- 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

Exact analytic characterization of the Autler-Townes doublet via the extended semiclassical Rabi model

András Csehi*

  • Department of Theoretical Physics, Faculty of Science and Technology, University of Debrecen, H-4002 Debrecen, PO Box 400, Hungary

  • *Contact author: csehi.andras@science.unideb.hu
PDF

Phys. Rev. A 113, 033121 – Published 25 March, 2026

DOI: https://doi.org/10.1103/3gz9-s6hv

Abstract

We study the resonance-enhanced multiphoton ionization (REMPI) of atoms by intense, short, XUV radiation fields. To characterize Rabi oscillations and the resulting Autler-Townes doublet in the photoelectron spectrum, a minimal-state effective description is utilized, incorporating energy-level shifts, multiphoton couplings, and irreversible losses. Extending the semiclassical Rabi model with such higher-order terms, the complex population amplitudes of the Rabi cycling atom are found in simple analytic forms. This allows for the exact analytic evaluation of the photoelectron spectrum under various simultaneous physical effects. The origins of the spectral characteristics, such as the splitting, shifting, asymmetry, and multipeak pattern, are identified through the found spectrum formula. Furthermore, the dependence of these features on the field intensity, frequency detuning, and interaction time are analyzed in detail. Finally, as the simplicity of the spectrum formula is preserved when further higher-order light-matter interaction terms are added, the presented general framework is suitable for studying arbitrary (n+m) photon REMPI phenomena.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (67)

  1. I. I. Rabi, Space quantization in a gyrating magnetic field, Phys. Rev. 51, 652 (1937).
  2. S. H. Autler and C. H. Townes, Stark effect in rapidly varying fields, Phys. Rev. 100, 703 (1955).
  3. R. Merlin, Rabi oscillations, Floquet states, Fermi's golden rule, and all that: Insights from an exactly solvable two-level model, Am J. Phys. 89, 26 (2021).
  4. D. Rogus and M. Lewenstein, Resonant ionisation by smooth laser pulses, J. Phys. B: At. Mol. Phys. 19, 3051 (1986).
  5. A. K. Kazansky and D. A. Telnov, Interference effects in the laser-pulse-induced autoionization spectrum of an atom, Sov. Phys.- JETP 67, 253 (1988).
  6. M. V. Fedorov and A. E. Kazakov, Resonances and saturation in multiphoton bound-free transitions, Prog. Quantum Electron. 13, 1 (1989).
  7. K. J. LaGattuta, Above-threshold ionization of atomic hydrogen via resonant intermediate states, Phys. Rev. A 47, 1560 (1993).
  8. C. Meier and V. Engel, Interference structure in the photoelectron spectra obtained from multiphoton ionization of Na2 with a strong femtosecond laser pulse, Phys. Rev. Lett. 73, 3207 (1994).
  9. M. G. Girju, K. Hristov, O. Kidun, and D. Bauer, Nonperturbative resonant strong field ionization of atomic hydrogen, J. Phys. B: At. Mol. Opt. Phys. 40, 4165 (2007).
  10. E. Allaria, R. Appio, L. Badano, W. A. Barletta, S. Bassanese, et al., Highly coherent and stable pulses from the FERMI seeded free-electron laser in the extreme ultraviolet, Nat. Photonics 6, 699 (2012).
  11. M. Flögel, J. Durá, B. Schütte, M. Ivanov, A. Rouzée, and M. J. J. Vrakking, Rabi oscillations in extreme ultraviolet ionization of atomic argon, Phys. Rev. A 95, 021401(R) (2017).
  12. S. Nandi, E. Olofsson, M. Bertolino, et al., Observation of Rabi dynamics with a short-wavelength free-electron laser, Nature (London) 608, 488 (2022).
  13. F. Richter, U. Saalmann, and E. Allaria, Strong-field quantum control in the extreme ultraviolet domain using pulse shaping, Nature (London) 636, 337 (2024).
  14. B. Major, O. Ghafur, K. Kovács, K. Varjú, V. Tosa, M. J. J. Vrakking, and B. Schütte, Compact intense extreme-ultraviolet source, Optica 8, 960 (2021).
  15. J. J. Cui, Y. Cheng, X. Wang, Z. Li, N. Rohringer, V. Kimberg, and S. B. Zhang, Proposal for observing XUV-induced Rabi oscillation using superfluorescent emission, Phys. Rev. Lett. 131, 043201 (2023).
  16. A. de las Heras, C. Hernández-García, J. Serrano, A. Prodanov, D. Popmintchev, T. Popmintchev, and L. Plaja, Attosecond Rabi oscillations in high harmonic generation resonantly driven by extreme ultraviolet laser fields, Phys. Rev. Res. 7, 023268 (2025).
  17. D. He, B. Shen, Y. Mao, Z. Zhang, M. He, Y. Li, and F. He, Quantum control of Rabi dynamics using chirped extreme ultraviolet pulses, New J. Phys. 27, 113204 (2025).
  18. Y.-J. Mao, H.-B. Yao, M. He, Z.-H. Zhang, Y. Li, and F. He, Unveiling Rabi dynamics through angle-resolved photoelectron momentum distributions using an ω−2ω pulse pair, Phys. Rev. A 108, 053117 (2023).
  19. L. Guangyin, J. Weichao, and P. Liangyou, Photoionization of atoms and molecules by strong XUV pulses, Acta Opt. Sin. 44, 1732001 (2024).
  20. S. Nandi, A. Stenquist, A. Papoulia, E. Olofsson, L. Badano, M. Bertolino, D. Busto, C. Callegari, S. Carlström, and M. B. Danailov, Generation of entanglement using a short-wavelength seeded free-electron laser, Sci. Adv. 10, eado0668 (2024).
  21. A. Stenquist and J. M. Dahlström, Harnessing time symmetry to fundamentally alter entanglement in photoionization, Phys. Rev. Res. 7, 013270 (2025).
  22. K. L. Ishikawa, K. C. Prince, and K. Ueda, Control of ion-photoelectron entanglement and coherence via Rabi oscillations, J. Phys. Chem. A 127, 10638 (2023).
  23. B.-R. Shen, Y.-J. Mao, Z.-H. Zhang, Y. Li, T. Sato, K. L. Ishikawa, and F. He, Coherent control of ion-photoelectron dynamics through Rabi oscillations: An ab initio study, Phys. Rev. A 111, 063113 (2025).
  24. L. Rico, M. Berkane, J. Dubois, J. Caillat, R. Taïeb, and C. Lévêque, Time-domain interferences as the source of electron-ion entanglement in Rabi-dressed photoemission, arXiv:2507.05850.
  25. M. Krug, T. Bayer, M. Wollenhaupt, C. Sarpe-Tudoran, T. Baumert, S. S. Ivanov, and N. V. Vitanov, Coherent strong-field control of multiple states by a single chirped femtosecond laser pulse, New J. Phys. 11, 105051 (2009).
  26. D. Younis and J. H. Eberly, Benchmark of few-level quantum theory vs ab initio numerical solutions for the strong-field Autler-Townes effect in photoionization of hydrogen, J. Phys. B: At. Mol. Opt. Phys. 55, 164001 (2022).
  27. B. Tóth, A. Tóth, and A. Csehi, Competition of multiphoton ionization pathways in lithium, J. Phys. B: At. Mol. Opt. Phys. 57, 055002 (2024).
  28. A. Yang, W.-Q. Jing, B.-S. Zhang, L.-B. Fan, L. Jia, and C.-C. Shu, Spectral phase control of an ultrashort pulse for exploring the Autler-Townes effect in hydrogen atomic ionization through a resonant state, Phys. Rev. A 112, 053116 (2025).
  29. W. Li, Y. Wang, X. Li, T. Yang, D. Zhang, and D. Ding, Ionization phase-shifts between two laser-dressed states of potassium atoms, Ann. Phys. 537, e00091 (2025).
  30. W. Li, Y. Wang, X. Li, T. Yang, D. Zhang, and D. Ding, Direct observation of two-photon-resonance Autler-Townes splitting of atomic sodium, Phys. Rev. A 111, 063102 (2025).
  31. Y.-J. Mao, Z.-H. Zhang, T. Chen, H.-B. Yao, Y. Li, and F. He, Decoupling the complex amplitudes of partial waves in two-photon ionization, Phys. Rev. A 109, 063117 (2024).
  32. H. Agueny, Quantum control and characterization of ultrafast ionization with orthogonal two-color laser pulses, Sci. Rep. 10, 239 (2020).
  33. D. Umarova, O. Umarov, A. Tóth, and A. Csehi, Spectral evidence of vibronic Rabi oscillations in the resonance-enhanced photodissociation of MgH+, Phys. Rev. A 110, 033112 (2024).
  34. P. V. Demekhin and L. S. Cederbaum, Coherent intense resonant laser pulses lead to interference in the time domain seen in the spectrum of the emitted particles, Phys. Rev. A 86, 063412 (2012).
  35. D. A. Tumakov, D. A. Telnov, G. Plunien, and V. M. Shabaev, Photoelectron spectra after multiphoton ionization of Li atoms in the one-photon Rabi-flopping regime, Phys. Rev. A 100, 023407 (2019).
  36. N. S. Simonović, D. B. Popović, and A. Bunjac, Manifestations of Rabi dynamics in the photoelectron energy spectra at resonant two-photon ionization of atom by intense short laser pulses, Atoms 11, 20 (2023).
  37. Y. Deng, X. Zhang, Y. Zhou, and P. Lu, Time- and energy-domain interference in the dynamically dressed Rabi-cycling atom, Phys. Rev. A 111, 043110 (2025).
  38. W.-q. Wan, G.-Y . Liu, M.-F . Xie, Q.-B . Meng, and W.-C . Jiang, Temporal interference picture for subpeak structures between Autler-Townes doublets in photoelectron energy spectra, Phys. Rev. A 111, 033117 (2025).
  39. A. Tóth, S. Borbély, and A. Csehi, Dynamic interference in below-threshold ionization, Phys. Rev. A 108, L061101 (2023).
  40. A. A. Elarabi and A. Csehi, Transition from Rabi oscillations to dynamic interference: The multipeak pattern of the Autler-Townes doublet revisited, Phys. Rev. A 111, 033107 (2025).
  41. A. Tóth and A. Csehi, Probing strong-field two-photon transitions through dynamic interference, J. Phys. B: At. Mol. Opt. Phys. 54, 035005 (2021).
  42. 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).
  43. 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).
  44. A. Bunjac, D. B. Popović, and N. S. Simonović, Analysis of the asymmetry of Autler-Townes doublets in the energy spectra of photoelectrons produced at two-photon ionization of atoms by strong laser pulses, Eur. Phys. J. D 76, 249 (2022).
  45. A. A. Elarabi and A. Csehi, Exactly solvable two-level model with time-dependent coupling, detuning, and loss: Application to strong-field resonance ionization, Phys. Rev. A 111, 063108 (2025).
  46. T. Bayer, K. Eickhoff, D. Köhnke, and M. Wollenhaupt, Phase control of the Autler-Townes doublet in multistate systems, Phys. Rev. A 108, 033111 (2023).
  47. U. Saalmann, S. K. Giri, and J. M. Rost, Adiabatic passage to the continuum: Controlling ionization with chirped laser pulses, Phys. Rev. Lett. 121, 153203 (2018).
  48. M. Wollenhaupt, A. Assion, O. Bazhan, Ch Horn, D. Liese, Ch Sarpe-Tudoran, M. Winter, and T. Baumert, Control of interferences in an Autler-Townes doublet: Symmetry of control parameters, Phys. Rev. A 68, 015401(R) (2003).
  49. M. Wollenhaupt, A. Präkelt, C. Sarpe-Tudoran, D. Liese, and T. Baumert, Quantum control by selective population of dressed states using intense chirped femtosecond laser pulses, Appl. Phys. B 82, 183 (2006).
  50. T. Bayer, M. Wollenhaupt, H. Braun, and T. Baumert, Ultrafast and efficient control of coherent electron dynamics via SPODS, Adv. Chem. Phys. 159, 235 (2016).
  51. E. Olofsson and J. M. Dahlström, Photoelectron signature of dressed-atom stabilization in an intense XUV field, Phys. Rev. Res. 5, 043017 (2023).
  52. E. Olofsson, E. L. Fulton, R. Tahouri, M. Bertolino, J. M. N. Djiokap, and J. M. Dahlström, Control of photoionization by resonant phase-locked pulse pairs, Phys. Rev. Res. 8, 013199 (2026).
  53. Y. Deng, Y. Liao, X. Zhang, M. Yu, P. Lu, and Y. Zhou, Control of Autler-Townes spectra by counter-rotating circularly polarized pulses, Phys. Rev. A 111, 023106 (2025).
  54. X. Zhang, Y. Liao, Y. Chen, M. Yu, K. Liu, P. Lu, and Y. Zhou, Characterization of the two-photon transition phase in the buildup process of Rabi oscillations, Phys. Rev. A 109, 033101 (2024).
  55. X. Zhang, Y. Liao, Y. Chen, M. Yu, S. Li, M. He, K. Liu, P. Lu, and Y. Zhou, Monitoring the ultrafast buildup of Rabi oscillations in the time domain, Phys. Rev. A 109, 063102 (2024).
  56. W.-C. Jiang, H. Liang, S. Wang, L.-Y. Peng, and J. Burgdörfer, Enhancing Autler-Townes splittings by ultrafast XUV pulses, Phys. Rev. Res. 3, L032052 (2021).
  57. L.-L. Wu, W.-C. Jiang, and L.-Y. Peng, Dynamically enhanced Autler-Townes splitting by orthogonal, Chin. Phys. B 33, 074204 (2024).
  58. F. Bassani, J. J. Forney, and A. Quattropani, Choice of gauge in two-photon transitions: 1s−2p transition in atomic hydrogen, Phys. Rev. Lett. 39, 1070 (1977).
  59. A. D. Bandrauk, F. Fillion-Gourdeau, and E. Lorin, Atoms and molecules in intense laser fields: gauge invariance of theory and models, J. Phys. B: At. Mol. Opt. Phys. 46, 153001 (2013).
  60. B. Kaufman, T. Rozgonyi, P. Marquetand, and T. C. Weinacht, Adiabatic elimination in strong-field light-matter coupling, Phys. Rev. A 102, 063117 (2020).
  61. P. V. Demekhin and L. S. Cederbaum, Strong interference effects in the resonant Auger decay of atoms induced by intense x-ray fields, Phys. Rev. A 83, 023422 (2011).
  62. E. Kreyszig, H. Kreyszig, and E. J. Norminton, Advanced Engineering Mathematics, 10th ed. (Wiley, New York, 2011).
  63. B. W. Shore and N. V. Vitanov, Overdamping of coherently driven quantum systems, Contemp. Phys. 47, 341 (2006).
  64. B. W. Shore, The Theory of Coherent Atomic Excitation (Wiley, New York, 1990).
  65. F. T. Hioe and C. E. Carroll, Two-state problems involving arbitrary amplitude and frequency modulations, Phys. Rev. A 32, 1541 (1985).
  66. S. Lee, J. Lim, J. Ahn, V. Hakobyan, and S. Guérin, Strong-field two-photon transition by phase shaping, Phys. Rev. A 82, 023408 (2010).
  67. The minimal-intensity energy positions of the spectral components in Eq. (25) are given by the condition (K−R±)T=n×(2π) (n integer). Making use of the definition of K in Eq. (18), the width of the spectral peaks (the energy separation of two neighboring minima) is found to be 2π/T. This value is modified at the two main AT peaks to 4π/T (see text for details).

Outline

Information

Phys. Rev. A 113, 033121– Published 25 March, 2026
  • Received 12 December 2025
  • Accepted 18 February 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