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Universal approach to saddle-point methods in attosecond science

Anne Weber1,*, Job Feldbrugge2, and Emilio Pisanty1

  • 1Attosecond Quantum Physics Laboratory, Department of Physics, King's College London, Strand Campus, London WC2R 2LS, United Kingdom
  • 2Higgs Centre for Theoretical Physics, University of Edinburgh, Edinburgh EH8 9YL, United Kingdom

  • *Contact author: anne.weber@kcl.ac.uk
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Phys. Rev. A 113, 063111 – Published 22 June, 2026

DOI: https://doi.org/10.1103/d2pt-xp7x

Abstract

Light-matter interactions within the strong-field regime, where intense laser fields can ionize a target via tunneling, give rise to fascinating phenomena such as the generation of high-order harmonic radiation (HHG) and, correspondingly, light pulses of attosecond duration. On the atomic scale, these strong-field processes are naturally described in terms of highly oscillatory time integrals which are often approximated using saddle-point methods. Those methods simultaneously simplify the calculations and let us understand the physical processes in terms of semiclassical electron trajectories, known as quantum orbits. However, applying saddle-point methods for HHG driven by polychromatic laser fields without clear dynamical symmetries has remained challenging. Here we introduce Picard–Lefschetz theory as a universal and robust link between the time integrals and the semiclassical trajectories, for arbitrary driving laser fields. The continuous deformation of the integration contour towards so-called Lefschetz thimbles allows for an exact evaluation of the integral, as well as the identification of relevant quantum orbits, independently of dynamical laser field symmetries or quantum-orbit classification heuristics. The latter is realized via the “necklace algorithm,” a solution to the open problem of determining the relevance of saddle points for a two-dimensional integral, which we introduce here. We demonstrate the versatility and rigor of Picard–Lefschetz methods by studying Stokes transitions and spectral caustics arising in HHG driven by two-color laser fields. For example, we showcase a quantum-orbit analysis of the color switchover, which links the regime of perturbative two-color fields with that of fully bichromatic driving fields. With this work, we set the foundation for a rigorous application of quantum-orbit-based approaches in attosecond science that enables the interpretation of state-of-the-art experimental setups, and guides the design of future ones.

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References (118)

  1. M. Lewenstein, Theory of high-harmonic generation by low-frequency laser fields, Phys. Rev. A 49, 2117 (1994).
  2. L. Keldysh, Ionization in the field of a strong electromagnetic wave, Zh. Eksp. Teor. Fiz. 47, 1945 (1965) [Sov. Phys.–JETP 20, 1307 (1965)].
  3. A. Perelomov, V. Popov, and M. Terent'ev, Ionization of atoms in an alternating electric field: II, Zh. Eksp. Teor. Fiz. 51, 309 (1967) [Sov. Phys.–JETP 24, 207 (1967)].
  4. W. Becker, F. Grasbon, R. Kopold, et al., Above-threshold ionization: From classical features to quantum effects, in Advances In Atomic, Molecular, and Optical Physics, edited by B. Bederson and H. Walther (Academic, New York, 2002), Vol. 48, pp. 35–98.
  5. R. Kopold, W. Becker, and M. Kleber, Quantum path analysis of high-order above-threshold ionization, dedicated to Marlan O. Scully on the occasion of his 60th birthday, Opt. Commun. 179, 39 (2000).
  6. P. Salières, B. Carré, L. Le Déroff, et al., Feynman's path-integral approach for intense-laser-atom interactions, Science 292, 902 (2001).
  7. S. Lefschetz, L'Analysis Situs et la Géométrie Algébrique (Gauthier-Villars, Paris, 1924).
  8. E. Picard and G. Simart, Théorie des Fonctions Algébriques de Deux Variables Indépendantes (Gauthier-Villars, Paris, 1897).
  9. F. Pham, Vanishing homologies and the n variables saddlepoint method, in Singularities, Part 2 (American Mathematical Society, Providence, RI, 1983), pp. 310–333.
