Foundations of Physics

, Volume 44, Issue 6, pp 678–688 | Cite as

On the Traversal Time of Barriers

Article

Abstract

Fifty years ago Hartman studied the barrier transmission time of wave packets (J Appl Phys 33:3427–3433, 1962). He was inspired by the tunneling experiments across thin insulating layers at that time. For opaque barriers he calculated faster than light propagation and a transmission time independent of barrier length, which is called the Hartman effect. A faster than light (FTL or superluminal) wave packet velocity was deduced in analog tunneling experiments with microwaves and with infrared light thirty years later. Recently, the conjectured zero time of electron tunneling was claimed to have been observed in ionizing helium inside the barrier. The calculated and measured short tunneling time arises at the barrier front. This tunneling time was found to be universal for elastic fields as well as for electromagnetic fields. Remarkable is that the delay time is the same for the reflected and the transmitted waves in the case of symmetric barriers. Several theoretical physicists predicted this strange nature of the tunneling process. However, even with this background many members of the physics community do not accept a FTL signal velocity interpretation of the experimental tunneling results. Instead a luminal front velocity was calculated to explain the FTL experimental results frequently. However, Brillouin stated in his book on wave propagation and group velocity that the front velocity is given by the group velocity of wave packets in the case of physical signals, which have only finite frequency bandwidths. Some studies assumed barriers to be cavities and the observed tunneling time does represent the cavity lifetime. We are going to discus these continuing misleading interpretations, which are found in journals and in textbooks till today.

Keywords

Faster than light velocity Front velocity Universal tunneling time Special theory of relativity Einstein causality Sommerfeld–Brillouin velocity analysis 

Mathematics Subject Classification

42.25.-p 42.50.Ct 03.65.-w 03.65.Ud 06.30.Gv 

References

  1. 1.
    Boyd, R.W., Guthier, D.J.: Controlling the velocity of light pulses. Science 326, 1074 (2009)ADSCrossRefGoogle Scholar
  2. 2.
    Nimtz, G.: Tunneling confronts special relativity. Found. Phys. 41, 1193–1199 (2011)ADSMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Longhi, S., Marano, M., Laporta, P., Belmonte, M.: Superluminal optical pulse Propagation at1.5 \(\mu \)m in periodic fiber Bragg gratings. Phys. Rev. E 64, 055602(R)(4) (2001)Google Scholar
  4. 4.
    Winful, H.G.: Energy storage in superluminal barrier tunneling: origin of the Hartman. Effect. Opt. Express 10, 1491–1496 (2002)ADSCrossRefGoogle Scholar
  5. 5.
    Winful, H.G.: Group delay, stored energy, and the tunneling of evanescent electromagnetic waves. Phys. Rev. E 68(1), 016615 (2003)ADSCrossRefGoogle Scholar
  6. 6.
    Winful, H.G.: Nature of superluminal barrier tunneling. Phys. Rev. Lett. 90(2), 023901 (2003)ADSCrossRefGoogle Scholar
  7. 7.
    Winful, H.G.: Delay time and the Hartman effect. Phys. Rev. Lett. 91, 260401 (2003)ADSCrossRefGoogle Scholar
  8. 8.
    Winful, H.G.: Mechanism for superluminal tunneling. Nature 424, 638–639 (2003)ADSCrossRefGoogle Scholar
  9. 9.
    Winful, H.G.: Apparent superluminality and the generalized Hartman effect in double-barrier tunneling. Phys. Rev. E 72, 046608 (2005)ADSCrossRefGoogle Scholar
  10. 10.
    Winful, H.G.: Tunneling time, the Hartman effect, and superluminality: a proposed resolution of an old paradox. Phys. Rep. 436, 1–69 (2006)ADSCrossRefGoogle Scholar
  11. 11.
    Yao, H.Y., Chen, N.C., Chang, T.H., Winful, H.G.: Frequency-dependent cavity lifetime and apparent superluminality in Fabry–Perot-like interferometers. Phys. Rev. A 86, 053832 (2012)ADSCrossRefGoogle Scholar
  12. 12.
    Hartman, T.E.: Tunneling of a Wave Packet. J. Appl. Phys. 33, 3427–3433 (1962)ADSCrossRefGoogle Scholar
  13. 13.
    Eckle, P., Pfeiffer, A., Cirelli, C., Staudte, A., Dörner, A., Müller, H., Büttiker, M., Keller, J.: Attosecond ionization and tunneling delay time measurements in helium. Science 322, 1525–1529 (2008)ADSCrossRefGoogle Scholar
  14. 14.
    Yakaboylu, E., Klaiber, M., Bauke, H., Hatsagortsyan, K., Keitel, C.: Relativistic features and time delay of laser-induced tunnel ionization. Phys. Rev. A 88, 063421 (2013)ADSCrossRefGoogle Scholar
  15. 15.
    Nimtz, G.: On virtual phonons, photons, and electrons. Found. Phys. 39, 1346 (2009)ADSMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Brillouin, L.: Wave Propagation in Periodic Structures, 2nd edn. Dover Publications, New York (1953)MATHGoogle Scholar
  17. 17.
    Brillouin, L.: Wave Propagation and Group Velocity. Academic Press, New York (1960)MATHGoogle Scholar
  18. 18.
    Harris, F.J.: On the use of windows for harmonic analysis with the discrete fourier transform. Proc. IEEE. 66, 51–84 (1978)ADSCrossRefGoogle Scholar
  19. 19.
    Hund, F.: Zur Deutung von Molekülspektren III. Z. Physik 43, 805–826 (1927)ADSCrossRefGoogle Scholar
  20. 20.
    Sommerfeld, A.: Vorlesungen über Theoretische Physik, vol. VI. Dieterich’sche Verlagsbuchhandlung, Berlin (1950)MATHGoogle Scholar
  21. 21.
    Kapuscik, E., Henryk Niewodniczanski Institute of Nuclear Physics, Krakow; Mielke, E, Universidad Autonoma Metropolitana-Iztapalapa, Mexico. Private communications.Google Scholar
  22. 22.
    de Carvalho, C.A.A., Nussenzveig, H.M.: Time delay. Phys. Rep. 364, 83174 (2002)CrossRefGoogle Scholar
  23. 23.
    McColl, L.A.: Note on transmission and reflection of wave packets by potential barriers. Phys. Rev. 40, 621–626 (1932)ADSCrossRefGoogle Scholar
  24. 24.
    Franz, W.: Duration of the tunneling single process. Phys. Status Solidi 22, K139–K140 (1967)ADSCrossRefGoogle Scholar
  25. 25.
    Fletcher, J.R.: Time delay in tunneling through a potential barrier. J. Phys. C 18, L55 (1985)ADSCrossRefGoogle Scholar
  26. 26.
    Collins, S., Lowe, D., Barker, J.E.: The quantum mechanical tunneling time problem—revisited. J. Phys. C 20, 6213–6232 (1987)ADSCrossRefGoogle Scholar
  27. 27.
    Low, F., Mende, P.: A note on the tunneling time problem. Ann. Phys. 210, 380–387 (1991)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    Wang, Z.-Y., Xiong, C.-D.: heoretical evidence for the superluminality of evanescent modes. Phys. Rev. A 75(4), 042105 (2007)ADSCrossRefGoogle Scholar
  29. 29.
    Chiao, R.Y., Steinberg, A.M.: Tunneling times and superluminality, pp. 345–405. Progress in Optics, XXXVII (1997)Google Scholar
  30. 30.
    Steinberg, A. M.: How much time does a tunneling particle spend in the barrier region? Phys. Rev. Lett. 74, 2405–2408 (1995)Google Scholar
  31. 31.
    Sexl, R.U., Urbantke, H.K.: Relativity, Groups. Particles. Springer, Wien (2001)MATHCrossRefGoogle Scholar
  32. 32.
    Fayngold, M.: Special Relativity and Motions Faster than Light. Wiley, Weinheim (2002)CrossRefGoogle Scholar
  33. 33.
    Chiao, R. Y., Kwiat, P. G., Steinberg A. M.: Faster than light? Scientific American, August, pp. 38–46 (1993)Google Scholar
  34. 34.
    Haibel, A., Nimtz, G.: Universal relationship of time and frequency in photonic tunneling. Ann. Phys. (Leipzig) 10, 707–712 (2001)ADSCrossRefGoogle Scholar
  35. 35.
    Esposito, S.: Universal photonic tunneling time. Phys. Rev. E 64(8), 026609 (2001)ADSCrossRefGoogle Scholar
  36. 36.
    Olkhovsky, V., Recami, E.: Recent developements in the time anlysis of tunneling processes. Phys. Rep. 214, 339 (1992)ADSCrossRefGoogle Scholar
  37. 37.
    Olkhovsky, V., Recami, E., Jakiel, J.: Unified time analysis of photon and particle tunneling. Phys. Rep. 398, 133 (2004)Google Scholar
  38. 38.
    Recami, E.: Superluminal tunneling through successive barriers: does QM predict infinite group-velocities? J. Mod. Opt. 51, 913 (2004)ADSMATHMathSciNetGoogle Scholar
  39. 39.
    Barbero, A., Hernandez-Figueroa, H., Recami, E.: Propagation speed of evanescent modes. Phys. Rev. E. 62, 8628 (2000)ADSCrossRefGoogle Scholar
  40. 40.
    Aharanov, Y., Erez, N., Reznik, B.: Superoscillations and tunneling times. Phys. Rev. A 65, 052124–1 (2002)ADSCrossRefGoogle Scholar
  41. 41.
    Merzbacher, E.: Quantum Mechanics. Wiley, New York (1970)Google Scholar
  42. 42.
    Twareque Ali, S.: Evanescent waves in quantum elecgrodynamics. Phys. Rev. D 7, 1668–1673 (1073)Google Scholar
  43. 43.
    Carniglia, C.K., Mandel, L.: Quantization of evanescent modes. Phys. Rev. D 3, 280–291 (1971)ADSCrossRefGoogle Scholar
  44. 44.
    Nimtz, G.: Do evanescent modes volate causality? Lect. Notes Phys. 702, 506–531 (2006)CrossRefGoogle Scholar
  45. 45.
    Steinberg, A.M., Kwiat, P.G., Chiao, R.Y.: Measurement of the single-photon tunneling time. Phys. Rev. Lett. 71, 708–711 (1993)ADSCrossRefGoogle Scholar
  46. 46.
    Nimtz, G., Heitmann, W.: Superluminal photonic tunneling and quantum electronics. Prog. Quantum Electron. 21, 81–108 (1997)ADSCrossRefGoogle Scholar
  47. 47.
    Anderson, M.: Light seems to defy its own speed limit. New Scientist, 16. August (2007)Google Scholar
  48. 48.
    Spielmann, Ch., Szipöcs, R., Stingl, A., Krausz, F.: Tunneling of optical pulses through photonic band gaps. Phys. Rev. Lett. 73, 2308 (1994)ADSCrossRefGoogle Scholar
  49. 49.
    Enders, A., Nimtz, G.: Evanescent-mode propagation and quantum tunneling. Phys. Rev. E 48, 632 (1993)ADSCrossRefGoogle Scholar
  50. 50.
    Longhi, S., Laporta, A., Belmonte, M., Recami, E.: Measurement of superluminal optical tunneling times in double-barrier photonic band gaps. Phys. Rev. E 65, 046610 (2002)ADSCrossRefGoogle Scholar
  51. 51.
    Aichmann, H., Nimtz, G., Spieker, H.: Photonische Tunnelzeiten. Verhandlungen der Deutschen Physikalischen Gesellschaft 7, 1258 (1995)Google Scholar
  52. 52.
    Vetter, R.-M.: Simulationen von Tunnelstrukturen. http://kups.ub.uni-koeln.de/910/
  53. 53.
    Nimtz, G.: On superluminal tunneling. Progr. Quantum Electron. 27, 417 (2003)ADSCrossRefGoogle Scholar
  54. 54.
    Goos, F., Hänchen, H.: Ein neuer und fundamentaler Versuch zur Totalreflexion. Annalen Physik 6, 333 (1947)ADSCrossRefGoogle Scholar
  55. 55.
    Haibel, A., Nimtz, G., Stahlhofen, A.A.: Frustrated total reflection: The double-prism revisited. Phys. Rev. E 61, 047601 (2003)Google Scholar
  56. 56.
    Feynman, R.P.: Quantum Electrodynamics. W A Benjamin, Reading (1961)Google Scholar
  57. 57.
    Gehring, G.M., Liapis, A.C., Lukishova, S.G., Boyd, R.W.: Time-domain measurements of reflection delay in frustrated total internal reflection. Phys. Rev. Lett. 111, 030404 (2013)ADSCrossRefGoogle Scholar
  58. 58.
    Gruschinski, Nimtz, G., Stahlhofen, A.: Resonance-like Goos-Hänchen shift induced by nano-metal films. Ann. Phys. (Berlin) 17, 917–921 (2008)Google Scholar
  59. 59.
    Shannon, C.E.: A mathematical theory of communication. Bell Sys. Tech. J. 27, 379 and 623 (1948)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Bad NauheimGermany
  2. 2.II. Physikalisches InstitutUniversität zu KölnKölnGermany

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