- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

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
  • Editors' Suggestion
  • Open Access
  • Access by Xinjiang University

Reconciliation of effective Hamiltonians for intense light-matter interaction

Jakob Nicolai Bruhnke and Jan Marcus Dahlström*

  • *Contact author: marcus.dahlstrom@fysik.lu.se
PDF

Phys. Rev. A 114, 033111 – Published 16 September, 2026

DOI: https://doi.org/10.1103/jkmy-8zwr

Abstract

Essential-state models are central for quantum control and technology in broad regimes of light-matter interaction. The canonical effective Hamiltonian is obtained equivalently from adiabatic elimination, the Markov approximation, and the pole approximation. These approximations are known to break down at high intensities, significantly limiting their applicability to moderate light-matter interaction. We show how this limitation can be addressed by applying quasidegenerate Rayleigh-Schrödinger perturbation theory (QDRSPT). We reconcile QDRSPT with adiabatic elimination and propose a quasidegenerate extension of adiabatic elimination that is robust when the detuning of the essential states is non-negligible. The accuracy of QDRSPT is demonstrated in both the low- and high-frequency regimes, showing excellent agreement with Floquet calculations at high intensities. The crucial corrections to adiabatic elimination make the eigenvectors of the effective Hamiltonian nonorthogonal. Physically, this allows us to account for the asymmetric strength with which different essential states couple to the nonessential states. We expect that our systematic approach to effective Hamiltonians from QDRSPT will constitute a new state of the art in intense light-matter interaction and quantum optics with novel forms of strong coupling and quantum control phenomena being conceivable.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (105)

