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Quantum phenomena in attosecond science

Abstract

The ability to manipulate and observe phenomena on attosecond timescales has yielded groundbreaking insights into electron dynamics and the behaviour of matter exposed to intense light fields. The interdisciplinary field of attosecond science connects various research areas, including quantum optics, quantum chemistry and quantum information science. However, the intrinsic quantum effects in attosecond science have been largely ignored. In this Perspective, we discuss the latest theoretical and experimental advances in exploring and understanding quantum phenomena within attosecond science. We focus on distinguishing genuinely quantum observations from classical phenomena in the context of high-harmonic generation and above-threshold ionization. Additionally, we illuminate the often overlooked yet important role of entanglement in attosecond processes, elucidating its influence on experimental outcomes.

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Fig. 1: Strong-field processes.
Fig. 2: Homodyne measurements to reconstruct the Wigner function.
Fig. 3: Examples of Wigner functions.
Fig. 4: The application of entanglement measures in attoscience.

References

  1. Agostini, P., Fabre, F., Mainfray, G., Petite, G. & Rahman, N. K. Free-free transitions following six-photon ionization of xenon atoms. Phys. Rev. Lett. 42, 1127 (1979).

    Article 
    ADS 

    Google Scholar 

  2. Becker, W. et al. Above-threshold ionization: from classical features to quantum effects. Adv. At. Mol. Opt. Phys. 48, 35–98 (2002).

    Article 
    ADS 

    Google Scholar 

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

    Article 

    Google Scholar 

  4. l’Huillier, A., Lompre, L., Mainfray, G. & Manus, C. Multiply charged ions induced by multiphoton absorption in rare gases at 0.53 μm. Phys. Rev. A 27, 2503 (1983).

    Article 
    ADS 

    Google Scholar 

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

    Article 
    ADS 

    Google Scholar 

  6. Hentschel, M. et al. Attosecond metrology. Nature 414, 509–513 (2001).

    Article 
    ADS 

    Google Scholar 

  7. Corkum, P. B. & Krausz, F. Attosecond science. Nat. Phys. 3, 381–387 (2007).

    Article 

    Google Scholar 

  8. Calegari, F., Sansone, G., Stagira, S., Vozzi, C. & Nisoli, M. Advances in attosecond science. J. Phys. B At. Mol. Opt. Phys. 49, 062001 (2016).

    Article 
    ADS 

    Google Scholar 

  9. Armstrong, G. S. et al. Dialogue on analytical and ab initio methods in attoscience. Eur. Phys. J. D At. Mol. Opt. Phys. 75, 209 (2021).

    ADS 

    Google Scholar 

  10. 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 (1994).

    Article 
    ADS 

    Google Scholar 

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

    Article 
    ADS 

    Google Scholar 

  12. Horodecki, R., Horodecki, P., Horodecki, M. & Horodecki, K. Quantum entanglement. Rev. Mod. Phys. 81, 865–942 (2009).

    Article 
    ADS 
    MathSciNet 

    Google Scholar 

  13. Vrakking, M. J. J. Ion-photoelectron entanglement in photoionization with chirped laser pulses. J. Phys. B At. Mol. Opt. Phys. 55, 134001 (2022).

    Article 
    ADS 

    Google Scholar 

  14. Koll, L.-M., Maikowski, L., Drescher, L., Witting, T. & Vrakking, M. J. Experimental control of quantum-mechanical entanglement in an attosecond pump-probe experiment. Phys. Rev. Lett. 128, 043201 (2022).

    Article 
    ADS 

    Google Scholar 

  15. Nishi, T., Lötstedt, E. & Yamanouchi, K. Entanglement and coherence in photoionization of H2 by an ultrashort XUV laser pulse. Phys. Rev. A 100, 013421 (2019).

    Article 
    ADS 

    Google Scholar 

  16. Busto, D. et al. Probing electronic decoherence with high-resolution attosecond photoelectron interferometry. Eur. Phys. J. D At. Mol. Opt. Phys. 76, 112 (2022).

    ADS 

    Google Scholar 

  17. Ruberti, M., Averbukh, V. & Mintert, F. Bell test of quantum entanglement in attosecond photoionization. Preprint at https://doi.org/10.48550/arXiv.2312.05036 (2023).

  18. Nabekawa, Y. & Midorikawa, K. Analysis of attosecond entanglement and coherence using feasible formulae. Phys. Rev. Res. 5, 033083 (2023).

    Article 

    Google Scholar 

  19. Christov, I. P. Phase-locking mechanism in non-sequential double ionization. Appl. Phys. B 125, 209 (2019).

    Article 
    ADS 

    Google Scholar 

  20. Maxwell, A. S., Madsen, L. B. & Lewenstein, M. Entanglement of orbital angular momentum in non-sequential double ionization. Nat. Commun. 13, 4706 (2022).

    Article 
    ADS 

    Google Scholar 

  21. Younis, D., Xie, S. & Eberly, J. H. Quantum entanglement during single-cycle nonsequential ionization. Preprint at https://doi.org/10.48550/arXiv.2403.09854 (2024).

  22. McKemmish, L. K., McKenzie, R. H., Hush, N. S. & Reimers, J. R. Quantum entanglement between electronic and vibrational degrees of freedom in molecules. J. Chem. Phys. 135, 244110 (2011).

    Article 
    ADS 

    Google Scholar 

  23. Vatasescu, M. Entanglement between electronic and vibrational degrees of freedom in a laser-driven molecular system. Phys. Rev. A 88, 063415 (2013).

    Article 
    ADS 

    Google Scholar 

  24. Vatasescu, M. Measures of electronic-vibrational entanglement and quantum coherence in a molecular system. Phys. Rev. A 92, 042323 (2015).

    Article 
    ADS 

    Google Scholar 

  25. Izmaylov, A. F. & Franco, I. Entanglement in the Born–Oppenheimer approximation. J. Chem. Theory Comput. 13, 20–28 (2016).

    Article 

    Google Scholar 

  26. Sanz-Vicario, J. L., Pérez-Torres, J. F. & Moreno-Polo, G. Electronic-nuclear entanglement in H2+: Schmidt decomposition of non-Born-Oppenheimer wave functions expanded in nonorthogonal basis sets. Phys. Rev. A 96, 022503 (2017).

    Article 
    ADS 

    Google Scholar 

  27. Blavier, M., Levine, R. D. & Remacle, F. Time evolution of entanglement of electrons and nuclei and partial traces in ultrafast photochemistry. Phys. Chem. Chem. Phys. 24, 17516–17525 (2022).

    Article 

    Google Scholar 

  28. Sundaram, B. & Milonni, P. W. High-order harmonic generation: simplified model and relevance of single-atom theories to experiment. Phys. Rev. A 41, 6571 (1990).

    Article 
    ADS 

    Google Scholar 

  29. Xu, H. Non-perturbative theory of harmonic generation under a high-intensity laser field. Z. Phys. D At. Mol. Clust. 28, 27–36 (1993).

    Article 
    ADS 

    Google Scholar 

  30. Compagno, G., Dietz, K. & Persico, F. QED theory of harmonic emission by a strongly driven atom. J. Phys. B At. Mol. Opt. Phys. 27, 4779 (1994).

    Article 
    ADS 

    Google Scholar 

  31. Becker, W., Lohr, A., Kleber, M. & Lewenstein, M. A unified theory of high-harmonic generation: application to polarization properties of the harmonics. Phys. Rev. A 56, 645 (1997).

    Article 
    ADS 

    Google Scholar 

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

    Article 
    ADS 

    Google Scholar 

  33. Diestler, D. Harmonic generation: quantum-electrodynamical theory of the harmonic photon-number spectrum. Phys. Rev. A 78, 033814 (2008).

    Article 
    ADS 

    Google Scholar 

  34. Bogatskaya, A., Volkova, E., Kharin, V. Y. & Popov, A. Polarization response in extreme nonlinear optics: when can the semiclassical approach be used? Laser Phys. Lett. 13, 045301 (2016).

    Article 
    ADS 

    Google Scholar 

  35. Yangaliev, D. N., Krainov, V. P. & Tolstikhin, O. I. Quantum theory of radiation by nonstationary systems with application to high-order harmonic generation. Phys. Rev. A 101, 013410 (2020).

    Article 
    ADS 

    Google Scholar 

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

    Article 
    ADS 

    Google Scholar 

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

    Article 

    Google Scholar 

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

    Article 
    ADS 

    Google Scholar 

  39. Gorlach, A. et al. High-harmonic generation driven by quantum light. Nat. Phys. 19, 1689–1696 (2023).

    Article 

    Google Scholar 

  40. Stammer, P. Absence of quantum optical coherence in high harmonic generation. Phys. Rev. Res. 6, L032033 (2024).

    Article 

    Google Scholar 

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

    Article 
    ADS 

    Google Scholar 

  42. Even Tzur, M. & Cohen, O. Motion of charged particles in bright squeezed vacuum. Light Sci. Appl. 13, 41 (2024).

    Article 
    ADS 

    Google Scholar 

  43. Tzur, M. E. et al. Generation of squeezed high-order harmonics. Phys. Rev. Res. 6, 033079 (2024).

    Article 

    Google Scholar 

  44. Stammer, P. On the limitations of the semi-classical picture in high harmonic generation. Nat. Phys. 20, 1040–1042 (2024).

    Article 

    Google Scholar 

  45. Bhattacharya, U. et al. Strong-laser–field physics, non-classical light states and quantum information science. Rep. Prog. Phys. 86, 094401 (2023).

    Article 
    ADS 
    MathSciNet 

    Google Scholar 

  46. Lewenstein, M. et al. Attosecond physics and quantum information science. In International Conference on Attosecond Science and Technology 27–44 (Springer, 2012).

  47. O’brien, J. L., Furusawa, A. & Vučković, J. Photonic quantum technologies. Nat. Photon. 3, 687–695 (2009).

    Article 
    ADS 

    Google Scholar 

  48. Gilchrist, A. et al. Schrödinger cats and their power for quantum information processing. J. Opt. B Quantum Semiclassical Opt. 6, S828 (2004).

    Article 

    Google Scholar 

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

  50. Stammer, P. & Lewenstein, M. Quantum optics as applied quantum electrodynamics is back in town. Acta Phys. Pol. A 143, S42 (2023).

    Article 
    ADS 

    Google Scholar 

  51. 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).

    Article 
    ADS 

    Google Scholar 

  52. Stammer, P. Theory of entanglement and measurement in high-order harmonic generation. Phys. Rev. A 106, L050402 (2022).

    Article 
    ADS 
    MathSciNet 

    Google Scholar 

  53. Stammer, P. et al. Entanglement and squeezing of the optical field modes in high harmonic generation. Phys. Rev. Lett. 132, 143603 (2024).

    Article 
    ADS 
    MathSciNet 

    Google Scholar 

  54. Rivera-Dean, J. et al. Strong laser fields and their power to generate controllable high-photon-number coherent-state superpositions. Phys. Rev. A 105, 033714 (2022).

    Article 
    ADS 

    Google Scholar 

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

    Article 

    Google Scholar 

  56. Lange, C. S., Hansen, T. & Madsen, L. B. Electron-correlation-induced nonclassicality of light from high-order harmonic generation. Phys. Rev. A 109, 033110 (2024).

    Article 
    ADS 
    MathSciNet 

    Google Scholar 

  57. Gerry, C. C. & Knight, P. L. Introductory Quantum Optics (Cambridge Univ. Press, 2023).

  58. Schleich, W. P. Quantum Optics in Phase Space (Wiley, 2011).

  59. 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).

    Article 

    Google Scholar 

  60. Mandel, L. Non-classical states of the electromagnetic field. Phys. Scr. 1986, 34 (1986).

    Article 

    Google Scholar 

  61. Loudon, R. & Knight, P. L. Squeezed light. J. Mod. Opt. 34, 709–759 (1987).

    Article 
    ADS 
    MathSciNet 

    Google Scholar 

  62. Hudson, R. L. When is the Wigner quasi-probability density non-negative? Rep. Prog. Phys. 6, 249–252 (1974).

    MathSciNet 

    Google Scholar 

  63. Lamprou, T., Rivera-Dean, J., Stammer, P., Lewenstein, M. & Tzallas, P. Nonlinear optics using intense optical Schrödinger “cat” states. Preprint at https://doi.org/10.48550/arXiv.2306.14480 (2023).

  64. Walls, D. F. Squeezed states of light. Nature 306, 141–146 (1983).

    Article 
    ADS 

    Google Scholar 

  65. Lemieux, S. et al. Photon bunching in high-harmonic emission controlled by quantum light. Preprint at https://doi.org/10.48550/arXiv.2404.05474 (2024).

  66. Rasputnyi, A. et al. High harmonic generation by bright squeezed vacuum. Preprint at https://doi.org/10.48550/arXiv.2403.15337 (2024).

  67. Sloan, J. et al. Entangling extreme ultraviolet photons through strong field pair generation. Preprint at https://doi.org/10.48550/arXiv.2309.16466 (2023).

  68. Yi, S., Babushkin, I., Smirnova, O. & Ivanov, M. Generation of massively entangled bright states of light during harmonic generation in resonant media. Preprint at https://doi.org/10.48550/arXiv.2401.02817 (2024).

  69. Theidel, D. et al. Evidence of the quantum-optical nature of high-harmonic generation. Preprint at https://doi.org/10.48550/arXiv.2405.15022 (2024).

  70. Reid, M. & Walls, D. Violations of classical inequalities in quantum optics. Phys. Rev. A 34, 1260 (1986).

    Article 
    ADS 

    Google Scholar 

  71. Wasak, T., Szańkowski, P., Ziń, P., Trippenbach, M. & Chwedeńczuk, J. Cauchy-Schwarz inequality and particle entanglement. Phys. Rev. A 90, 033616 (2014).

    Article 
    ADS 

    Google Scholar 

  72. Hillery, M. & Zubairy, M. S. Entanglement conditions for two-mode states. Phys. Rev. Lett. 96, 050503 (2006).

    Article 
    ADS 
    MathSciNet 

    Google Scholar 

  73. Rivera-Dean, J. et al. Light-matter entanglement after above-threshold ionization processes in atoms. Phys. Rev. A 106, 063705 (2022).

    Article 
    ADS 

    Google Scholar 

  74. Milošević, D. Quantum theory of photon emission during strong-laser-field-induced ionization. Phys. Rev. A 108, 033110 (2023).

    Article 
    ADS 
    MathSciNet 

    Google Scholar 

  75. Fang, Y., Sun, F.-X., He, Q. & Liu, Y. Strong-field ionization of hydrogen atoms with quantum light. Phys. Rev. Lett. 130, 253201 (2023).

    Article 
    ADS 
    MathSciNet 

    Google Scholar 

  76. Wang, S. & Lai, X. High-order above-threshold ionization of an atom in intense quantum light. Phys. Rev. A 108, 063101 (2023).

    Article 
    ADS 

    Google Scholar 

  77. Sansone, G. et al. Electron localization following attosecond molecular photoionization. Nature 465, 763–766 (2010).

    Article 
    ADS 

    Google Scholar 

  78. Goulielmakis, E. et al. Real-time observation of valence electron motion. Nature 466, 739–743 (2010).

    Article 
    ADS 

    Google Scholar 

  79. Calegari, F. et al. Ultrafast electron dynamics in phenylalanine initiated by attosecond pulses. Science 346, 336–339 (2014).

    Article 
    ADS 

    Google Scholar 

  80. Kienberger, R. et al. Steering attosecond electron wave packets with light. Science 297, 1144–1148 (2002).

    Article 
    ADS 

    Google Scholar 

  81. Cederbaum, L. & Zobeley, J. Ultrafast charge migration by electron correlation. Chem. Phys. Lett. 307, 205–210 (1999).

    Article 
    ADS 

    Google Scholar 

  82. Despré, V. et al. Attosecond hole migration in benzene molecules surviving nuclear motion. J. Phys. Chem. Lett. 6, 426–431 (2015).

    Article 

    Google Scholar 

  83. Du, H., Covington, C., Leone, S. R. & Varga, K. Excited-state electronic coherence in vinyl bromide ions. Phys. Rev. A 100, 053412 (2019).

    Article 
    ADS 

    Google Scholar 

  84. Matselyukh, D. T., Despré, V., Golubev, N. V., Kuleff, A. I. & Wörner, H. J. Decoherence and revival in attosecond charge migration driven by non-adiabatic dynamics. Nat. Phys. 18, 1206–1213 (2022).

    Article 

    Google Scholar 

  85. Folorunso, A. S. et al. Attochemistry regulation of charge migration. J. Phys. Chem. A 127, 1894–1900 (2023).

    Article 

    Google Scholar 

  86. Guillemin, R. et al. Postcollision interaction effects in KLL Auger spectra following argon 1s photoionization. Phys. Rev. A 92, 012503 (2015).

    Article 
    ADS 

    Google Scholar 

  87. Carlström, S., Mauritsson, J., Schafer, K. J., L’Huillier, A. & Gisselbrecht, M. Quantum coherence in photo-ionisation with tailored XUV pulses. J. Phys. B At. Mol. Opt. Phys. 51, 015201 (2017).

    Article 
    ADS 

    Google Scholar 

  88. Ruberti, M., Patchkovskii, S. & Averbukh, V. Quantum coherence in molecular photoionization. Phys. Chem. Chem. Phys. 24, 19673–19686 (2022).

    Article 

    Google Scholar 

  89. Smirnova, O. Attosecond prints of electrons. Nature 466, 701–702 (2010).

    Article 
    ADS 

    Google Scholar 

  90. Pabst, S., Greenman, L., Ho, P. J., Mazziotti, D. A. & Santra, R. Decoherence in attosecond photoionization. Phys. Rev. Lett. 106, 053003 (2011).

    Article 
    ADS 

    Google Scholar 

  91. Karamatskou, A., Goetz, R. E., Koch, C. P. & Santra, R. Suppression of hole decoherence in ultrafast photoionization. Phys. Rev. A 101, 043405 (2020).

    Article 
    ADS 

    Google Scholar 

  92. Vrakking, M. J. Control of attosecond entanglement and coherence. Phys. Rev. Lett. 126, 113203 (2021).

    Article 
    ADS 

    Google Scholar 

  93. Bell, J. S. On the Einstein Podolsky Rosen paradox. Phys. Phys. Fiz. 1, 195–200 (1964).

    MathSciNet 

    Google Scholar 

  94. Laurell, H. et al. Continuous-variable quantum state tomography of photoelectrons. Phys. Rev. Res. 4, 033220 (2022).

    Article 

    Google Scholar 

  95. Remacle, F. & Levine, R. D. An electronic time scale in chemistry. Proc. Natl Acad. Sci. USA 103, 6793–6798 (2006).

    Article 
    ADS 

    Google Scholar 

  96. Yong, H., Sun, S., Gu, B. & Mukamel, S. Attosecond charge migration in molecules imaged by combined X-ray and electron diffraction. J. Am. Chem. Soc. 144, 20710–20716 (2022).

    Article 

    Google Scholar 

  97. Vacher, M., Bearpark, M. J., Robb, M. A. & Malhado, Ja. P. Electron dynamics upon ionization of polyatomic molecules: coupling to quantum nuclear motion and decoherence. Phys. Rev. Lett. 118, 083001 (2017).

    Article 
    ADS 

    Google Scholar 

  98. Arnold, C., Vendrell, O. & Santra, R. Electronic decoherence following photoionization: full quantum-dynamical treatment of the influence of nuclear motion. Phys. Rev. A 95, 033425 (2017).

    Article 
    ADS 

    Google Scholar 

  99. Arnold, C., Vendrell, O., Welsch, R. & Santra, R. Control of nuclear dynamics through conical intersections and electronic coherences. Phys. Rev. Lett. 120, 123001 (2018).

    Article 
    ADS 

    Google Scholar 

  100. Jia, D., Manz, J. & Yang, Y. De- and recoherence of charge migration in ionized iodoacetylene. J. Phys. Chem. Lett. 10, 4273–4277 (2019).

    Article 

    Google Scholar 

  101. Dey, D., Kuleff, A. I. & Worth, G. A. Quantum interference paves the way for long-lived electronic coherences. Phys. Rev. Lett. 129, 173203 (2022).

    Article 
    ADS 

    Google Scholar 

  102. Weber, T. et al. Correlated electron emission in multiphoton double ionization. Nature 405, 658–661 (2000).

    Article 
    ADS 

    Google Scholar 

  103. Becker, W., Liu, X., Ho, P. J. & Eberly, J. H. Theories of photoelectron correlation in laser-driven multiple atomic ionization. Rev. Mod. Phys. 84, 1011 (2012).

    Article 
    ADS 

    Google Scholar 

  104. Akoury, D. et al. The simplest double slit: interference and entanglement in double photoionization of H2. Science 318, 949–952 (2007).

    Article 
    ADS 

    Google Scholar 

  105. Ho, P. J., Panfili, R., Haan, S. L. & Eberly, J. H. Nonsequential double ionization as a completely classical photoelectric effect. Phys. Rev. Lett. 94, 093002 (2005).

    Article 
    ADS 

    Google Scholar 

  106. Hao, X. et al. Quantum effects in double ionization of argon below the threshold intensity. Phys. Rev. Lett. 112, 073002 (2014).

    Article 
    ADS 

    Google Scholar 

  107. Quan, W. et al. Quantum interference in laser-induced nonsequential double ionization. Phys. Rev. A 96, 032511 (2017).

    Article 
    ADS 

    Google Scholar 

  108. Maxwell, A. & de Morisson Faria, C. F. Controlling below-threshold nonsequential double ionization via quantum interference. Phys. Rev. Lett. 116, 143001 (2016).

    Article 
    ADS 

    Google Scholar 

  109. Harvey, T. R., Grillo, V. & McMorran, B. J. Stern-Gerlach-like approach to electron orbital angular momentum measurement. Phys. Rev. A 95, 021801 (2017).

    Article 
    ADS 

    Google Scholar 

  110. Noguchi, Y., Nakayama, S., Ishida, T., Saitoh, K. & Uchida, M. Efficient measurement of the orbital-angular-momentum spectrum of an electron beam via a Dammann vortex grating. Phys. Rev. Appl. 12, 064062 (2019).

    Article 
    ADS 

    Google Scholar 

  111. Schmidt-Böcking, H., Ullrich, J., Dörner, R. & Cocke, C. L. The COLTRIMS reaction microscope — the spyhole into the ultrafast entangled dynamics of atomic and molecular systems. Ann. Phys. 533, 2100134 (2021).

    Article 

    Google Scholar 

  112. Baldelli, N., Bhattacharya, U., González-Cuadra, D., Lewenstein, M. & Graß, T. Detecting Majorana zero modes via strong field dynamics. ACS Omega 7, 47424–47430 (2022).

    Article 

    Google Scholar 

  113. Bera, M. L. et al. Topological phase detection through high-harmonic spectroscopy in extended Su-Schrieffer-Heeger chains. Phys. Rev. B 108, 214104 (2023).

    Article 
    ADS 

    Google Scholar 

  114. Alcalà, J. et al. High-harmonic spectroscopy of quantum phase transitions in a high-Tc superconductor. Proc. Natl Acad. Sci. USA 119, e2207766119 (2022).

    Article 

    Google Scholar 

  115. Rivera-Dean, J. et al. Quantum-optical analysis of high-order harmonic generation in H2+ molecules. Phys. Rev. A 109, 033706 (2024).

    Article 
    ADS 

    Google Scholar 

  116. Lamprou, T. et al. Quantum-optical spectrometry in relativistic laser–plasma interactions using the high-harmonic generation process: a proposal. Photonics 8, 192 (2021).

    Article 

    Google Scholar 

  117. Rivera-Dean, J. et al. Nonclassical states of light after high-harmonic generation in semiconductors: a Bloch-based perspective. Phys. Rev. B 109, 035203 (2024).

    Article 
    ADS 

    Google Scholar 

  118. Rivera-Dean, J. et al. Quantum optical analysis of high-harmonic generation in solids within a Wannier-Bloch picture. Preprint at https://doi.org/10.48550/arXiv.2211.00033 (2022).

  119. Gonoskov, I. et al. Nonclassical light generation and control from laser-driven semiconductor intraband excitations. Phys. Rev. B 109, 125110 (2024).

    Article 
    ADS 

    Google Scholar 

  120. Stammer, P., Martos, T. F., Lewenstein, M. & Rajchel-Mieldzioć, G. Metrological robustness of high photon number optical cat states. Quantum Sci. Technol. 9, 045047 (2024).

    Article 

    Google Scholar 

  121. Pladevall, X. O. & Mompart, J. Applied Bohmian Mechanics: From Nanoscale Systems to Cosmology (CRC, 2019).

  122. Elsayed, T. A., Mølmer, K. & Madsen, L. B. Entangled quantum dynamics of many-body systems using Bohmian trajectories. Sci. Rep. 8, 12704 (2018).

    Article 
    ADS 

    Google Scholar 

  123. Weber, A., Khokhlova, M. & Pisanty, E. Quantum tunnelling without a barrier. Preprint at https://doi.org/10.48550/arXiv.2311.14826 (2023).

  124. Argüello-Luengo, J. et al. Analog simulation of high-harmonic generation in atoms. PRX Quantum 5, 010328 (2024).

    Article 
    ADS 

    Google Scholar 

  125. Buluta, I. & Nori, F. Quantum simulators. Science 326, 108–111 (2009).

    Article 
    ADS 

    Google Scholar 

  126. Ruberti, M. Onset of ionic coherence and ultrafast charge dynamics in attosecond molecular ionisation. Phys. Chem. Chem. Phys. 21, 17584–17604 (2019).

    Article 

    Google Scholar 

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Acknowledgements

The authors thank the organizers of the ‘Quantum Battles in Attoscience 2023’ conference for the opportunity and support during the preparation for the discussion. L.C.-R. acknowledges financial support from the AQuA DIP project, grant number EP/J019143/1. A.F. acknowledges financial support from the Cluster of Excellence ‘CUI: Advanced Imaging of Matter’ of the Deutsche Forschungsgemeinschaft (DFG) – EXC 2056 – project ID 390715994, from the International Max Planck Graduate School for Ultrafast imaging & Structural Dynamics (IMPRS-UFAST) and from the Christiane Nüsslein-Vollhard-Foundation. D.D. acknowledges funding from the National Science Foundation under grant number CHE-2347622 and the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant agreement number 892554. P.S. acknowledges funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement number 847517. Institut de Ciencies Fotoniques (ICFO) group acknowledges support from the following: Ministerio de Ciencia y Innovation Agencia Estatal de Investigaciones (R&D project CEX2019-000910-S, AEI/10.13039/501100011033, Plan National FIDEUA PID2019-106901GB-I00, FPI), Fundació Privada Cellex, Fundació Mir-Puig, Generalitat de Catalunya (AGAUR Grant number 2017 SGR 1341, CERCA programme), MICIIN with funding from European Union NextGenerationEU (PRTR-C17.I1), Generalitat de Catalunya, and EU Horizon 2020 FET-OPEN OPTOlogic (grant number 899794) and ERC AdG NOQIA. Views and opinions expressed are, however, those of the authors only and do not necessarily reflect those of the European Union, European Commission, European Climate, Infrastructure and Environment Executive Agency (CINEA), nor any other granting authority. Neither the European Union nor any granting authority can be held responsible for them.

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Authors and Affiliations

  1. Department of Physics and Astronomy, University College London, London, UK

    Lidice Cruz-Rodriguez

  2. Department of Chemistry, University of Warwick, Coventry, UK

    Lidice Cruz-Rodriguez

  3. Department of Chemistry, Northwestern University, Evanston, IL, USA

    Diptesh Dey

  4. Department of Physics, University of Hamburg, Hambury, Germany

    Antonia Freibert

  5. ICFO — Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Barcelona, Spain

    Philipp Stammer

  6. Atominstitut, Technische Universität Wien, Vienna, Austria

    Philipp Stammer

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Cruz-Rodriguez, L., Dey, D., Freibert, A. et al. Quantum phenomena in attosecond science.
Nat Rev Phys (2024). https://doi.org/10.1038/s42254-024-00769-2

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  • Accepted: 11 September 2024

  • Published: 07 October 2024

  • DOI: https://doi.org/10.1038/s42254-024-00769-2

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