  10. E. Witten, Analytic continuation of Chern-Simons theory, AMS/IP Stud. Adv. Math. 50, 347 (2011).
  11. J. Feldbrugge and N. Turok, Existence of real time quantum path integrals, Ann. Phys. 454, 169315 (2023).
  12. O. Raz, O. Pedatzur, B. D. Bruner, and N. Dudovich, Spectral caustics in attosecond science, Nat. Photon. 6, 170 (2012).
  13. V. A. Birulia and V. V. Strelkov, Spectral caustic in two-color high-order harmonic generation: Role of Coulomb effects, Phys. Rev. A 99, 043413 (2019).
  14. A.-K. Raab, M. Redon, S. R. Abbing, et al., XUV yield optimization of two-color high-order harmonic generation in gases, Nanophotonics 14, 3947 (2025).
  15. F. Dong, Q. Xia, and J. Liu, Caustic effects on high-order harmonic generation in graphene, Phys. Rev. A 109, L041102 (2024).
  16. D. Faccialà, S. Pabst, B. D. Bruner, et al., High-order harmonic generation spectroscopy by recolliding electron caustics, J. Phys. B: At. Mol. Opt. Phys. 51, 134002 (2018).
  17. D. Faccialà, S. Pabst, B. D. Bruner, et al., Probe of multielectron dynamics in xenon by caustics in high-order harmonic generation, Phys. Rev. Lett. 117, 093902 (2016).
  18. P. B. Corkum, Plasma perspective on strong field multiphoton ionization, Phys. Rev. Lett. 71, 1994 (1993).
  19. K. C. Kulander, K. J. Schafer, and J. L. Krause, Dynamics of short-pulse excitation, ionization and harmonic conversion, in Super-Intense Laser–Atom Physics, NATO ASI Series (Springer, New York, NY, 1993), pp. 95–110.
  20. C. Jin, G. Wang, H. Wei, et al., Waveforms for optimal sub-keV high-order harmonics with synthesized two- or three-color laser fields, Nat. Commun. 5, 4003 (2014).
  21. G. Cirmi, R. E. Mainz, M. A. Silva-Toledo, et al., Optical waveform synthesis and its applications, Laser Photon. Rev. 17, 2200588 (2023).
  22. S. Mitra, S. Biswas, J. Schötz, et al., Suppression of individual peaks in two-color high harmonic generation, J. Phys. B: At. Mol. Opt. Phys. 53, 134004 (2020).
  23. E. Mansten, J. M. Dahlström, P. Johnsson, et al., Spectral shaping of attosecond pulses using two-color laser fields, New J. Phys. 10, 083041 (2008).
  24. L. E. Chipperfield, J. S. Robinson, J. W. G. Tisch, and J. P. Marangos, Ideal waveform to generate the maximum possible electron recollision energy for any given oscillation period, Phys. Rev. Lett. 102, 063003 (2009).
  25. O. Kneller, D. Azoury, Y. Federman, et al., A look under the tunnelling barrier via attosecond-gated interferometry, Nat. Photonics 16, 304 (2022).
  26. X. He, J. M. Dahlström, R. Rakowski, et al., Interference effects in two-color high-order harmonic generation, Phys. Rev. A 82, 033410 (2010).
  27. D. Shafir, H. Soifer, B. D. Bruner, et al., Resolving the time when an electron exits a tunnelling barrier, Nature (London) 485, 343 (2012).
  28. N. Dudovich, O. Smirnova, J. Levesque, et al., Measuring and controlling the birth of attosecond XUV pulses, Nat. Phys. 2, 781 (2006).
  29. J. Zhao and M. Lein, Determination of ionization and tunneling times in high-order harmonic generation, Phys. Rev. Lett. 111, 043901 (2013).
  30. N. Eicke and M. Lein, Attoclock with counter-rotating bicircular laser fields, Phys. Rev. A 99, 031402 (2019).
  31. J. Mauritsson, J. M. Dahlström, E. Mansten, and T. Fordell, Sub-cycle control of attosecond pulse generation using two-colour laser fields, J. Phys. B: At. Mol. Opt. Phys. 42, 134003 (2009).
