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Photon-statistics force in ultrafast electron dynamics


In strong-field physics and attosecond science, intense light induces ultrafast electron dynamics. Such ultrafast dynamics of electrons in matter is at the core of phenomena such as high-harmonic generation, where these dynamics lead to the emission of extreme-ultraviolet bursts with attosecond duration. So far, all ultrafast dynamics of matter were understood to purely originate from the classical vector potential of the driving light, disregarding the influence of the quantum nature of light. Here we show theoretically that the dynamics of matter driven by bright (intense) light significantly depend on the quantum state of the driving light through its quantum noise, which induces an effective photon-statistics force. To provide a unified framework for the analysis and control over such a force, we extend the strong-field approximation theory to account for non-classical driving light. Our quantum strong-field approximation theory shows that in high-harmonic generation, experimentally feasible squeezing of the driving light can shift and shape electronic trajectories and attosecond pulses at the scale of hundreds of attoseconds. Our work presents a new degree of freedom for attosecond spectroscopy, by relying on non-classical electromagnetic fields, and more generally, introduces a direct connection between attosecond science and quantum optics.

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Fig. 1: Squeezing-dependent generation of attosecond pulses and high-order harmonics.
Fig. 2: Three-step model in a quantized field and the resulting electronic trajectories.
Fig. 3: Dependence of attosecond pulses on squeezing of the driving field.
Fig. 4: Vacuum fluctuations induce yoctosecond time delays in electronic trajectories in HHG.
Fig. 5: Influence of the effective photon-statistics force: Newtonian trajectories.

Data availability

The data supporting the findings of this study are available from the corresponding authors upon reasonable request.

Code availability

The code supporting the findings of this study is available from the corresponding authors upon reasonable request.


  1. Villeneuve, D. M. Attosecond science. Contemp. Phys. 59, 47–61 (2018).

    ADS  Google Scholar 

  2. Dudovich, N. et al. Measuring and controlling the birth of attosecond XUV pulses. Nat. Phys. 2, 781–786 (2006).

    Google Scholar 

  3. Grundmann, S. et al. Zeptosecond birth time delay in molecular photoionization. Science 370, 339–341 (2020).

    ADS  Google Scholar 

  4. Schultze, M. et al. Attosecond band-gap dynamics in silicon. Science 346, 1348–1352 (2014).

    ADS  Google Scholar 

  5. Ayuso, D. et al. Synthetic chiral light for efficient control of chiral light–matter interaction. Nat. Photon. 13, 866–871 (2019).

    ADS  Google Scholar 

  6. Mayer, N., Patchkovskii, S., Morales, F., Ivanov, M. & Smirnova, O. Imprinting chirality on atoms using synthetic chiral light fields. Phys. Rev. Lett. 129, 243201 (2022).

  7. Peng, P. et al. Coherent control of ultrafast extreme ultraviolet transient absorption. Nat. Photon. 16, 45–51 (2022).

    ADS  Google Scholar 

  8. Sainadh, U. S. et al. Attosecond angular streaking and tunnelling time in atomic hydrogen. Nature 568, 75–77 (2019).

    ADS  Google Scholar 

  9. McPherson, A. et al. Studies of multiphoton production of vacuum-ultraviolet radiation in the rare gases. J. Opt. Soc. Am. B 4, 595–601 (1987).

    Google Scholar 

  10. Ferray, M. et al. Multiple-harmonic conversion of 1064 nm radiation in rare gases. J. Phys. B: At. Mol. Opt. Phys. 21, L31–L35 (1988).

    Google Scholar 

  11. Krause, J. L., Schafer, K. J. & Kulander, K. C. High-order harmonic generation from atoms and ions in the high intensity regime. Phys. Rev. Lett. 68, 3535–3538 (1992).

    ADS  Google Scholar 

  12. Corkum, P. B. Plasma perspective on strong field multiphoton ionization. Phys. Rev. Lett. 71, 1994–1997 (1993).

    ADS  Google Scholar 

  13. Lewenstein, M., Balcou, P., Ivanov, M. Y., L’Huillier, A. & Corkum, P. B. Theory of high-harmonic generation by low-frequency laser fields. Phys. Rev. A 49, 2117–2132 (1994).

    ADS  Google Scholar 

  14. Luu, T. T. et al. Extreme–ultraviolet high–harmonic generation in liquids. Nat. Commun. 9, 3723 (2018).

    ADS  Google Scholar 

  15. Ghimire, S. et al. Observation of high-order harmonic generation in a bulk crystal. Nat. Phys. 7, 138–141 (2011).

    Google Scholar 

  16. Dromey, B. et al. High harmonic generation in the relativistic limit. Nat. Phys. 2, 456–459 (2006).

    Google Scholar 

  17. Paul, P. M. et al. Observation of a train of attosecond pulses from high harmonic generation. Science 292, 1689–1692 (2001).

    ADS  Google Scholar 

  18. Vampa, G., McDonald, C. R., Orlando, G., Corkum, P. B. & Brabec, T. Semiclassical analysis of high harmonic generation in bulk crystals. Phys. Rev. B 91, 64302 (2015).

    ADS  Google Scholar 

  19. Li, L. et al. Reciprocal-space-trajectory perspective on high-harmonic generation in solids. Phys. Rev. Lett. 122, 193901 (2019).

    ADS  Google Scholar 

  20. Amini, K. et al. Symphony on strong field approximation. Rep. Prog. Phys. 82, 116001 (2019).

    ADS  Google Scholar 

  21. Scully, M. O. & Zubairy, M. S. Quantum Optics (Cambridge Univ. Press, 1997).

  22. Jechow, A., Seefeldt, M., Kurzke, H., Heuer, A. & Menzel, R. Enhanced two-photon excited fluorescence from imaging agents using true thermal light. Nat. Photon. 7, 973–976 (2013).

    ADS  Google Scholar 

  23. Manceau, M., Spasibko, K. Y., Leuchs, G., Filip, R. & Chekhova, M. V. Indefinite-mean Pareto photon distribution from amplified quantum noise. Phys. Rev. Lett. 123, 123606 (2019).

    ADS  Google Scholar 

  24. Spasibko, K. Y. et al. Multiphoton effects enhanced due to ultrafast photon-number fluctuations. Phys. Rev. Lett. 119, 223603 (2017).

    ADS  Google Scholar 

  25. Qu, Y. & Singh, S. Photon correlation effects in second harmonic generation. Opt. Commun. 90, 111–114 (1992).

    ADS  Google Scholar 

  26. Iskhakov, T. S., Pérez, A. M., Spasibko, K. Y., Chekhova, M. V. & Leuchs, G. Superbunched bright squeezed vacuum state. Opt. Lett. 37, 1919–1921 (2012).

    ADS  Google Scholar 

  27. Pérez, A. M. et al. Bright squeezed-vacuum source with 1.1 spatial mode. Opt. Lett. 39, 2403–2406 (2014).

    ADS  Google Scholar 

  28. Finger, M. A., Iskhakov, T. S., Joly, N. Y., Chekhova, M. V. & Russell, P. S. J. Raman-free, noble-gas-filled photonic-crystal fiber source for ultrafast, very bright twin-beam squeezed vacuum. Phys. Rev. Lett. 115, 143602 (2015).

    ADS  Google Scholar 

  29. Gao, J., Shen, F. & Eden, J. G. Quantum electrodynamic treatment of harmonic generation in intense optical fields. Phys. Rev. Lett. 81, 1833–1836 (1998).

    ADS  Google Scholar 

  30. Gao, J., Shen, F. & Eden, J. G. Interpretation of high-order harmonic generation in terms of transitions between quantum Volkov states. Phys. Rev. A 61, 043812 (2000).

    ADS  Google Scholar 

  31. Tsatrafyllis, N., Kominis, I. K., Gonoskov, I. A. & Tzallas, P. High-order harmonics measured by the photon statistics of the infrared driving-field exiting the atomic medium. Nat. Commun. 8, 15170 (2017).

    ADS  Google Scholar 

  32. Lewenstein, M. et al. Generation of optical Schrödinger cat states in intense laser–matter interactions. Nat. Phys. 17, 1104–1108 (2021).

    Google Scholar 

  33. Lewenstein, M. et al. Attosecond physics and quantum information science. Preprint at (2022).

  34. Stammer, P. et al. Quantum electrodynamics of intense laser-matter interactions: a tool for quantum state engineering. PRX Quantum 4, 010201 (2023).

  35. Rivera-Dean, J. et al. New schemes for creating large optical Schrödinger cat states using strong laser fields. J. Comput. Electron. 20, 2111–2123 (2021).

    Google Scholar 

  36. Stammer, P. et al. High photon number entangled states and coherent state superposition from the extreme ultraviolet to the far infrared. Phys. Rev. Lett. 128, 123603 (2022).

  37. Gorlach, A. et al. High harmonic generation driven by quantum light: general formalism and extended cutoff. In Conference on Lasers and Electro-Optics FM3B.1 (Optica Publishing Group, 2022).

  38. Tsur, M. E. et al. High harmonic generation driven by quantum light: strong-field approximation & attosecond pulses. In Conference on Lasers and Electro-Optics FW4B.1 (Optica Publishing Group, 2022).

  39. Kim, M. S., de Oliveira, F. A. M. & Knight, P. L. Properties of squeezed number states and squeezed thermal states. Phys. Rev. A 40, 2494–2503 (1989).

    ADS  Google Scholar 

  40. Grynberg, G., Aspect, A., Fabre, C. & Cohen-Tannoudji, C. Introduction to Quantum Optics: From the Semi-Classical Approach to Quantized Light (Cambridge Univ. Press, 2010).

  41. Janszky, J. & Yushin, Y. Many-photon processes with the participation of squeezed light. Phys. Rev. A 36, 1288 (1987).

    ADS  Google Scholar 

  42. Lecompte, C., Mainfray, G., Manus, C. & Sanchez, F. Laser temporal-coherence effects on multiphoton ionization processes. Phys. Rev. A 11, 1009 (1975).

    ADS  Google Scholar 

  43. Lamprou, T., Liontos, I., Papadakis, N. C. & Tzallas, P. A perspective on high photon flux nonclassical light and applications in nonlinear optics. High Power Laser Sci. Eng. 8, e42 (2020).

    Google Scholar 

  44. Heimerl, J. et al. Statistics of multiphoton photoemission under coherent and non-classical illumination. In Conference on Lasers and Electro-Optics, Technical Digest Series FM5D.6 (Optica Publishing Group, 2022).

  45. Paris, M. G. A. Displacement operator by beam splitter. Phys. Lett. A 217, 78–80 (1996).

    ADS  Google Scholar 

  46. Tzur, M. E., Neufeld, O., Bordo, E., Fleischer, A. & Cohen, O. Selection rules in symmetry-broken systems by symmetries in synthetic dimensions. Nat. Commun. 13, 1312 (2022).

    ADS  Google Scholar 

  47. Gorlach, A., Neufeld, O., Rivera, N., Cohen, O. & Kaminer, I. The quantum-optical nature of high harmonic generation. Nat. Commun. 11, 4598 (2020).

    ADS  Google Scholar 

  48. Pizzi, A., Gorlach, A., Rivera, N., Nunnenkamp, A. & Kaminer, I. Light emission from strongly driven many-body systems. Nat. Phys. 19, 551–561 (2023).

  49. Pedatzur, O. et al. Attosecond tunnelling interferometry. Nat. Phys. 11, 815–819 (2015).

    Google Scholar 

  50. Milošević, D. B. & Becker, W. Role of long quantum orbits in high-order harmonic generation. Phys. Rev. A 66, 063417 (2002).

    ADS  Google Scholar 

  51. Kira, M., Koch, S. W., Smith, R. P., Hunter, A. E. & Cundiff, S. T. Quantum spectroscopy with Schrödinger-cat states. Nat. Phys. 7, 799–804 (2011).

    Google Scholar 

  52. Lee, R.-K. Squeezed light illustration by QuantumStateDistributions.jl. GitHub (2022).

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We thank the Helen Diller Quantum Center for their support. This work was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (819440-TIMP), and by the Israel Science Foundation (grant no. 1781/18). M.E.T. gratefully acknowledges support from the Council for Higher Education scholarship for excellence in quantum science and technology and the Helen Diller scholarship for excellence in quantum science and technology.

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Authors and Affiliations



M.E.T. and O.C. initiated this research. A.G. derived the theory presented in Supplementary Section I. M.E.T., M.B. and A.G. derived the theory in Supplementary Section II. M.E.T. and M.B. derived the theory in Supplementary Sections IIIB and IVA. M.E.T. derived the remaining theory, performed the numerical calculations and wrote the first draft of the paper. The project was supervised by O.C., I.K. and M.K. All authors discussed the results and contributed to the writing of the manuscript.

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Correspondence to Matan Even Tzur or Oren Cohen.

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Nature Photonics thanks Maciej Lewenstein, Dejan B. Milosevic and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Supplementary Sections I–VII and derivations.

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Even Tzur, M., Birk, M., Gorlach, A. et al. Photon-statistics force in ultrafast electron dynamics. Nat. Photon. 17, 501–509 (2023).

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