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

Unraveling interchannel quantum interference in below-threshold nonsequential double ionization with statistical measures

S. Hashim and C. Figueira de Morisson Faria*

  • *Contact author: c.faria@ucl.ac.uk

Phys. Rev. Research 8, 013156 – Published 11 February, 2026

DOI: https://doi.org/10.1103/zkf1-j2bg

Abstract

We present a systematic study of interchannel quantum interference in laser-induced nonsequential double ionization within the strong-field approximation. Focusing on the below-threshold intensity regime where the recollision excitation with subsequent ionization (RESI) pathway dominates, we derive analytical phase conditions governing interference between distinct excitation channels for arbitrary driving fields. To quantify the interplay between channels resulting from a vast number of interfering processes, we introduce statistical metrics based on the earth mover's distance, allowing us to assess the relative weight of each channel's contribution to the two-electron photoelectron momentum distributions. We identify key factors that determine whether interchannel interference is appreciable, such as comparable channel intensities, strong spatial overlap between the excited-state wave functions, and the energy difference between contributing channels. We demonstrate that for linearly polarized few-cycle pulses, the typical intrachannel interference features associated with exchange, temporal shifts, and combined exchange-temporal interference are retained with interchannel interference. Our findings establish a hierarchy of interference mechanisms in RESI and may provide practical guidelines for enhancing or suppressing interference in different regions of the momentum plane. The tool kit presented in this work is transferable to a wide range of interferometric schemes involving different excitation channels.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (83)

  1. F. Krausz and M. Ivanov, Attosecond physics, Rev. Mod. Phys. 81, 163 (2009).
  2. S. Haessler, J. Caillat, W. Boutu, C. Giovanetti-Teixeira, T. Ruchon, T. Auguste, Z. Diveki, P. Breger, A. Maquet, B. Carré, R. Taïeb, and P. Salières, Attosecond imaging of molecular electronic wavepackets, Nat. Phys. 6, 200 (2010).
  3. J. P. Marangos, Development of high harmonic generation spectroscopy of organic molecules and biomolecules, J. Phys. B: At. Mol. Opt. Phys. 49, 132001 (2016).
  4. P. Stammer, J. Rivera-Dean, A. S. Maxwell, T. Lamprou, J. Argüello-Luengo, P. Tzallas, M. F. Ciappina, and M. Lewenstein, Entanglement and squeezing of the optical field modes in high harmonic generation, Phys. Rev. Lett. 132, 143603 (2024).
  5. T. W. Hänsch, Nobel lecture: Passion for precision, Rev. Mod. Phys. 78, 1297 (2006).
  6. J. L. Hall, Nobel lecture: Defining and measuring optical frequencies, Rev. Mod. Phys. 78, 1279 (2006).
  7. C. M. Caves, Quantum-mechanical noise in an interferometer, Phys. Rev. D 23, 1693 (1981).
  8. V. Giovannetti, S. Lloyd, and L. Maccone, Quantum-enhanced measurements: Beating the standard quantum limit, Science 306, 1330 (2004).
  9. Y. Huismans, A. Gijsbertsen, A. S. Smolkowska, J. H. Jungmann, A. Rouzée, P. S. W. M. Logman, F. Lépine, C. Cauchy, S. Zamith, T. Marchenko, J. M. Bakker, G. Berden, B. Redlich, A. F. G. van der Meer, M. Y. Ivanov, T.-M. Yan, D. Bauer, O. Smirnova, and M. J. J. Vrakking, Scaling laws for photoelectron holography in the midinfrared wavelength regime, Phys. Rev. Lett. 109, 013002 (2012).
  10. S. Skruszewicz, J. Tiggesbäumker, K.-H. Meiwes-Broer, M. Arbeiter, T. Fennel, and D. Bauer, Two-color strong-field photoelectron spectroscopy and the phase of the phase, Phys. Rev. Lett. 115, 043001 (2015).
  11. D. Ayuso, P. Decleva, S. Patchkovskii, and O. Smirnova, Chiral dichroism in bi-elliptical high-order harmonic generation, J. Phys. B: At. Mol. Opt. Phys. 51, 06LT01 (2018).
  12. X. L. Hao, J. Chen, W. D. Li, B. Wang, X. Wang, and W. Becker, Quantum effects in double ionization of argon below the threshold intensity, Phys. Rev. Lett. 112, 073002 (2014).
  13. A. S. Maxwell and C. Figueira de Morisson Faria, Quantum interference in time-delayed nonsequential double ionization, Phys. Rev. A 92, 023421 (2015).
  14. A. S. Maxwell and C. Figueira de Morisson Faria, Controlling below-threshold nonsequential double ionization via quantum interference, Phys. Rev. Lett. 116, 143001 (2016).
  15. W. Quan, X. L. Hao, Y. L. Wang, Y. J. Chen, S. G. Yu, S. P. Xu, Z. L. Xiao, R. P. Sun, X. Y. Lai, S. L. Hu, M. Q. Liu, Z. Shu, X. Wang, W. D. Li, W. Becker, X. J. Liu, and J. Chen, Quantum interference in laser-induced nonsequential double ionization, Phys. Rev. A 96, 032511 (2017).
  16. C. Figueira de Morisson Faria and X. Liu, Electron–electron correlation in strong laser fields, J. Mod. Opt. 58, 1076 (2011).
  17. T.-L. Ho, X. Cui, and W. Li, Alternative route to strong interaction: Narrow Feshbach resonance, Phys. Rev. Lett. 108, 250401 (2012).
  18. P. B. Corkum, Plasma perspective on strong field multiphoton ionization, Phys. Rev. Lett. 71, 1994 (1993).
  19. W. Becker, X. J. Liu, P. J. Ho, and J. H. Eberly, Theories of photoelectron correlation in laser-driven multiple atomic ionization, Rev. Mod. Phys. 84, 1011 (2012).
  20. L. B. Fu, G. G. Xin, D. F. Ye, and J. Liu, Recollision dynamics and phase diagram for nonsequential double ionization with circularly polarized laser fields, Phys. Rev. Lett. 108, 103601 (2012).
  21. N. Dudovich, O. Smirnova, J. Levesque, Y. Mairesse, M. Y. Ivanov, D. M. Villeneuve, and P. B. Corkum, Measuring and controlling the birth of attosecond XUV pulses, Nat. Phys. 2, 781 (2006).
  22. D. Shafir, H. Soifer, B. D. Bruner, M. Dagan, Y. Mairesse, S. Patchkovskii, M. Y. Ivanov, O. Smirnova, and N. Dudovich, Resolving the time when an electron exits a tunnelling barrier, Nature (London) 485, 343 (2012).
  23. S. Hashim, R. Tenney, and C. Figueira de Morisson Faria, Detangling the quantum tapestry of intrachannel interference in below-threshold nonsequential double ionization with few-cycle laser pulses, Phys. Rev. A 109, 063110 (2024).
  24. S. Hashim, D. Habibović, and C. F. d. M. Faria, Below-threshold nonsequential double ionization with linearly polarized two-color fields. II. Quantum interference, Phys. Rev. A 112, 023119 (2025).
  25. O. Smirnova, Y. Mairesse, S. Patchkovskii, N. Dudovich, D. Villeneuve, P. Corkum, and M. Y. Ivanov, High harmonic interferometry of multi-electron dynamics in molecules, Nature (London) 460, 972 (2009).
  26. B. B. Augstein and C. Figueira de Morisson Faria, High-order harmonic generation in diatomic molecules: Quantum interference, nodal structures and multiple orbitals, Mod. Phys. Lett. B 26, 1130002 (2012).
  27. O. Hassouneh, A. C. Brown, and H. W. van der Hart, Multichannel interference in high-order-harmonic generation from Ne+ driven by an ultrashort intense laser pulse, Phys. Rev. A 89, 033409 (2014).
  28. P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Augé, P. Balcou, H. G. Muller, and P. Agostini, Observation of a train of attosecond pulses from high harmonic generation, Science 292, 1689 (2001).
  29. J. M. Dahlström, A. L’Huillier, and A. Maquet, Introduction to attosecond delays in photoionization, J. Phys. B: At. Mol. Opt. Phys. 45, 183001 (2012).
  30. P. Agostini, Nobel lecture: Genesis and applications of attosecond pulse trains, Rev. Mod. Phys. 96, 030501 (2024).
  31. M. Nisoli, P. Decleva, F. Calegari, A. Palacios, and F. Martín, Attosecond electron dynamics in molecules, Chem. Rev. 117, 10760 (2017).
  32. S. Ghimire and D. A. Reis, High-harmonic generation from solids, Nat. Phys. 15, 10 (2018).
  33. S. Y. Kruchinin, F. Krausz, and V. S. Yakovlev, Colloquium: Strong-field phenomena in periodic systems, Rev. Mod. Phys. 90, 021002 (2018).
  34. Y. Rubner, C. Tomasi, and L. J. Guibas, The earth mover's distance as a metric for image retrieval, Int. J. Comput. Vision 40, 99 (2000).
  35. C. Zhang, Y. Cai, G. Lin, and C. Shen, Deepemd: Few-shot image classification with differentiable Earth Mover's Distance and structured classifiers, in IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR 2020), Seattle, WA (IEEE, Piscataway, NJ, 2020), pp. 12200–12210.
  36. P. T. Komiske, E. M. Metodiev, and J. Thaler, Metric space of collider events, Phys. Rev. Lett. 123, 041801 (2019).
  37. D. Y. Orlova, N. Zimmerman, S. Meehan, C. Meehan, J. Waters, E. E. B. Ghosn, A. Filatenkov, G. A. Kolyagin, Y. Gernez, S. Tsuda, W. Moore, R. B. Moss, L. A. Herzenberg, and G. Walther, Earth Mover's Distance (EMD): A true metric for comparing biomarker expression levels in cell populations, PLoS One 11, e0151859 (2016).
  38. Z. Wang, J. Liang, and M. Head-Gordon, Earth Mover's Distance as a metric to evaluate the extent of charge transfer in excitations using discretized real-space densities, J. Chem. Theory Comput. 19, 7704 (2023).
  39. L.-W. Pi and A. S. Landsman, Attosecond time delay in photoionization of noble-gas and halogen atoms, Appl. Sci. 8, 322 (2018).
  40. J. Venzke, A. Becker, and A. Jaron-Becker, Asymmetries in ionization of atomic superposition states by ultrashort laser pulses, Sci. Rep. 10, 16164 (2020).
  41. N. Böwering, T. Lischke, B. Schmidtke, N. Müller, T. Khalil, and U. Heinzmann, Asymmetry in photoelectron emission from chiral molecules induced by circularly polarized light, Phys. Rev. Lett. 86, 1187 (2001).
  42. S. Beaulieu, A. Comby, A. Clergerie, J. Caillat, D. Descamps, N. Dudovich, B. Fabre, R. Géneaux, F. Légaré, S. Petit, B. Pons, G. Porat, T. Ruchon, R. Taïeb, V. Blanchet, and Y. Mairesse, Attosecond-resolved photoionization of chiral molecules, Science 358, 1288 (2017).
  43. A. G. Harvey, Z. Mašín, and O. Smirnova, General theory of photoexcitation induced photoelectron circular dichroism, J. Chem. Phys. 149, 064104 (2018).
  44. D. Habibović, K. R. Hamilton, O. Neufeld, and L. Rego, Emerging tailored light sources for studying chirality and symmetry, Nat. Rev. Phys. 6, 663 (2024).
  45. A. F. Ordonez, D. Ayuso, P. Decleva, and O. Smirnova, Geometric magnetism and anomalous enantio-sensitive observables in photoionization of chiral molecules, Commun. Phys. 6, 257 (2023).
  46. A. S. Maxwell, A. Serafini, S. Bose, and C. F. d. M. Faria, Quantum estimation in strong fields: In situ ponderomotive sensing, Phys. Rev. A 103, 043519 (2021).
  47. B. T. Kiani, G. De Palma, M. Marvian, Z.-W. Liu, and S. Lloyd, Learning quantum data with the quantum earth mover's distance, Quantum Sci. Technol. 7, 045002 (2022).
  48. T. Shaaran, M. T. Nygren, and C. Figueira de Morisson Faria, Laser-induced nonsequential double ionization at and above the recollision-excitation-tunneling threshold, Phys. Rev. A 81, 063413 (2010).
  49. T. Shaaran and C. Figueira de Morisson Faria, Laser-induced nonsequential double ionization: Kinematic constraints for the recollision-excitation-tunneling mechanism, J. Mod. Opt. 57, 984 (2010).
  50. T. Shaaran, C. Figueira de Morisson Faria, and H. Schomerus, Causality and quantum interference in time-delayed laser-induced nonsequential double ionization, Phys. Rev. A 85, 023423 (2012).
  51. T. Rook, L. C. Rodriguez, and C. F. d. M. Faria, Influence of catastrophes and hidden dynamical symmetries on ultrafast backscattered photoelectrons, Phys. Rev. Res. 6, 023329 (2024).
  52. S. Hashim, D. Habibović, and C. F. d. M. Faria, Below-threshold nonsequential double ionization with linearly polarized two-color fields. I. Symmetry and dominance, Phys. Rev. A 112, 023118 (2025).
  53. T. Shaaran, A rigorous treatment of excitation and quantum interference in laser-induced nonsequential double ionization of atoms and molecules, Ph.D. thesis, University College London, 2012.
  54. 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).
  55. For a monochromatic field or long enough pulses, the direct (rescattered) ATI cutoff energy is 2Up (10Up), where Up is the ponderomotive energy. These numbers relate to the maximal kinetic energy that a direct or rescattered ATI electron may acquire from the external field, and are approximate for few-cycle pulses. Bichromatic or polychromatic fields will lead to different cutoff laws, but the final kinetic energy of the first electron will always be higher. For NSDI RESI, there is an inelastic collision, so that the energy that gap E2g−E2e should be subtracted from the rescattered ATI cutoff.
  56. T. Rook and C. F. de Morisson Faria, Exploring symmetries in photoelectron holography with two-color linearly polarized fields, J. Phys. B: At. Mol. Opt. Phys. 55, 165601 (2022).
  57. C. Figueira de Morisson Faria, T. Shaaran, and M. T. Nygren, Time-delayed nonsequential double ionization with few-cycle laser pulses: Importance of the carrier-envelope phase, Phys. Rev. A 86, 053405 (2012).
  58. C. Figueira de Morisson Faria, M. Dörr, W. Becker, and W. Sandner, Time-frequency analysis of two-color high-harmonic generation, Phys. Rev. A 60, 1377 (1999).
  59. Note that this channel-only phase difference, αl,l(Cn,Cm), differs from the αl,l(m,n) phase difference employed in Ref. [24] where mandn refer to different events within a channel, whereas this work neglects multiple ionization events per half cycle for simplicity but this convention can be extended in principle.
  60. E. Clementi and D. L. Raimondi, The correlation coefficient: An overview, J. Chem. Phys. 38, 2686 (1963).
  61. B. Kranstauber, M. Smolla, and K. Safi, Similarity in spatial utilization distributions measured by the earth mover's distance, Methods Ecol. Evol. 8, 155 (2017).
  62. A. G. Asuero, A. Sayago, and A. G. González, The correlation coefficient: An overview, Crit. Rev. Anal. Chem. 36, 41 (2006).
  63. J. Aasi, J. Abadie, B. P. Abbott, R. Abbott, and T. D. Abbott, Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light, Nat. Photonics 7, 613 (2013).
  64. C. A. Casacio, L. S. Madsen, A. Terrasson, M. Waleed, K. Barnscheidt, B. Hage, M. A. Taylor, and W. P. Bowen, Quantum-enhanced nonlinear microscopy, Nature (London) 594, 201 (2021).
  65. A. Rasputnyi, Z. Chen, M. Birk, O. Cohen, I. Kaminer, M. Krüger, D. Seletskiy, M. Chekhova, and F. Tani, High-harmonic generation by a bright squeezed vacuum, Nat. Phys. 20, 1960 (2024).
  66. M. Lewenstein, M. F. Ciappina, E. Pisanty, J. Rivera-Dean, P. Stammer, Th. Lamprou, and P. Tzallas, Generation of optical Schrödinger cat states in intense laser-matter interactions, Nat. Phys. 17, 1104 (2021).
  67. Y. Fang, F.-X. Sun, Q. He, and Y. Liu, Strong-field ionization of hydrogen atoms with quantum light, Phys. Rev. Lett. 130, 253201 (2023).
  68. S. Wang and X. Lai, High-order above-threshold ionization of an atom in intense quantum light, Phys. Rev. A 108, 063101 (2023).
  69. J. Rivera-Dean, P. Stammer, C. F. d. M. Faria, and M. Lewenstein, Microscopic analysis of above-threshold ionization driven by squeezed light, Phys. Rev. A 112, 063101 (2025).
  70. H. Liu, H. Zhang, X. Wang, and J. Yuan, Atomic double ionization with quantum light, Phys. Rev. Lett. 134, 123202 (2025).
  71. U. Bhattacharya, T. Lamprou, A. S. Maxwell, A. Ordóñez, E. Pisanty, J. Rivera-Dean, P. Stammer, M. F. Ciappina, M. Lewenstein, and P. Tzallas, Strong-laser-field physics, non-classical light states and quantum information science, Rep. Prog. Phys. 86, 094401 (2023).
  72. J. Rivera-Dean, P. Stammer, A. S. Maxwell, T. Lamprou, P. Tzallas, M. Lewenstein, and M. F. Ciappina, Light-matter entanglement after above-threshold ionization processes in atoms, Phys. Rev. A 106, 063705 (2022).
  73. 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).
  74. M. J. J. Vrakking, Control of attosecond entanglement and coherence, Phys. Rev. Lett. 126, 113203 (2021).
  75. M. J. J. Vrakking, Ion-photoelectron entanglement in photoionization with chirped laser pulses, J. Phys. B: At. Mol. Opt. Phys. 55, 134001 (2022).
  76. I. A. Ivanov and K. T. Kim, Entanglement in photo-ionization process, Sci. Rep. 14, 11378 (2024).
  77. S. Majorosi, M. G. Benedict, and A. Czirják, Quantum entanglement in strong-field ionization, Phys. Rev. A 96, 043412 (2017).
  78. M. Ruberti, V. Averbukh, and F. Mintert, Bell test of quantum entanglement in attosecond photoionization, Phys. Rev. X 14, 041042 (2024).
  79. F. Shobeiry, P. Fross, H. Srinivas, T. Pfeifer, R. Moshammer, and A. Harth, Emission control of entangled electrons in photoionisation of a hydrogen molecule, Sci. Rep. 14, 19630 (2024).
  80. D. Younis, S. Xie, and J. H. Eberly, Quantum entanglement during single-cycle nonsequential ionization, arXiv:2403.09854 [physics.atom-ph].
  81. A. S. Maxwell, L. B. Madsen, and M. Lewenstein, Entanglement of orbital angular momentum in non-sequential double ionization, Nat. Commun. 13, 4706 (2022).
  82. J. Zhang, Z. Cao, X. Hao, W. Li, and J. Chen, Enhanced electron–electron entanglement in few-cycle laser-atom interactions, Chin. Phys. Lett. 42, 110301 (2025).
  83. C. R. Givens and R. M. Shortt, A class of Wasserstein metrics for probability distributions, Mich. Math. J. 31, 231 (1984).

Outline

Information

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation