The absorption of multiple photons when there is no resonant intermediate state is a well-known nonlinear process in atomic vapours, dyes and semiconductors. The N-photon absorption (NPA) rate for donors in semiconductors scales proportionally from hydrogenic atoms in vacuum with the dielectric constant and inversely with the effective mass, factors that carry exponents 6N and 4N, respectively, suggesting that extremely large enhancements are possible. We observed 1PA, 2PA and 3PA in Si:P with a terahertz free-electron laser. The 2PA coefficient for 1s–2s at 4.25 THz was 400,000,000 GM (=4 × 10−42 cm4 s), many orders of magnitude larger than is available in other systems. Such high cross-sections allow us to enter a regime where the NPA cross-section exceeds that of 1PA—that is, when the intensity approaches the binding energy per Bohr radius squared divided by the uncertainty time (only 3.84 MW cm2 in silicon)—and will enable new kinds of terahertz quantum control.

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The authors acknowledge the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO) for support to the FELIX Laboratory, and financial support from the UK Engineering and Physical Sciences Research Council (COMPASSS/ADDRFSS, grant no. EP/M009564/1). B.N.M. is grateful for a Royal Society Wolfson Research Merit Award. The authors also thank S. Pavlov, M. Thewalt, G. Davies and E. Linfield for useful discussions.

Author information


  1. Radboud University, Institute for Molecules and Materials, FELIX Laboratory, Nijmegen, The Netherlands

    • M. A. W. van Loon
    • , N. Stavrias
    • , K. Saeedi
    •  & B. Redlich
  2. Advanced Technology Institute and SEPNet, University of Surrey, Guildford, UK

    • Nguyen H. Le
    • , K. L. Litvinenko
    •  & B. N. Murdin
  3. London Centre for Nanotechnology and Department of Physics and Astronomy, University College London, London, UK

    • P. T. Greenland
  4. Institute of Photonics and Quantum Science, SUPA, Heriot Watt University, Edinburgh, UK

    • C. R. Pidgeon
  5. Laboratory for Solid State Physics, ETH Zurich, Zurich, Switzerland

    • G. Aeppli
  6. Institut de Physique, EPF Lausanne, Lausanne, Switzerland

    • G. Aeppli
  7. Photon Science Division, Paul Scherrer Institut, Villigen PSI, Switzerland

    • G. Aeppli


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B.N.M. designed the project. M.A.W.L., N.S., K.S., B.R. and P.T.G. performed the experiments. N.H.L. and B.N.M. provided theoretical methods and calculations. B.N.M., K.L.L., C.R.P. and G.A. wrote the paper.

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The authors declare no competing interests.

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Correspondence to B. N. Murdin.

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