  1. M. Wollenhaupt, A. Assion, O. Bazhan, C. Horn, D. Liese, C. Sarpe-Tudoran, M. Winter, and T. Baumert, Control of interferences in an Autler-Townes doublet: Symmetry of control parameters, Phys. Rev. A 68, 015401 (2003).
  2. T. Bayer, D. Gräfing, S. Kerbstadt, D. Pengel, K. Eickhoff, L. Englert, and M. Wollenhaupt, Time-resolved 3D imaging of ultrafast spin–orbit wave packet dynamics, New J. Phys. 21, 033001 (2019).
  3. W. Li, J. Zhang, Y. Jin, D. Zhang, D. Ding, and K. Ueda, Partial-wave resolved spin-orbit dynamics, Phys. Rev. Lett. 135, 143201 (2025).
  4. L. Zhang, S. Carlström, O. Smirnova, M. Ivanov, and D. Ye, Spin polarization in strong-field ionization as sensor of trapped electron orbits, Phys. Rev. Lett. 135, 193201 (2025).
  5. M. Anand, S. Pabst, O. Kwon, and D. E. Kim, Attosecond counter-rotating-wave effect in xenon driven by strong fields, Phys. Rev. A 95, 053420 (2017).
  6. M. Bertolino, S. Carlström, J. Peschel, F. Zapata, E. Lindroth, and J. M. Dahlström, Thomas-Reiche-Kuhn correction for truncated configuration-interaction spaces: Case of laser-assisted dynamical interference, Phys. Rev. A 106, 043108 (2022).
  7. F. Vismarra et al., Dynamic interference of chirped photoelectrons, Phys. Rev. Lett. 135, 033202 (2025).
  8. M. Li, M.-F. Xie, H. Wang, L. Jia, J. Li, W. Wang, J. Cai, X. Hong, X. Shi, Y. Lv, X. Zhao, S. Luo, W.-C. Jiang, L.-Y. Peng, and D. Ding, Observation of laser-assisted dynamic interference by attosecond controlled photoelectron spectroscopy, Phys. Rev. Lett. 133, 253201 (2024).
  9. A. Tóth and A. Csehi, Probing strong-field two-photon transitions through dynamic interference, J. Phys. B 54, 035005 (2021).
  10. M. Ferray, A. L'Huillier, X. F. Li, L. A. Lompre, G. Mainfray, and C. Manus, Multiple-harmonic conversion of 1064 nm radiation in rare gases, J. Phys. B 21, L31 (1988).
  11. F. Krausz and M. Ivanov, Attosecond physics, Rev. Mod. Phys. 81, 163 (2009).
  12. P. Emma et al., First lasing and operation of an ångstrom-wavelength free-electron laser, Nat. Photon. 4, 641 (2010).
  13. E. Allaria et al., Highly coherent and stable pulses from the FERMI seeded free-electron laser in the extreme ultraviolet, Nat. Photon. 6, 699 (2012).
  14. N. S. Mirian et al., Generation and measurement of intense few-femtosecond superradiant extreme-ultraviolet free-electron laser pulses, Nat. Photon. 15, 523 (2021).
  15. S. Nandi et al., Observation of Rabi dynamics with a short-wavelength free-electron laser, Nature (London) 608, 488 (2022).
  16. S. Nandi et al., Generation of entanglement using a short-wavelength seeded free-electron laser, Sci. Adv. 10, eado0668 (2024).
  17. F. Richter et al., Strong-field quantum control in the extreme ultraviolet domain using pulse shaping, Nature (London) 636, 337 (2024).
  18. E. Olofsson and J. M. Dahlström, Photoelectron signature of dressed-atom stabilization in an intense XUV field, Phys. Rev. Res. 5, 043017 (2023).
  19. E. Olofsson, E. L. Fulton, R. Tahouri, M. Bertolino, J. M. N. Djiokap, and J. M. Dahlström, Coherent control of ionization via stabilization by resonant pulse pairs, Phys. Rev. Res. 8, 013199 (2026).
  20. M. Dörr, O. Latinne, and C. J. Joachain, Time evolution of two-photon population transfer between the 1s and 2s states of a hydrogen atom, Phys. Rev. A 55, 3697 (1997).
  21. L. P. Yatsenko, B. W. Shore, T. Halfmann, K. Bergmann, and A. Vardi, Source of metastable H(2s) atoms using the Stark chirped rapid-adiabatic-passage technique, Phys. Rev. A 60, R4237 (1999).
  22. J. N. Bruhnke, E. Olofsson, A. Stenquist, and J. M. Dahlström, Giant counter-rotating oscillations on the attosecond timescale, Phys. Rev. Res. 7, L042019 (2025).
  23. S. Kumar, J. D. Piel, C. H. Greene, and N. Shivaram, Broadband femtosecond lasers enable efficient two-photon excitation of the ultranarrow linewidth singlet 1s2s state in helium, Phys. Rev. Res. 8, 013009 (2026).
  24. T. M. Linker et al., Attosecond inner-shell lasing at ångström wavelengths, Nature (London) 642, 934 (2025).
  25. F. H. M. Faisal, Theory of Multiphoton Processes (Springer, New York, 1987).
  26. B. W. Shore, The Theory of Coherent Atomic Excitation (Wiley-VCH, New York, 1990).
  27. C. Cohen-Tannoudji, G. Grynberg, and J. Dupont-Roc, Atom-Photon Interactions: Basic Processes and Applications (Wiley-VCH, New York, 1998).
  28. L. Huang, J. Luneau, J. Schirk, F. Wallner, C. M. F. Schneider, S. Filipp, K. Liegener, and P. Rabl, Theory of multiphoton processes for applications in quantum control Phys. Rev. A 113, 032620 (2026).
  29. N. V. Vitanov and S. Stenholm, Population transfer via a decaying state, Phys. Rev. A 56, 1463 (1997).
  30. A. Kinos, L. Rippe, S. Kröll, and A. Walther, Designing gate operations for single-ion quantum computing in rare-earth-ion-doped crystals, Phys. Rev. A 104, 052624 (2021).
  31. J. Kumlin, C. Braun, C. Tresp, N. Stiesdal, S. Hofferberth, and A. Paris-Mandoki, Quantum optics with Rydberg superatoms, J. Phys. Commun. 7, 052001 (2023).
  32. V. Paulisch, H. Rui, H. K. Ng, and B.-G. Englert, Beyond adiabatic elimination: A hierarchy of approximations for multi-photon processes, Eur. Phys. J. Plus 129, 12 (2014).
  33. E. Brion, L. H. Pedersen, and K. Mølmer, Adiabatic elimination in a lambda system, J. Phys. A: Math. Theor. 40, 1033 (2007).
  34. H. C. Baker, Non-Hermitian quantum theory of multiphoton ionization, Phys. Rev. A 30, 773 (1984).
  35. B. T. Torosov and N. V. Vitanov, Adiabatic elimination of a nearly resonant quantum state, J. Phys. B 45, 135502 (2012).
  36. M. Sanz, E. Solano, and Í. L. Egusquiza, in Applications + Practical Conceptualization + Mathematics = Fruitful Innovation, edited by R. S. Anderssen, P. Broadbridge, Y. Fukumoto, K. Kajiwara, T. Takagi, E. Verbitskiy, and M. Wakayama, Mathematics for Industry Vol. 11 (Springer, Tokyo, 2016), pp. 127–142.
  37. K. N. Zlatanov, G. S. Vasilev, and N. V. Vitanov, Morris-Shore transformation for nondegenerate systems, Phys. Rev. A 102, 063113 (2020).
  38. I. Lindgren and J. Morrison, Atomic Many-Body Theory, edited by V. I. Goldanskii, R. Gomer, F. P. Schäfer, and J. P. Toennies, Springer Series in Chemical Physics Vol. 13 (Springer, Berlin, 1982).
  39. D. F. James and J. Jerke, Effective Hamiltonian theory and its applications in quantum information, Can. J. Phys. 85, 625 (2007).
  40. O. Gamel and D. F. V. James, Time-averaged quantum dynamics and the validity of the effective Hamiltonian model, Phys. Rev. A 82, 052106 (2010).
  41. N. Macrì, L. Giannelli, E. Paladino, and G. Falci, Coarse-grained effective Hamiltonian via the Magnus expansion for a three-level system, Entropy 25, 234 (2023).
  42. K. D. Barajas and W. C. Campbell, Quantum gate dynamics beyond the rotating wave approximation using multitimescale quantum averaging theory, Phys. Rev. A 113, L030403 (2026).
  43. K. D. Barajas and W. C. Campbell, Quantum averaging theory for multi-timescale driven quantum systems, Phys. Rev. A 113, 032210 (2026).
  44. J. R. Schrieffer and P. A. Wolff, Relation between the Anderson and Kondo Hamiltonians, Phys. Rev. 149, 491 (1966).
  45. I. Shavitt and L. T. Redmon, Quasidegenerate perturbation theories. A canonical van Vleck formalism and its relationship to other approaches, J. Chem. Phys. 73, 5711 (1980).
  46. L. S. Cederbaum, J. Schirmer, and H.-D. Meyer, Block diagonalisation of Hermitian matrices, J. Phys. A: Math. Gen. 22, 2427 (1989).
  47. A. B. Klimov and L. L. Sanchez-Soto, Method of small rotations and effective Hamiltonians in nonlinear quantum optics, Phys. Rev. A 61, 063802 (2000).
  48. A. B. Klimov, L. L. Sánchez-Soto, A. Navarro, and E. C. Yustas, Effective Hamiltonians in quantum optics: A systematic approach, J. Mod. Opt. 49, 2211 (2002).
  49. J. R. Morris and B. W. Shore, Reduction of degenerate two-level excitation to independent two-state systems, Phys. Rev. A 27, 906 (1983).
  50. L. G. Hanson, J. Zhang, and P. Lambropoulos, Manifestations of atomic and core resonances in photoelectron energy spectra, Phys. Rev. A 55, 2232 (1997).
  51. L. P. Yatsenko, V. I. Romanenko, B. W. Shore, T. Halfmann, and K. Bergmann, Two-photon excitation of the metastable 2s state of hydrogen assisted by laser-induced chirped Stark shifts and continuum structure, Phys. Rev. A 71, 033418 (2005).
  52. X. Zhang, Y. Zhou, Y. Liao, Y. Chen, J. Liang, Q. Ke, M. Li, A. Csehi, and P. Lu, Effect of nonresonant states in near-resonant two-photon ionization of hydrogen, Phys. Rev. A 106, 063114 (2022).
  53. A. Tóth, S. Borbély, Y. Zhou, and A. Csehi, Role of dynamic Stark shifts in strong-field excitation and subsequent ionization, Phys. Rev. A 107, 053101 (2023).
  54. S.-I. Chu, in Advances in Atomic and Molecular Physics, edited by D. R. Bates and B. Bederson (Academic, New York, 1985), Vol. 21, pp. 197–253.
  55. W. Schweizer, P. Faßbinder, and R. González-Férez, Model potentials for alkali metal atoms and Li-like ions, At. Data Nucl. Data Tables 72, 33 (1999).
  56. J. B. Foresman, M. Head-Gordon, J. A. Pople, and M. J. Frisch, Toward a systematic molecular orbital theory for excited states, J. Phys. Chem. 96, 135 (1992).
  57. A. Dreuw and M. Head-Gordon, Single-reference ab initio methods for the calculation of excited states of large molecules, Chem. Rev. 105, 4009 (2005).
  58. I. Lindgren, The Rayleigh-Schrodinger perturbation and the linked-diagram theorem for a multi-configurational model space, J. Phys. B 7, 2441 (1974).
  59. P. Durand, Direct determination of effective Hamiltonians by wave-operator methods. I. General formalism, Phys. Rev. A 28, 3184 (1983).
  60. C. Brouder, G. H. Duchamp, F. Patras, and G. Z. Tóth, The Rayleigh-Schrödinger perturbation series of quasi-degenerate systems, Int. J. Quantum Chem. 112, 2256 (2012).
  61. K. Suzuki and R. Okamoto, Degenerate perturbation theory in quantum mechanics, Prog. Theor. Phys. 70, 439 (1983).
  62. B. H. Brandow, Formal theory of effective π-electron Hamiltonians, Int. J. Quantum Chem. 15, 207 (1979).
  63. J. P. Killingbeck and G. Jolicard, The Bloch wave operator: Generalizations and applications: Part I. The time-independent case, J. Phys. A: Math. Gen. 36, R105 (2003).
  64. C. M. Bender and T. T. Wu, Anharmonic oscillator. II. A study of perturbation theory in large order, Phys. Rev. D 7, 1620 (1973).
  65. E. R. Vrscay and J. Cizek, Continued fractions and Rayleigh–Schrödinger perturbation theory at large order, J. Math. Phys. 27, 185 (1986).
  66. U. D. Jentschura, Resummation of the divergent perturbation series for a hydrogen atom in an electric field, Phys. Rev. A 64, 013403 (2001).
  67. J. H. Shirley, Solution of the Schrödinger equation with a Hamiltonian periodic in time, Phys. Rev. 138, B979 (1965).
  68. S. Guérin, F. Monti, J.-M. Dupont, and H. R. Jauslin, On the relation between cavity-dressed states, Floquet states, RWA and semiclassical models, J. Phys. A: Math. Gen. 30, 7193 (1997).
  69. M. Lewenstein, M. F. Ciappina, E. Pisanty, J. Rivera-Dean, P. Stammer, T. Lamprou, and P. Tzallas, Generation of optical Schrödinger cat states in intense laser–matter interactions, Nat. Phys. 17, 1104 (2021).
  70. P. Stammer, J. Rivera-Dean, A. Maxwell, T. Lamprou, A. Ordóñez, M. F. Ciappina, P. Tzallas, and M. Lewenstein, Quantum electrodynamics of intense laser-matter interactions: A tool for quantum state engineering, PRX Quantum 4, 010201 (2023).
  71. N. Tsatrafyllis, I. K. Kominis, I. A. Gonoskov, and P. Tzallas, High-order harmonics measured by the photon statistics of the infrared driving-field exiting the atomic medium, Nat. Commun. 8, 15170 (2017).
  72. A. Gorlach, O. Neufeld, N. Rivera, O. Cohen, and I. Kaminer, The quantum-optical nature of high harmonic generation, Nat. Commun. 11, 4598 (2020).
  73. R. V. Gothelf, C. S. Lange, and L. B. Madsen, High-order harmonic generation in a crystal driven by quantum light, Phys. Rev. A 111, 063105 (2025).
  74. A. Stenquist, J. N. Bruhnke, F. Zapata, and J. M. Dahlström, Entanglement transfer in a composite electron–ion–photon system, Rep. Prog. Phys. 88, 080502 (2025).
  75. S.-I. Chu and D. A. Telnov, Beyond the Floquet theorem: Generalized Floquet formalisms and quasienergy methods for atomic and molecular multiphoton processes in intense laser fields, Phys. Rep. 390, 1 (2004).
  76. C. J. Joachain, N. J. Kylstra, and R. M. Potvliege, Atoms in Intense Laser Fields (Cambridge University Press, Cambridge, 2011).
  77. S.-I. Chu, Quasienergy formalism for intense field multiphoton ionization of atoms induced by circularly polarized radiation, Chem. Phys. Lett. 54, 367 (1978).
  78. T.-S. Ho, S.-I. Chu, and J. V. Tietz, Semiclassical many-mode Floquet theory, Chem. Phys. Lett. 96, 464 (1983).
