantum 7, 010357 (2026).
  48. J. M. Martinis, Superconducting phase qubits, Quantum Inf. Process. 8, 81 (2009).
  49. T. P. Orlando, J. E. Mooij, L. Tian, C. van der Wal, L. Levitov, S. Lloyd, and J. J. Mazo, Superconducting persistent-current qubit, Phys. Rev. B 60, 15398 (1999).
  50. J. E. Mooij, T. P. Orlando, L. Levitov, L. Tian, C. H. van der Wal, and S. Lloyd, Josephson persistent-current qubit, Science 285, 1036 (1999).
  51. J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schuster, J. Majer, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Charge-insensitive qubit design derived from the Cooper pair box, Phys. Rev. A 76, 042319 (2007).
  52. P. Magnard, P. Kurpiers, B. Royer, T. Walter, J.-C. Besse, S. Gasparinetti, M. Pechal, J. Heinsoo, S. Storz, A. Blais, and A. Wallraff, Fast and unconditional all-microwave reset of a superconducting qubit, Phys. Rev. Lett. 121, 060502 (2018).
  53. J. Yang, M. Khanahmadi, I. Strandberg, A. Gaikwad, C. Castillo-Moreno, A. Frisk Kockum, M. A. Ullah, G. Johansson, A. M. Eriksson, and S. Gasparinetti, Deterministic generation of frequency-bin-encoded microwave photons, Phys. Rev. Lett. 134, 240803 (2025).
  54. J. Yang, A. M. Eriksson, M. A. Aamir, I. Strandberg, C. Castillo-Moreno, D. Perez Lozano, P. Persson, and S. Gasparinetti, Deterministic generation of shaped single microwave photons using a parametrically driven coupler, Phys. Rev. Appl. 20, 054018 (2023).
  55. 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).
  56. 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).
  57. C. R. Harris, K. J. Millman, S. J. Van Der Walt, R. Gommers, P. Virtanen, D. Cournapeau, E. Wieser, J. Taylor, S. Berg, N. J. Smith, R. Kern, M. Picus, S. Hoyer, M. H. Van Kerkwijk, M. Brett, A. Haldane, J. F. Del Río, M. Wiebe, P. Peterson, P. Gérard-Marchant, et al., Array programming with NumPy, Nature (London) 585, 357 (2020).
  58. J. D. Hunter, Matplotlib: A 2D graphics environment, Comput. Sci. Eng. 9, 90 (2007).

Outline

Information

Phys. Rev. A 114, 042605– Published 6 October, 2026
  • Received 8 October 2025
  • Revised 1 April 2026
  • Accepted 2 September 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. Funded by Bibsam.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation