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Foundations of Physics

, Volume 39, Issue 7, pp 829–846 | Cite as

Tunneling Times with Covariant Measurements

  • J. Kiukas
  • A. Ruschhaupt
  • R. F. Werner
Article

Abstract

We consider the time delay of massive, non-relativistic, one-dimensional particles due to a tunneling potential. In this setting the well-known Hartman effect asserts that often the sub-ensemble of particles going through the tunnel seems to cross the tunnel region instantaneously. An obstacle to the utilization of this effect for getting faster signals is the exponential damping by the tunnel, so there seems to be a trade-off between speedup and intensity. In this paper we prove that this trade-off is never in favor of faster signals: the probability for a signal to reach its destination before some deadline is always reduced by the tunnel, for arbitrary incoming states, arbitrary positive and compactly supported tunnel potentials, and arbitrary detectors. More specifically, we show this for several different ways to define “the same incoming state” and “the same detector” when comparing the settings with and without tunnel potential. The arrival time measurements are expressed in the time-covariant approach, but we also allow the detection to be a localization measurement at a later time.

Keywords

Arrival time Tunneling Hartman effect 

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References

  1. 1.
    Akhiezer, N.I., Glazman, I.M.: Theory of Linear Operator in Hilbert Space, vol. I. Dover, New York (1993) Google Scholar
  2. 2.
    Ali, S.T.: Stochastic localization, quantum mechanics on phase space and quantum space time. Riv. Nuovo Cim. 8, 1–128 (1985) CrossRefGoogle Scholar
  3. 3.
    Busch, P., Grabowski, M., Lahti, P.: Operational Quantum Physics, 2nd edn. Springer, Berlin (1997) Google Scholar
  4. 4.
    Büttiker, M., Landauer, R.: Traversal time for tunneling. Phys. Rev. Lett. 49, 1742–1739 (1982) CrossRefGoogle Scholar
  5. 5.
    Carmeli, C., Heinonen, T., Toigo, A.: Position and momentum observables on ℝ and on ℝ3. J. Math. Phys. 45, 2526–2539 (2004) zbMATHCrossRefADSMathSciNetGoogle Scholar
  6. 6.
    Chiao, R.Y., Steinberg, A.M.: Prog. Opt. 37, 345 (1997) CrossRefGoogle Scholar
  7. 7.
    Christ, M., Kiselev, A.: WKB asymptotics of generalized eigenfunctions of one-dimensional Schrödinger operators. J. Funct. Anal. 179, 426–447 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Christ, M., Kiselev, A.: WKB and spectral analysis of one-dimensional Schrödinger operators with slowly varying potentials. Commun. Math. Phys. 218, 245–262 (2001) zbMATHCrossRefADSMathSciNetGoogle Scholar
  9. 9.
    Davies, E.B.: Quantum Theory of Open Systems. Academic Press, London (1976) zbMATHGoogle Scholar
  10. 10.
    Deift, P., Trubowitz, E.: Inverse scattering on the line. Commun. Pure Appl. Math. XXXII, 121–251 (1979) CrossRefMathSciNetGoogle Scholar
  11. 11.
    Deift, P., Killip, R.: On the absolutely continuous spectrum of one-dimensional Schrödinger operators with square summable potentials. Commun. Math. Phys. 203, 341–347 (1999) zbMATHCrossRefADSMathSciNetGoogle Scholar
  12. 12.
    Dollard, J.D.: Scattering into cones I: potential scattering. Commun. Math. Phys. 12, 193–203 (1969) CrossRefADSMathSciNetGoogle Scholar
  13. 13.
    Dym, H., McKean, H.P.: Fourier Series and Integrals. Academic Press, San Diego (1972) zbMATHGoogle Scholar
  14. 14.
    Enders, A., Nimtz, G.: On superluminal barrier traversal. J. Phys. I France 2, 1698–1693 (1992) CrossRefGoogle Scholar
  15. 15.
    Hartman, T.E.: Tunneling of a wave packet. J. Appl. Phys. 33, 3433–3427 (1962) CrossRefADSGoogle Scholar
  16. 16.
    Holevo, A.S.: Probabilistic and Statistical Aspects of Quantum Theory. North-Holland, Amsterdam (1982) zbMATHGoogle Scholar
  17. 17.
    Holevo, A.S.: Generalized imprimitivity systems for Abelian groups. Russ. Math. 27, 49–71 (1983) Google Scholar
  18. 18.
    Kijowski, J.: On the time operator in quantum mechanics and the Heisenberg uncertainty relation for energy and time. Rep. Math. Phys. 6, 361–386 (1974) CrossRefADSMathSciNetGoogle Scholar
  19. 19.
    Ludwig, G.: Foundations of Quantum Mechanics, vol. I. Springer, Berlin (1983) zbMATHGoogle Scholar
  20. 20.
    Muga, J.G., Sala Mayato, R., Egusquiza, I.L. (eds.): Time in Quantum Mechanics, 2nd edn. Lecture Notes in Physics, vol. 734. Springer, Berlin (2008) zbMATHGoogle Scholar
  21. 21.
    Nimtz, G., Heitmann, W.: Superluminal photonic tunneling and quantum electronics. Prog. Quantum Electron. 21, 81–108 (1997) CrossRefADSGoogle Scholar
  22. 22.
    Pauli, W.: General Principles of Quantum Theory. Springer, Berlin (1980) Google Scholar
  23. 23.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics II. Academic Press, San Diego (1975) zbMATHGoogle Scholar
  24. 24.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics III. Academic Press, San Diego (1979) zbMATHGoogle Scholar
  25. 25.
    Steinberg, A.M., Kwiat, P.G., Chiao, R.Y.: Measurement of a single photon tunneling time. Phys. Rev. Lett. 71, 708–711 (1993) CrossRefADSGoogle Scholar
  26. 26.
    Titchmarsh, E.C.: Eigenfunction Expansions, 2nd edn. Oxford University Press, Oxford (1962) zbMATHGoogle Scholar
  27. 27.
    Werner, R.: Screen observables in relativistic and nonrelativistic quantum mechanics. J. Math. Phys. 27, 793–803 (1986) CrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Institut für Mathematische PhysikTU BraunschweigBraunschweigGermany

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