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:

Attosecond nonlinear polarization and light–matter energy transfer in solids


Electric-field-induced charge separation (polarization) is the most fundamental manifestation of the interaction of light with matter and a phenomenon of great technological relevance. Nonlinear optical polarization1,2 produces coherent radiation in spectral ranges inaccessible by lasers and constitutes the key to ultimate-speed signal manipulation. Terahertz techniques3,4,5,6,7,8 have provided experimental access to this important observable up to frequencies of several terahertz9,10,11,12,13. Here we demonstrate that attosecond metrology14 extends the resolution to petahertz frequencies of visible light. Attosecond polarization spectroscopy allows measurement of the response of the electronic system of silica to strong (more than one volt per ångström) few-cycle optical (about 750 nanometres) fields. Our proof-of-concept study provides time-resolved insight into the attosecond nonlinear polarization and the light–matter energy transfer dynamics behind the optical Kerr effect and multi-photon absorption. Timing the nonlinear polarization relative to the driving laser electric field with sub-30-attosecond accuracy yields direct quantitative access to both the reversible and irreversible energy exchange between visible–infrared light and electrons. Quantitative determination of dissipation within a signal manipulation cycle of only a few femtoseconds duration (by measurement and ab initio calculation) reveals the feasibility of dielectric optical switching at clock rates above 100 terahertz. The observed sub-femtosecond rise of energy transfer from the field to the material (for a peak electric field strength exceeding 2.5 volts per ångström) in turn indicates the viability of petahertz-bandwidth metrology with a solid-state device.

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

Figure 1: Attosecond spectroscopy of the nonlinear polarization.
Figure 2: Sub-femtosecond-resolved optical Kerr effect in silica.
Figure 3: The nonlinear optical polarization response of silica at critical field strengths.
Figure 4: Energy exchange between strong optical fields and electrons in real time.

Similar content being viewed by others


  1. Boyd, R. W. Nonlinear Optics (Academic Press, Elsevier, 2008)

  2. Wegener, M. Extreme Nonlinear Optics: an Introduction (Springer, 2005)

  3. Valdmanis, J. A., Mourou, G. & Gabel, C. W. Picosecond electro-optic sampling system. Appl. Phys. Lett. 41, 211–212 (1982)

    Article  ADS  Google Scholar 

  4. Wu, Q. & Zhang, X. C. Free-space electro-optic sampling of terahertz beams. Appl. Phys. Lett. 67, 3523 (1995)

    Article  CAS  ADS  Google Scholar 

  5. Sell, A., Leitenstorfer, A. & Huber, R. Phase-locked generation and field-resolved detection of widely tunable terahertz pulses with amplitudes exceeding 100 MV/cm. Opt. Lett. 33, 2767–2769 (2008)

    Article  CAS  ADS  Google Scholar 

  6. Hebling, J., Lo Yeh, K., Hoffmann, M. C. & Nelson, K. A. High-power THz generation, THz nonlinear optics, and THz nonlinear spectroscopy. IEEE J. Sel. Top. Quantum Electron. 14, 345–353 (2008)

    Article  CAS  ADS  Google Scholar 

  7. Huber, R. et al. Switching ultrastrong light–matter coupling on a subcycle scale. J. Appl. Phys. 109, 102418 (2011)

    Article  ADS  Google Scholar 

  8. Leitenstorfer, A., Nelson, K. A., Reimann, K. & Tanaka, K. Focus on nonlinear terahertz studies. New J. Phys. 16, 045016 (2014)

    Article  ADS  Google Scholar 

  9. Kuehn, W. et al. Terahertz-induced interband tunneling of electrons in GaAs. Phys. Rev. B 82, 075204 (2010)

    Article  ADS  Google Scholar 

  10. Junginger, F. et al. Nonperturbative interband response of a bulk InSb semiconductor driven off resonantly by terahertz electromagnetic few-cycle pulses. Phys. Rev. Lett. 109, 147403 (2012)

    Article  CAS  ADS  Google Scholar 

  11. Somma, C., Reimann, K., Flytzanis, C., Elsaesser, T. & Woerner, M. High-field terahertz bulk photovoltaic effect in lithium niobate. Phys. Rev. Lett. 112, 146602 (2014)

    Article  CAS  ADS  Google Scholar 

  12. Ulbricht, R., Hendry, E., Shan, J., Heinz, T. F. & Bonn, M. Carrier dynamics in semiconductors studied with time-resolved terahertz spectroscopy. Rev. Mod. Phys. 83, 543–586 (2011)

    Article  CAS  ADS  Google Scholar 

  13. Kampfrath, T., Tanaka, K. & Nelson, K. A. Resonant and nonresonant control over matter and light by intense terahertz transients. Nature Photon. 7, 680–690 (2013)

    Article  CAS  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  15. Taur, Y. & Ning, T. H. Fundamentals of Modern VLSI Devices (Cambridge Univ. Press, 2009)

  16. Markov, I. L. Limits on fundamental limits to computation. Nature 512, 147–154 (2014)

    Article  CAS  ADS  Google Scholar 

  17. Ionescu, A. M. & Riel, H. Tunnel field-effect transistors as energy-efficient electronic switches. Nature 479, 329–337 (2011)

    Article  CAS  ADS  Google Scholar 

  18. Schiffrin, A. et al. Optical-field-induced current in dielectrics. Nature 493, 70–74 (2012)

    Article  CAS  ADS  Google Scholar 

  19. Schultze, M. et al. Controlling dielectrics with the electric field of light. Nature 493, 75–78 (2012)

    Article  CAS  ADS  Google Scholar 

  20. Itatani, J. et al. Attosecond streak camera. Phys. Rev. Lett. 88, 173903 (2002)

    Article  CAS  ADS  Google Scholar 

  21. Kienberger, R. et al. Atomic transient recorder. Nature 427, 817–821 (2004)

    Article  CAS  ADS  Google Scholar 

  22. Pati, A. P., Wahyutama, I. S. & Pfeiffer, A. N. Subcycle-resolved probe retardation in strong-field pumped dielectrics. Nature Commun. 6, 7746 (2015)

    Article  ADS  Google Scholar 

  23. Loriot, V., Hertz, E., Faucher, O. & Lavorel, B. Measurement of high order Kerr refractive index of major air components. Opt. Express 17, 13429–13434 (2009); erratum 18, 3011–3012 (2010)

    Article  ADS  Google Scholar 

  24. Brée, C., Demircan, A. & Steinmeyer, G. Saturation of the all-optical Kerr effect. Phys. Rev. Lett. 106, 183902 (2011)

    Article  ADS  Google Scholar 

  25. Geissler, M. et al. Light propagation in field-ionizing media: extreme nonlinear optics. Phys. Rev. Lett. 83, 2930–2933 (1999)

    Article  CAS  ADS  Google Scholar 

  26. Yabana, K., Sugiyama, T., Shinohara, Y., Otobe, T. & Bertsch, G. Time-dependent density functional theory for strong electromagnetic fields in crystalline solids. Phys. Rev. B 85, 045134 (2012)

    Article  ADS  Google Scholar 

  27. Yablonovitch, E., Heritage, J. P., Aspnes, D. E. & Yafet, Y. Virtual photoconductivity. Phys. Rev. Lett. 63, 976–979 (1989)

    Article  CAS  ADS  Google Scholar 

  28. Cavaleri, A. L. et al. Intense 1.5-cycle near infrared laser waveforms and their use for the generation of ultra-broadband soft-x-ray harmonic continua. New J. Phys. 9, 242 (2007)

    Article  ADS  Google Scholar 

  29. Milam, D. Review and assessment of measured values of the nonlinear refractive-index coefficient of fused silica. Appl. Opt. 37, 546–550 (1998)

    Article  CAS  ADS  Google Scholar 

Download references


We acknowledge discussions with M. Stockman and V. Apalkov. This work was supported by the Max Planck Society and the Deutsche Forschungsgemeinschaft Cluster of Excellence: Munich Centre for Advanced Photonics ( M.S. was supported by a Marie Curie International Outgoing Fellowship (FP7-PEOPLE-2011-IOF). E.M.B. acknowledges funding from the European Community’s Seventh Framework Programme (FP7/2007-2013) under grant agreement number 290605 (PSI-FELLOW/COFUND) and from the Swiss National Science Foundation through NCCR MUST. This research is based upon work supported by the US Air Force Office of Scientific Research under award number FA9550-16-1-0073 and used computational resources of the K computer provided by the RIKEN Advanced Institute for Computational Science through the HPCI System Research project (Project ID: hp140103).

Author information

Authors and Affiliations



F.K. and M.S. initiated, conceived and supervised the study. A.S. and E.M.B. developed the experimental method. A.S., E.M.B. and C.J. (in close cooperation with T.L., O.R., M.J., W.S. and V.S.) prepared and performed the experiment. S.A.S., H.F., K.Y. and N.K. accomplished the theoretical modelling. A.S., E.M.B., V.S.Y., R.K., N.K., M.S. and F.K. analysed and interpreted the experimental data. All authors discussed the results and contributed to the final manuscript.

Corresponding authors

Correspondence to M. Schultze or F. Krausz.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Supplementary information

Supplementary Information

This file contains Supplementary text and Data, Supplementary Figures 1-16 and Supplementary References. (PDF 2139 kb)

PowerPoint slides

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sommer, A., Bothschafter, E., Sato, S. et al. Attosecond nonlinear polarization and light–matter energy transfer in solids. Nature 534, 86–90 (2016).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

This article is cited by


By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.


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