Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Letter
  • Published:

Generation of optical Schrödinger cat states in intense laser–matter interactions


The physics of intense laser–matter interactions1,2 is described by treating the light pulses classically, anticipating no need to access optical measurements beyond the classical limit. However, the quantum nature of the electromagnetic fields is always present3. Here we demonstrate that intense laser–atom interactions may lead to the generation of highly non-classical light states. This was achieved by using the process of high-harmonic generation in atoms4,5, in which the photons of a driving laser pulse of infrared frequency are upconverted into photons of higher frequencies in the extreme ultraviolet spectral range. The quantum state of the fundamental mode after the interaction, when conditioned on the high-harmonic generation, is a so-called Schrödinger cat state, which corresponds to a superposition of two distinct coherent states: the initial state of the laser and the coherent state reduced in amplitude that results from the interaction with atoms. The results open the path for investigations towards the control of the non-classical states, exploiting conditioning approaches on physical processes relevant to high-harmonic generation.

This is a preview of subscription content, access via your institution

Access options

Buy this article

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Schematic representation of the generation of optical ‘cat’ states.
Fig. 2: Calculated Wigner functions of a Schrödinger ‘kitten’ and a ‘cat’ state.
Fig. 3: Experimental approach and Wigner function of the coherent state of the laser.
Fig. 4: Measurement of the Wigner function of the genuine Schrödinger ‘cat’ state.

Similar content being viewed by others

Data availability

Source data are provided with this paper. All other data that support the plots within this paper are available from the corresponding authors on reasonable request.

Code availability

The codes used in this study are available from the corresponding authors upon request.


  1. Mourou, G. Nobel Lecture: Extreme light physics and application. Rev. Mod. Phys. 91, 030501 (2019).

    Article  ADS  MathSciNet  Google Scholar 

  2. Strickland, D. Nobel Lecture: Generating high-intensity ultrashort optical pulses. Rev. Mod. Phys. 91, 030502 (2019).

    Article  ADS  MathSciNet  Google Scholar 

  3. Glauber, R. J. Nobel Lecture: One hundred years of light quanta. Rev. Mod. Phys. 78, 1267–1278 (2006).

    Article  ADS  Google Scholar 

  4. 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).

    Article  ADS  Google Scholar 

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

    Article  Google Scholar 

  6. Maiman, T. H. Stimulated optical radiation in ruby. Nature 187, 493–494 (1960).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  8. Delone, N. B. & Krainov, V. P. Multiphoton Processes in Atoms 2nd edn (Springer Series on Atomic, Optical, and Plasma Physics, Springer-Verlag, 2000).

  9. Protopapas, M., Keitel, C. H. & Knight, P. L. Atomic physics with super-high intense lasers. Rep. Prog. Phys. 60, 389–486 (1997).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  11. Kulander, K. C., Schafer, K. J. & Krause, J. L. in Super-Intense Laser Atom Physics (eds Piraux, B. et al.) 95–110 (NATO ASI Series B: Physics, Vol. 316, Plenum, 1993).

  12. Lewenstein, M. et al. Theory of high-harmonic generation by low-frequency laser fields. Phys. Rev. A 49, 2117–2132 (1994).

    Article  ADS  Google Scholar 

  13. Leonhardt, U. in Measuring the Quantum State of Light 1st edn (eds Knight, P. L. & Miller, A.) 98–143 (Cambridge Studies in Modern Optics, Cambridge Univ. Press, 1997).

  14. Ourjoumtsev, A., Jeong, H., Tualle-Brouri, R. & Grangier, P. Generation of optical ‘Schrödinger cats’ from photon number states. Nature 448, 784–786 (2007).

    Article  ADS  Google Scholar 

  15. Zavatta, A., Viciani, S. & Bellini, M. Quantum-to-classical transition with single-photon-added coherent states of light. Science 306, 660–662 (2004).

    Article  ADS  Google Scholar 

  16. Ourjoumtsev, A., Tualle-Brouri, R., Laurat, J. & Grangier, P. Generating optical Schrödinger kittens for quantum information processing. Science 312, 83–86 (2006).

    Article  ADS  Google Scholar 

  17. Wakui, K., Takahashi, H., Furusawa, A. & Sasaki, M. Photon subtracted squeezed states generated with periodically poled KTiOPO4. Opt. Express 15, 3568–3574 (2007).

    Article  ADS  Google Scholar 

  18. Acín, A. et al. The quantum technologies roadmap: a European community view. New J. Phys. 20, 080201 (2018).

    Article  Google Scholar 

  19. Walmsley, I. A. Quantum optics: science and technology in a new light. Science 348, 525–530 (2015).

    Article  ADS  MathSciNet  Google Scholar 

  20. Deutsch, I. H. Harnessing the power of the second quantum revolution. PRX Quantum 1, 020101 (2020).

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

  22. Gonoskov, I. A., Tsatrafyllis, N., Kominis, I. K. & Tzallas, P. Quantum optical signatures in strong-field laser physics: infrared photon counting in high-order harmonic generation. Sci. Rep. 6, 32821 (2016).

