The delay time for a quantum tunneling particle is independent of the thickness of the opaque barrier. This is called the Hartman effect, after Thomas Hartman who discovered it in 1962. In 2007, Nimtz and Stahlhofen demonstrated quantum tunneling of "evanescent modes" across a gap might result in virtual particles traveling faster than light.
The Hartman effect is the tunnelling effect through a barrier where the tunnelling time tends to a constant for large barriers. This was first described by Thomas E. Hartman in 1962. The Hartman effect was demonstrated with evanescent microwaves by Achim Enders and Günter Nimtz. and with infrared pulses by Longhi et al. This could, for instance, be the gap between two prisms. When the prisms are in contact, the light passes straight through, but when there is a gap, the light is refracted. There is a finite probability that a photon will tunnel across the gap rather than follow the refracted path. For large gaps between the prisms the tunnelling time approaches a constant and thus the photons appear to have crossed with a superluminal speed. Alfons Stahlhofen and Günter Nimtz developed a mathematical approach and interpretation of the nature of evanescent modes as virtual particles, which confirms the theory of the Hartmann effect.
However, an analysis by Herbert Winful suggests that the Hartman effect cannot actually be used to violate relativity by transmitting signals faster than c, because the tunnelling time "should not be linked to a velocity since evanescent waves do not propagate". Winful means by this that the photons crossing the barrier are virtual photons, only existing in the interactions and could not be propagated into the outside world.
- T.E. Hartman (1962). "Tunneling of a wave packet". Journal of Applied Physics. 33 (12): 3427. Bibcode:1962JAP....33.3427H. doi:10.1063/1.1702424.
- G. Nimtz, A.A. Stahlhofen (2007). "Macroscopic violation of special relativity". arXiv: [quant-ph].
The experimental result is that the reflected and the transmitted (tunneled) signal are received by the detectors at the same time. ... violation of the Einstein energy relation, the zero time spreading, and the non observability of evanescent modes - can be explained by identifying evanescent modes with virtual photons
- J.C. Martinez, E. Polatdemir (2006). "Origin of the Hartman effect". Physics Letters A. 351 (1–2): 31–36. Bibcode:2006PhLA..351...31M. doi:10.1016/j.physleta.2005.10.076.
- A. Enders, G.Nimtz; Nimtz, G. (1993). "Evanescent-mode propagation and quantum tunneling". Physical Review E. 48 (1): 632–634. Bibcode:1993PhRvE..48..632E. doi:10.1103/PhysRevE.48.632.
- S. Longhi; et al. (2002). "Measurement of superluminal optical tunneling times in double-barrier photonic band gaps". Physical Review E. 65 (4): 046610. arXiv:. Bibcode:2002PhRvE..65d6610L. doi:10.1103/PhysRevE.65.046610. PMID 12006050.
- A.A. Stahlhofen, G. Nimtz (2006). "Evanescent modes are virtual photons". Europhysics Letters. 76 (2): 189–195. Bibcode:2006EL.....76..189S. doi:10.1209/epl/i2006-10271-9.
- H. Winful (2006). "Tunneling time, the Hartman effect, and superluminality: A proposed resolution of an old paradox" (PDF). Physics Reports. 436 (1-2): 1–69. Bibcode:2006PhR...436....1W. doi:10.1016/j.physrep.2006.09.002.