  79. S.-I. Chu and W. P. Reinhardt, Intense field multiphoton ionization via complex dressed states: Application to the H atom, Phys. Rev. Lett. 39, 1195 (1977).
  80. A. Maquet, S.-I. Chu, and W. P. Reinhardt, Stark ionization in dc and ac fields: An L2 complex-coordinate approach, Phys. Rev. A 27, 2946 (1983).
  81. B. Simon, The definition of molecular resonance curves by the method of exterior complex scaling, Phys. Lett. 71A, 211 (1979).
  82. R. M. Potvliege and R. Shakeshaft, Time-independent theory of multiphoton ionization of an atom by an intense field, Phys. Rev. A 38, 4597 (1988).
  83. N. Moiseyev, Non-Hermitian Quantum Mechanics (Cambridge University Press, Cambridge, 2011).
  84. C. R. Holt, M. G. Raymer, and W. P. Reinhardt, Time dependences of two-, three-, and four-photon ionization of atomic hydrogen in the ground 12S and metastable 22S states, Phys. Rev. A 27, 2971 (1983).
  85. C. Buth, R. Santra, and L. S. Cederbaum, Non-Hermitian Rayleigh-Schrödinger perturbation theory, Phys. Rev. A 69, 032505 (2004).
  86. K. J. Schafer, B. Yang, L. F. DiMauro, and K. C. Kulander, Above threshold ionization beyond the high harmonic cutoff, Phys. Rev. Lett. 70, 1599 (1993).
  87. J. B. Watson, A. Sanpera, D. G. Lappas, P. L. Knight, and K. Burnett, Nonsequential double ionization of helium, Phys. Rev. Lett. 78, 1884 (1997).
  88. H. G. Muller and F. C. Kooiman, Bunching and focusing of tunneling wave packets in enhancement of high-order above-threshold ionization, Phys. Rev. Lett. 81, 1207 (1998).
  89. P. Virtanen et al., SciPy 1.0: Fundamental algorithms for scientific computing in Python, Nat. Methods 17, 261 (2020).
  90. B. L. Beers and L. Armstrong, Exact solution of a realistic model for two-photon ionization, Phys. Rev. A 12, 2447 (1975).
  91. J. N. Bruhnke, Code for the paper "Reconciliation of effective Hamiltonians for intense light-matter interaction", Dataset: Zenodo, Geneva, 2026, doi: https://zenodo.org/records/22009587.
  92. J. N. Bruhnke and J. M. Dahlström, Quantitative analysis of resonant ionization by smooth laser pulses: Connection between effective Hamiltonian theory and strong-field dressed continua, arXiv:2606.21526.
  93. M. Dörr, R. M. Potvliege, and R. Shakeshaft, Multiphoton processes in an intense laser field: III. Resonant ionization of hydrogen by subpicosecond pulses, Phys. Rev. A 41, 558 (1990).
  94. K. Suzuki and R. Okamoto, Effective operators in time-independent approach, Prog. Theor. Phys. 93, 905 (1995).
  95. A. Gorlach, M. E. Tzur, M. Birk, M. Krüger, N. Rivera, O. Cohen, and I. Kaminer, High-harmonic generation driven by quantum light, Nat. Phys. 19, 1689 (2023).
  96. J. M. González-Monge and J. Feist, High-harmonic generation driven by temporal-mode quantum states of light, arXiv:2512.06602.
  97. C. S. Lange and L. B. Madsen, Hierarchy of approximations for describing quantum light from high-harmonic generation: A Fermi-Hubbard-model study, Phys. Rev. A 111, 013113 (2025).
  98. C. Bloch and J. Horowitz, Sur la détermination des premiers états d'un système de fermions dans le cas dégénéré, Nucl. Phys. 8, 91 (1958).
  99. H. Feshbach, A unified theory of nuclear reactions. II, Ann. Phys. (NY) 19, 287 (1962).
  100. P.-O. Löwdin, A note on the quantum-mechanical perturbation theory, J. Chem. Phys. 19, 1396 (1951).
  101. L. G. Hanson, J. Zhang, and P. Lambropoulos, Theory of core-resonant ionization, Europhys. Lett. 30, 81 (1995).
  102. G. A. Voth and R. A. Marcus, Adiabatically reduced coupled equations for intramolecular dynamics calculations, J. Chem. Phys. 84, 2254 (1986).
  103. L. Allen and C. R. Stroud, Broadening and saturation in n-photon absorption, Phys. Rep. 91, 1 (1982).
  104. E. M. Krenciglowa and T. T. S. Kuo, Convergence of effective Hamiltonian expansion and partial summations of folded diagrams, Nucl. Phys. A 235, 171 (1974).
  105. K. Takayanagi, Effective Hamiltonian in the extended Krenciglowa–Kuo method, Nucl. Phys. A 864, 91 (2011).

Outline

Information

Phys. Rev. A 114, 033111– Published 16 September, 2026
  • Received 6 May 2026
  • Accepted 20 August 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