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.
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Akhiezer, N.I., Glazman, I.M.: Theory of Linear Operator in Hilbert Space, vol. I. Dover, New York (1993)
Ali, S.T.: Stochastic localization, quantum mechanics on phase space and quantum space time. Riv. Nuovo Cim. 8, 1–128 (1985)
Busch, P., Grabowski, M., Lahti, P.: Operational Quantum Physics, 2nd edn. Springer, Berlin (1997)
Büttiker, M., Landauer, R.: Traversal time for tunneling. Phys. Rev. Lett. 49, 1742–1739 (1982)
Carmeli, C., Heinonen, T., Toigo, A.: Position and momentum observables on ℝ and on ℝ3. J. Math. Phys. 45, 2526–2539 (2004)
Chiao, R.Y., Steinberg, A.M.: Prog. Opt. 37, 345 (1997)
Christ, M., Kiselev, A.: WKB asymptotics of generalized eigenfunctions of one-dimensional Schrödinger operators. J. Funct. Anal. 179, 426–447 (2001)
Christ, M., Kiselev, A.: WKB and spectral analysis of one-dimensional Schrödinger operators with slowly varying potentials. Commun. Math. Phys. 218, 245–262 (2001)
Davies, E.B.: Quantum Theory of Open Systems. Academic Press, London (1976)
Deift, P., Trubowitz, E.: Inverse scattering on the line. Commun. Pure Appl. Math. XXXII, 121–251 (1979)
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)
Dollard, J.D.: Scattering into cones I: potential scattering. Commun. Math. Phys. 12, 193–203 (1969)
Dym, H., McKean, H.P.: Fourier Series and Integrals. Academic Press, San Diego (1972)
Enders, A., Nimtz, G.: On superluminal barrier traversal. J. Phys. I France 2, 1698–1693 (1992)
Hartman, T.E.: Tunneling of a wave packet. J. Appl. Phys. 33, 3433–3427 (1962)
Holevo, A.S.: Probabilistic and Statistical Aspects of Quantum Theory. North-Holland, Amsterdam (1982)
Holevo, A.S.: Generalized imprimitivity systems for Abelian groups. Russ. Math. 27, 49–71 (1983)
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)
Ludwig, G.: Foundations of Quantum Mechanics, vol. I. Springer, Berlin (1983)
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)
Nimtz, G., Heitmann, W.: Superluminal photonic tunneling and quantum electronics. Prog. Quantum Electron. 21, 81–108 (1997)
Pauli, W.: General Principles of Quantum Theory. Springer, Berlin (1980)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics II. Academic Press, San Diego (1975)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics III. Academic Press, San Diego (1979)
Steinberg, A.M., Kwiat, P.G., Chiao, R.Y.: Measurement of a single photon tunneling time. Phys. Rev. Lett. 71, 708–711 (1993)
Titchmarsh, E.C.: Eigenfunction Expansions, 2nd edn. Oxford University Press, Oxford (1962)
Werner, R.: Screen observables in relativistic and nonrelativistic quantum mechanics. J. Math. Phys. 27, 793–803 (1986)
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Dedicated to Pekka Lahti on the occasion of his 60th birthday.
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Kiukas, J., Ruschhaupt, A. & Werner, R.F. Tunneling Times with Covariant Measurements. Found Phys 39, 829–846 (2009). https://doi.org/10.1007/s10701-009-9275-z
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DOI: https://doi.org/10.1007/s10701-009-9275-z