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Giant multiphoton absorption for THz resonances in silicon hydrogenic donors


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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Fig. 1: Si:P level scheme, with some relevant even (blue) and odd (red) parity states.
Fig. 2: The experiment.
Fig. 3: N-photon photoconductance spectra for various laser attenuations.
Fig. 4: Photoconductance signal strength for some transitions from Fig. 3.


  1. Gontier, Y. & Trahin, M. On the multiphoton absorption in atomic hydrogen. Phys. Lett. A 36, 463–464 (1971).

    Article  ADS  Google Scholar 

  2. Foot, C. J., Couillaud, B., Beausoleil, R. G. & Hänsch, T. W. Continuous-wave two-photon spectroscopy of the 1S–2S transition in hydrogen. Phys. Rev. Lett. 54, 1913–1916 (1985).

    Article  ADS  Google Scholar 

  3. Žitnik, M. et al. High resolution multiphoton spectroscopy by a tunable free-electron-laser light. Phys. Rev. Lett. 113, 193201 (2014).

    Article  ADS  Google Scholar 

  4. Albota, M. et al. Design of organic molecules with large two-photon absorption cross sections. Science 281, 1653–1656 (1998).

    Article  ADS  Google Scholar 

  5. Schneider, H., Maier, T., Liu, H. C., Walther, M. & Koidl, P. Ultrasensitive femtosecond two-photon detector with resonantly enhanced nonlinear absorption. Opt. Lett. 30, 287–289 (2005).

    Article  ADS  Google Scholar 

  6. Schneider, H. et al. Terahertz two-photon quantum well infrared photodetector. Opt. Lett. 17, 12279–12284 (2009).

    Google Scholar 

  7. Venkataraman, V., Saha, K., Londero, P. & Gaeta, A. L. Few-photon all-optical modulation in a photonic band-gap fiber. Phys. Rev. Lett. 107, 193902 (2011).

    Article  ADS  Google Scholar 

  8. Atanasov, R., Haché, A., Hughes, J. L. P., van Driel, H. M. & Sipe, J. E. Coherent control of photocurrent generation in bulk semiconductors. Phys. Rev. Lett. 76, 1703–1706 (1996).

    Article  ADS  Google Scholar 

  9. Hayat, A., Ginzburg, P. & Orenstein, M. Observation of two-photon emission from semiconductors. Nat. Photon. 2, 238–241 (2008).

    Article  Google Scholar 

  10. Stufler, S. et al. Two-photon Rabi oscillations in a single In x Ga1–xAs/GaAs quantum dot. Phys. Rev. B 73, 125304 (2006).

    Article  ADS  Google Scholar 

  11. Hendrickson, S. M., Lai, M. M., Pittman, T. B. & Franson, J. D. Observation of two-photon absorption at low power levels using tapered optical fibers in rubidium vapor. Phys. Rev. Lett. 105, 173602 (2010).

    Article  ADS  Google Scholar 

  12. Gullans, M. J. & Taylor, J. M. Optical control of donor spin qubits in silicon. Phys. Rev. B 92, 195411 (2015).

    Article  ADS  Google Scholar 

  13. Murdin, B. N. et al. Infrared free-electron laser measurement of power limiting by two-photon absorption in InSb. Opt. Quantum Electron. 25, 171–175 (1993).

    Article  Google Scholar 

  14. Ganichev, S. D. et al. Multiphoton absorption in semiconductors at submillimeter wavelengths. Sov. Phys. JETP 64, 729–737 (1986).

    Google Scholar 

  15. Böhm, W., Ettlinger, E. & Prettl, W. Far-infrared two-photon transitions in n-GaAs. Phys. Rev. Lett. 47, 1198–1201 (1981).

    Article  ADS  Google Scholar 

  16. Planken, P. C. M. et al. Using far-infrared two-photon excitation to measure the resonant-polaron effect in the Reststrahlen band of GaAs:Si. Opt. Commun. 124, 258–262 (1996).

    Article  ADS  Google Scholar 

  17. Zeuner, S., Allen, S. J., Maranowski, K. D. & Gossard, A. C. Photon-assisted tunneling in GaAs/AlGaAs superlattices up to room temperature. Appl. Phys. Lett. 69, 2689–2691 (1996).

    Article  ADS  Google Scholar 

  18. Golka, J. & Mostowski, J. Two-photon spectroscopy of shallow donor states in semiconductors. Phys. Rev. B 18, 2755–2760 (1978).

    Article  ADS  Google Scholar 

  19. Bassani, F. & Quattropani, A. Two-photon spectroscopy of shallow centers in semiconductors. Solid State Commun. 53, 1077–1081 (1985).

    Article  ADS  Google Scholar 

  20. Bassani, F., Forney, J.-J. & Quattropani, A. Choice of gauge in two-photon transitions: 1s−2s transition in atomic hydrogen. Phys. Rev. Lett. 39, 1070–1073 (1977).

    Article  ADS  Google Scholar 

  21. Thayyullathil, R. B., Radhakrishnan, R. & Seema, M. Three-photon transitions from ground state to bound states in atomic hydrogen. J. Phys. A 36, 8473–8478 (2003).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Kohn, W. & Luttinger, J. M. Theory of donor states in silicon. Phys. Rev. 98, 915–922 (1955).

    Article  ADS  MATH  Google Scholar 

  23. Pajot, B. in Optical Absorption of Impurities and Defects in Semiconducting Crystals: Hydrogen-like Centres (eds Cardona, M et al.) Ch. 6 (Springer: Berlin, 2009). .

  24. Murdin, B. N. et al. Si:P as a laboratory analogue for hydrogen on high magnetic field white dwarf stars. Nat. Commun. 4, 1469 (2013).

    Article  Google Scholar 

  25. Clauws, P., Broeckx, J., Rotsaert, E. & Vennik, J. Oscillator strengths of shallow impurity spectra in germanium and silicon. Phys. Rev. B 38, 12377–12382 (1988).

    Article  ADS  Google Scholar 

  26. Chang, Y. C., McGill, T. C. & Smith, D. L. Model Hamiltonian of donors in indirect-gap materials. Phys. Rev. B 23, 4169–4182 (1981).

    Article  ADS  Google Scholar 

  27. Sauer, R. Optical determination of highly excited s-like donor states in silicon. J. Luminesc. 12, 495–499 (1976).

    Article  ADS  Google Scholar 

  28. Greenland, P. T. et al. Coherent control of Rydberg states in silicon. Nature 465, 1057–1061 (2010).

    Article  ADS  Google Scholar 

  29. Karaiskaj, D., Stotz, J. A. H., Meyer, T., Thewalt, M. L. W. & Cardona, M. Impurity absorption spectroscopy in 28Si: the importance of inhomogeneous isotope broadening. Phys. Rev. Lett. 90, 186402 (2003).

    Article  ADS  Google Scholar 

  30. Ganichev, S. D., Prettl, W. & Huggard, P. G. Phonon assisted tunnel ionization of deep impurities in the electric field of far-infrared radiation. Phys. Rev. Lett. 71, 3882–3885 (1993).

    Article  ADS  Google Scholar 

  31. Ganichev, S. D. et al. Carrier tunneling in high-frequency electric fields. Phys. Rev. Lett. 80, 2409–2412 (1998).

    Article  ADS  Google Scholar 

  32. Saha, K., Venkataraman, V., Londero, P. & Gaeta, A. L. Enhanced two-photon absorption in a hollow-core photonic-band-gap fiber. Phys. Rev. A 83, 033833 (2011).

    Article  ADS  Google Scholar 

  33. So, P. T. C., Dong, C. Y., Masters, B. R. & Berland, K. M. Two-photon excitation fluorescence microscopy. Annu. Rev. Biomed. Eng. 2, 399–429 (2000).

    Article  Google Scholar 

  34. Li, X., van Embden, J., Chon, W. M. J. & Gu, M. Enhanced two-photon absorption of CdS nanocrystal rods. Appl. Phys. Lett. 94, 103117 (2009).

  35. Feng, X., Li, Z., Li, X. & Liu, Y. Giant two-photon absorption in circular graphene quantum dots in infrared region. Sci. Rep. 6, 33260 (2016).

  36. Bristow, A. D., Rotenberg, N. & van Driel, H. M. Two-photon absorption and Kerr coefficients of silicon for 850–2200 nm. Appl. Phys. Lett. 90, 191104 (2007).

    Article  ADS  Google Scholar 

  37. Tuncel, E. et al. Free-electron laser studies of direct and indirect two-photon absorption in germanium. Phys. Rev. Lett. 70, 4146–4149 (1993).

    Article  ADS  Google Scholar 

  38. Zavriyev, A., Dupont, E., Corkum, P. B., Liu, H. C. & Biglov, Z. Direct autocorrelation measurements of mid-infrared picosecond pulses by quantum-well devices. Opt. Lett. 20, 1886–1888 (1995).

    Article  ADS  Google Scholar 

  39. Wang, T. et al. Multi-photon absorption and third-order nonlinearity in silicon at mid-infrared wavelengths. Opt. Express 21, 32192–32198 (2013).

    Article  ADS  Google Scholar 

  40. Chick, S. et al. Coherent superpositions of three states for phosphorous donors in silicon prepared using THz radiation. Nat. Commun. 8, 16038 (2017).

    Article  ADS  Google Scholar 

  41. Litvinenko, K. L. et al. Weak probe readout of coherent impurity orbital superpositions in silicon. Phys. Rev. B 94, 235207 (2016).

    Article  ADS  Google Scholar 

  42. Greenland, P. T. et al. Quantitative analysis of electrically detected Ramsey fringes in P-doped Si. Phys. Rev. B 92, 165310 (2015).

    Article  ADS  Google Scholar 

  43. Litvinenko, K. L. et al. Coherent creation and destruction of orbital wavepackets in Si:P with electrical and terahertz read-out. Nat. Commun. 6, 6549 (2015).

    Article  Google Scholar 

  44. Li, L. H. et al. Multi-watt high-power THz frequency quantum cascade lasers. Electron. Lett. 53, 799–800 (2017).

    Article  Google Scholar 

  45. Ohtani, K. et al. High performance 4.7 THz GaAs quantum cascade lasers based on four quantum wells. New J. Phys. 18, 123004 (2016).

  46. Tochitsky, S. Y., Sung, C., Trubnick, S. E., Joshi, C. & Vodopyanov, K. L. High-power tunable, 0.5–3 THz radiation source based on nonlinear difference frequency mixing of CO2 laser lines. J. Opt. Soc. Am. B 24, 2509–2516 (2007).

    Article  ADS  Google Scholar 

  47. Knippels, G. M. H. et al. Generation and complete electric-field characterization of intense ultrashort tunable far-infrared laser pulses. Phys. Rev. Lett. 83, 1578 (1999).

    Article  ADS  Google Scholar 

  48. Vinh, N. Q. et al. Silicon as a model ion trap: time domain measurements of donor Rydberg states. Proc. Natl Acad. Sci. USA 105, 10649–10653 (2008).

    Article  ADS  Google Scholar 

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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.

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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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Correspondence to B. N. Murdin.

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van Loon, M.A.W., Stavrias, N., Le, N.H. et al. Giant multiphoton absorption for THz resonances in silicon hydrogenic donors. Nature Photon 12, 179–184 (2018).

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