  32. C. Ruiz, D. J. Hoffmann, R. Torres, et al., Control of the polarization of attosecond pulses using a two-color field, New J. Phys. 11, 113045 (2009).
  33. S. Roscam Abbing, F. Campi, F. S. Sajjadian, et al., Divergence control of high-harmonic generation, Phys. Rev. Appl. 13, 054029 (2020).
  34. S. Haessler, T. Balčiunas, G. Fan, et al., Optimization of quantum trajectories driven by strong-field waveforms, Phys. Rev. X 4, 021028 (2014).
  35. D. Baykusheva and H. J. Wörner, Chiral discrimination through bielliptical high-harmonic spectroscopy, Phys. Rev. X 8, 031060 (2018).
  36. D. Ayuso, O. Neufeld, A. F. Ordonez, et al., Synthetic chiral light for efficient control of chiral light–matter interaction, Nat. Photon. 13, 866 (2019).
  37. J. Feldbrugge, J.-L. Lehners, and N. Turok, Lorentzian quantum cosmology, Phys. Rev. D 95, 103508 (2017).
  38. Y. Tanizaki and T. Koike, Real-time Feynman path integral with Picard–Lefschetz theory and its applications to quantum tunneling, Ann. Phys. 351, 250 (2014).
  39. R. Bharathkumar and A. Joseph, Lefschetz thimbles and quantum phases in zero-dimensional bosonic models, Eur. Phys. J. C 80, 923 (2020).
  40. J. Feldbrugge, U.-L. Pen, and N. Turok, Oscillatory path integrals for radio astronomy, Ann. Phys. 451, 169255 (2023).
  41. J. Feldbrugge and J. Y. L. Jones, Efficient evaluation of real-time path integrals, Phys. Rev. D 111, 083524 (2025).
  42. A. Weber, M. Khokhlova, and E. Pisanty, Quantum tunneling without a barrier, Phys. Rev. A 111, 043103 (2025).
  43. A. Weber, PicardLefschetz.jl, https://github.com/anneaux/PicardLefschetz.jl, Zenodo, 2026, doi:10.5281/zenodo.18504409.
  44. A. Weber, ATIsimPL, https://github.com/anneaux/ATIsimPL, Zenodo, 2026, doi:10.5281/zenodo.18504217.
  45. A. Weber, HHGsimPL, https://github.com/anneaux/HHGsimPL, Zenodo, 2026, doi:10.5281/zenodo.18504394.
  46. A. Weber, Figure-maker notebooks and code for Picard–Lefschetz methods for ATI and HHG integrals, Zenodo, 2026, doi:10.5281/zenodo.17298777.
  47. A.-T. Le, H. Wei, C. Jin, and C. D. Lin, Strong-field approximation and its extension for high-order harmonic generation with mid-infrared lasers, J. Phys. B: At. Mol. Opt. Phys. 49, 053001 (2016).
  48. S. V. Popruzhenko, Keldysh theory of strong field ionization: History, applications, difficulties and perspectives, J. Phys. B: At. Mol. Opt. Phys. 47, 204001 (2014).
  49. O. Smirnova and M. Ivanov, Multielectron high harmonic generation: Simple man on a complex plane, in Attosecond and XUV Physics: Ultrafast Dynamics and Spectroscopy, edited by T. Schultz and M. Vrakking (Wiley, Weinheim, 2014), pp. 201–256.
  50. K. Amini, J. Biegert, F. Calegari, et al., Symphony on strong field approximation, Rep. Prog. Phys. 82, 116001 (2019).
  51. N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals (Ardent Media, Ottawa, 1975).
  52. R. Wong, Asymptotic and Computational Analysis: Conference in Honor of Frank W.J. Olver's 65th Birthday (CRC Press, Boca Raton, FL, 2020).
  53. A. Jašarević, E. Hasović, R. Kopold, et al., Application of the saddle-point method to strong-laser-field ionization, J. Phys. A: Math. Theor. 53, 125201 (2020).
  54. D. B. Milošević and W. Becker, Role of long quantum orbits in high-order harmonic generation, Phys. Rev. A 66, 063417 (2002).
  55. C. Figueira de Morisson Faria, D. B. Milošević, and G. G. Paulus, Phase-dependent effects in bichromatic high-order harmonic generation, Phys. Rev. A 61, 063415 (2000).
  56. C. Figueira de Morisson Faria, H. Schomerus, and W. Becker, High-order above-threshold ionization: The uniform approximation and the effect of the binding potential, Phys. Rev. A 66, 043413 (2002).
  57. E. Pisanty, M. F. Ciappina, and M. Lewenstein, The imaginary part of the high-harmonic cutoff, J. Phys.: Photonics 2, 034013 (2020).
  58. D. B. Milošević, Application of the uniform approximation to integrals occurring in ionization by a strong elliptically polarized laser field, Phys. Rev. A 111, 053105 (2025).
  59. D. Habibović and D. B. Milošević, Complete classification and additional saddle-point solutions for high-order above-threshold ionization induced by a strong laser field, Phys. Rev. A 111, 023103 (2025).
  60. A. Gibbs, D. P. Hewett, and D. Huybrechs, Numerical evaluation of oscillatory integrals via automated steepest descent contour deformation, J. Comput. Phys. 501, 112787 (2024).
  61. A. V. Shanin, A. I. Korolkov, and K. S. Kniazeva, Saddle point method for transient processes in waveguides, J. Theor. Comp. Acout. 30, 2150018 (2022).
  62. F. H. M. Faisal, Multiple absorption of laser photons by atoms, J. Phys. B: At. Mol. Phys. 6, L89 (1973).
  63. H. R. Reiss, Effect of an intense electromagnetic field on a weakly bound system, Phys. Rev. A 22, 1786 (1980).
  64. A. Nayak, M. Dumergue, S. Kühn, et al., Saddle point approaches in strong field physics and generation of attosecond pulses, Phys. Rep. 833, 1 (2019).
  65. D. B. Milošević, D. Bauer, and W. Becker, Quantum-orbit theory of high-order atomic processes in intense laser fields, J. Mod. Opt. 53, 125 (2006).
  66. D. B. Milošević and W. Becker, X-ray harmonic generation by orthogonally polarized two-color fields: Spectral shape and polarization, Phys. Rev. A 100, 031401 (2019).
  67. V. I. Arnold, A. N. Varchenko, and S. M. Gusein-Zade, Singularities of Differentiable Maps: Volume II Monodromy and Asymptotic Integrals (Birkhäuser, Basel, 2012).
  68. J. Feldbrugge, J.-L. Lehners, and N. Turok, No smooth beginning for spacetime, Phys. Rev. Lett. 119, 171301 (2017).
  69. J. Feldbrugge, Multiplane lensing in wave optics, Mon. Not. R. Astron. Soc. 520, 2995 (2023).
  70. B. Bonga, J. Feldbrugge, and A. R. Metidieri, Wave optics for rotating stars, Phys. Rev. D 111, 063061 (2025).
  71. H. Fujii, D. Honda, M. Kato, et al., Hybrid Monte Carlo on Lefschetz thimbles – A study of the residual sign problem, J. High Energy Phys. 10 (2013) 147.
  72. M. Cristoforetti, F. D. Renzo, and L. Scorzato (Collaboration, AuroraScience), High density QCD on a Lefschetz thimble? Phys. Rev. D 86, 074506 (2012).
  73. M. Cristoforetti, F. D. Renzo, A. Mukherjee, and L. Scorzato, Monte Carlo simulations on the Lefschetz thimble: Taming the sign problem, Phys. Rev. D 88, 051501 (2013).
  74. J. Feldbrugge and U.-L. Pen, The real-time Feynman path integral for step potentials, Ann. Phys. 490, 170490 (2026).
  75. J. Feldbrugge, D. L. Jow, and U.-L. Pen, Complex classical paths in quantum reflections and tunneling, Phys. Rev. D 111, 085027 (2025).
  76. A. Weber, Caustics and catastrophes in strong-field physics. Picard–Lefschetz theory as a universal approach to saddle-point methods in attosecond science, Ph.D. thesis, King's College London, 2026, arXiv:2605.03794, https://kclpure.kcl.ac.uk/portal/en/studentTheses/caustics-and-catastrophes-in-strong-field-physics/.
  77. E. Delabaere and C. J. Howls, Global asymptotics for multiple integrals with boundaries, Duke Math. J. 112, 251 (2002).
  78. A. Alexandru, G. Başar, and P. Bedaque, Monte Carlo algorithm for simulating fermions on Lefschetz thimbles, Phys. Rev. D 93, 014504 (2016).
  79. J. Nishimura and S. Shimasaki, Combining the complex Langevin method and the generalized Lefschetz-thimble method, J. High Energy Phys. 06 (2017) 023.
  80. J. L. Feldbrugge, U.-L. Pen, and N. Turok, Picard–Lefschetz path integrals, GitHub, https://p-lpi.github.io/.
  81. G. E. Bartholomew, Numerical integration over the triangle, Math. Tables Other Aids Comput. 13, 295 (1959).
  82. M. Han, Z. Huang, H. Liu, et al., Spinfoam on a Lefschetz thimble: Markov chain Monte Carlo computation of a Lorentzian spinfoam propagator, Phys. Rev. D 103, 084026 (2021).
  83. A. Alexandru, G. Başar, P. F. Bedaque, and N. C. Warrington, Complex paths around the sign problem, Rev. Mod. Phys. 94, 015006 (2022).
  84. Y. Shoji and K. Trailović, Stable evaluation of Lefschetz thimble intersection numbers: Towards real-time path integrals, Phys. Lett. B 873, 140198 (2026).
  85. S. K. Lando, Geometry of the Stokes sets for families of functions of one variable, J. Math. Sci. 83, 534 (1997).
  86. L. E. Chipperfield, L. N. Gaier, P. L. Knight, et al., Conditions for the reliable production of attosecond pulses using ultra-short laser-generated high harmonics, J. Mod. Opt. 52, 243 (2005).
  87. F. J. Wright, The Stokes set of the cusp diffraction catastrophe, J. Phys. A: Math. Gen. 13, 2913 (1980).
  88. M. V. Berry and C. J. Howls, Stokes surfaces of diffraction catastrophes with codimension three, Nonlinearity 3, 281 (1990).
  89. J. Feldbrugge, R. van de Weygaert, J. Hidding, and J. Feldbrugge, Caustic skeleton & cosmic web, J. Cosmol. Astropart. Phys. 05 (2018) 027.
  90. T. Poston and I. Stewart, Catastrophe Theory and its Applications, 1st ed., Surveys and Reference Works in Mathematics, Vol. 2 (Pitman Publishing Ltd., Bath, UK, 1978).
  91. P. T. Saunders, An Introduction to Catastrophe Theory (Cambridge University Press, Cambridge, 1980).
  92. NIST Handbook of Mathematical Functions, edited by F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark (Cambridge University Press, Cambridge, 2010).
  93. C. Chester, B. Friedman, and F. Ursell, An extension of the method of steepest descents, Math. Proc. Cambridge Philos. Soc. 53, 599 (1957).
  94. M. V. Berry, Uniform asymptotic smoothing of Stokes's discontinuities, Proc. R. Soc. A. Math. Phys. Sci. 422, 7 (1989).
  95. H. Schomerus and M. Sieber, Bifurcations of periodic orbits and uniform approximations, J. Phys. A: Math. Gen. 30, 4537 (1997).
  96. J. J. Stamnes and B. Spjelkavik, Evaluation of the field near a cusp of a caustic, Optica Acta: Int. J. Opt. 30, 1331 (1983).
  97. L. E. Chipperfield, P. L. Knight, J. W. G. Tisch, and J. P. Marangos, Tracking individual electron trajectories in a high harmonic spectrum, Opt. Commun. 264, 494 (2006).
  98. O. Pedatzur, G. Orenstein, V. Serbinenko, et al., Attosecond tunnelling interferometry, Nat. Phys. 11, 815 (2015).
  99. X. Zou, L. Jurkovičová, A. Weber, et al., 2D quantum-path interference in high-harmonic generation driven by highly-bichromatic fields, arXiv:2604.12838.
  100. V. V. Strelkov, M. A. Khokhlova, A. A. Gonoskov, et al., High-order harmonic generation by atoms in an elliptically polarized laser field: Harmonic polarization properties and laser threshold ellipticity, Phys. Rev. A 86, 013404 (2012).
  101. E. Pisanty and Á. Jiménez-Galán, Strong-field approximation in a rotating frame: High-order harmonic emission from p states in bicircular fields, Phys. Rev. A 96, 063401 (2017).
  102. E. Pisanty, D. D. Hickstein, B. R. Galloway, et al., High harmonic interferometry of the Lorentz force in strong mid-infrared laser fields, New J. Phys. 20, 053036 (2018).
  103. J. Itatani, J. Levesque, D. Zeidler, et al., Tomographic imaging of molecular orbitals, Nature (London) 432, 867 (2004).
  104. A. Zaïr, M. Holler, A. Guandalini, et al., Quantum path interferences in high-order harmonic generation, Phys. Rev. Lett. 100, 143902 (2008).
  105. D. J. Hoffmann, C. Hutchison, A. Zaïr, and J. P. Marangos, Control of temporal mapping and harmonic intensity modulation using two-color orthogonally polarized fields, Phys. Rev. A 89, 023423 (2014).
  106. L. Brugnera, D. J. Hoffmann, T. Siegel, et al., Trajectory selection in high harmonic generation by controlling the phase between orthogonal two-color fields, Phys. Rev. Lett. 107, 153902 (2011).
  107. T.-M. Yan, S. V. Popruzhenko, M. J. J. Vrakking, and D. Bauer, Low-energy structures in strong field ionization revealed by quantum orbits, Phys. Rev. Lett. 105, 253002 (2010).
  108. A. J. Uzan, G. Orenstein, Á. Jiménez-Galán, et al., Attosecond spectral singularities in solid-state high-harmonic generation, Nat. Photon. 14, 183 (2020).
  109. S. P. Goreslavskii and S. V. Popruzhenko, Tunneling limit in the theory of photoelectron rescattering by the parent ion, J. Exp. Theor. Phys. 90, 778 (2000).
  110. T. Rook and C. Figueira De Morisson Faria, Exploring symmetries in photoelectron holography with two-color linearly polarized fields, J. Phys. B: At. Mol. Opt. Phys. 55, 165601 (2022).
  111. D. Habibović, W. Becker, and D. B. Milošević, Complete classification and additional saddle-point solutions for high-order above-threshold ionization induced by a strong laser field. II. Classical considerations, Phys. Rev. A 111, 053110 (2025).
  112. M. V. Berry, Focusing and twinkling: Critical exponents from catastrophes in non-Gaussian random short waves, J. Phys. A: Math. Gen. 10, 2061 (1977).
  113. M. V. Berry and C. J. Howls, Hyperasymptotics for integrals with saddles, Proc. R. Soc. Lond. Ser. A 434, 657 (1991).
  114. D. Dorigoni, An introduction to resurgence, trans-series and alien calculus, Ann. Phys. 409, 167914 (2019).
  115. P. V. Demekhin and L. S. Cederbaum, Dynamic interference of photoelectrons produced by high-frequency laser pulses, Phys. Rev. Lett. 108, 253001 (2012).
  116. F. Vismarra, M. Bertolino, E. Appi, et al., Dynamic interference of chirped photoelectrons, Phys. Rev. Lett. 135, 033202 (2025).
  117. J. Itatani, F. Quéré, G. L. Yudin, et al., Attosecond streak camera, Phys. Rev. Lett. 88, 173903 (2002).
  118. D. Moos, H. Jürß, and D. Bauer, Intense-laser-driven electron dynamics and high-order harmonic generation in solids including topological effects, Phys. Rev. A 102, 053112 (2020).

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Phys. Rev. A 113, 063111– Published 22 June, 2026
  • Received 22 October 2025
  • Revised 30 March 2026
  • Accepted 22 May 2026
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