    Article  ADS  Google Scholar 

  23. Gombkötő, Á., Varró, S., Mati, P. & Földi, P. High-order harmonic generation as induced by a quantized field: phase-space picture. Phys. Rev. A 101, 013418 (2020).

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

  25. 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 

  26. 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).

    Article  ADS  Google Scholar 

  27. Tsatrafyllis, N. et al. Quantum optical signatures in a strong laser pulse after interaction with semiconductors. Phys. Rev. Lett. 122, 193602 (2019).

    Article  ADS  Google Scholar 

  28. Lvovsky, A. I. & Raymer, M. G. Continuous-variable optical quantum-state tomography. Rev. Mod. Phys. 81, 299–332 (2009).

    Article  ADS  Google Scholar 

  29. Schleich, W. P. Quantum Optics in Phase Space (Wiley-VHC Verlag, 2001).

  30. Breitenbach, G., Schiller, S. & Mlynek, J. Measurement of the quantum states of squeezed light. Nature 387, 471–475 (1997).

    Article  ADS  Google Scholar 

  31. Krausz, F. & Ivanov, M. Y. Attosecond physics. Rev. Mod. Phys. 81, 163–234 (2009).

    Article  ADS  Google Scholar 

  32. Nayak, A. et al. Saddle point approaches in strong field physics and generation of attosecond pulses. Phys. Rep. 833, 1–52 (2019).

    Article  ADS  MathSciNet  Google Scholar 

  33. Wünsche, A. Quantization of Gauss-Hermite and Gauss-Laguerre beams in free space. J. Opt. B 6, S47–S59 (2004).

    Article  ADS  Google Scholar 

  34. Grynberg, G., Aspect, A. & Fabre, C. Introduction to Quantum Optics (Cambridge Univ. Press, 2010).

Download references


We dedicate this work to the memory of Roy J. Glauber, the inventor of coherent states, also a wonderful mentor. We thank J. Biegert, I. Kaminer and P. Salières for enlightening discussions. We also thank I. Liontos, E. Skantzakis and S. Karsch from Max Plank Institute for Quantum Optics for their assistance on maintaining the performance of the Ti:Sa laser system and N. Pappadakis for his contribution on the development of the data acquisition and data analysis system. M.L. group acknowledges the European Research Council (ERC AdG) NOQIA, Spanish Ministry MINECO and State Research Agency AEI (FIDEUA PID2019-106901GB-I00/10.13039/501100011033, SEVERO OCHOA No. SEV-2015-0522 and CEX2019-000910-S, FPI), European Social Fund, Fundació Cellex, Fundació Mir-Puig, Generalitat de Catalunya (AGAUR grant no. 2017 SGR 1341, CERCA programme, QuantumCAT_U16-011424, co-funded by ERDF Operational Program of Catalonia 2014-2020), MINECO-EU QUANTERA MAQS (funded by State Research Agency (AEI) PCI2019-111828-2/10.13039/501100011033), EU Horizon 2020 FET-OPEN OPTOLogic (grant no. 899794), and the National Science Centre, Poland-Symfonia grant no. 2016/20/W/ST4/00314. M.F.C. acknowledges the Grantová agentura Ceské Republiky (GACR grant 20-24805J). J.R.-D. has received funding from the Secretaria d’Universitats i Recerca del Departament d’Empresa i Coneixement de la Generalitat de Catalunya, as well as the European Social Fund (L’FSE inverteix en el teu futur)–FEDER. P.T. group acknowledges LASERLABEUROPE (H2020-EU. grant ID 654148), FORTH Synergy Grant AgiIDA (grant no. 00133), the European Union’s Horizon 2020 framework programme for research and innovation under the NFFA-Europe-Pilot project (grant no. 101007417), the HELLAS-CH (MIS grant no. 5002735) (which is implemented under the Action for Strengthening Research and Innovation Infrastructures, funded by the Operational Program Competitiveness, Entrepreneurship and Innovation (NSRF 2014-2020) and co-financed by Greece and the European Union (European Regional Development Fund)), and the European Union’s Horizon 2020 research. ELI-ALPS is supported by the European Union and co-financed by the European Regional Development Fund (GINOP grant no. 2.3.6-15-2015-00001).

Author information

Authors and Affiliations



M.L. supervised the theoretical part of the work; M.F.C., J.R.-D. and E.P. equally contributed to the manuscript preparation and the development of the theoretical approach; P.S. contributed to the theoretical calculations; Th.L. contributed in the experimental runs and data analysis; P.T. supervised the experimental part of the work.

Corresponding authors

Correspondence to M. Lewenstein or P. Tzallas.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature Physics thanks the anonymous reviewers for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

Supplementary Discussion and Figs. 1–3.

Source data

Source Data Fig. 2

Calculated Wigner function of the optical kitten (Fig. 2a,b) and cat (Fig. 2c,d) states.

Source Data Fig. 3

Measured Wigner function of the laser coherent state (Fig. 3b).

Source Data Fig. 4

Measured Wigner function of the cat state.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

This article is cited by


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing