Superluminal tunneling through two successive barriers

Abstract

-We study the phenomenon of one-dimensional non-resonant tunneling through two successive (opaque) potential barriers, separated by an intermediate free region R, by analyzing the relevant solutions to the Schroedinger equation. We find that the total traversal time does not depend not only on the barrier widths (the so-called "Hartman effect"), but also on the R width: so that the effective velocity in the region R, between the two barriers, can be regarded as practically infinite. This agrees with the results known from the corresponding waveguide experiments, which simulated the tunneling experiment herein considered due to the known formal identity between the Schroedinger and the Helmholtz equation.

Reflectionless tunneling of light in gradient optics

“…One obtains here the result already mentioned (or observed) by several authors [17][18][19][20] that despite the fact that the "phase time" seems to indicate a superluminal propagation, the signal velocity is always subluminal, confirming a result previously obtained on the grounds of an independent calculation of the group velocity of the evanescent wave in such inhomogeneous films. 16 As a conclusion, we have shown that some gradient index films presenting a cut-off frequency -provided by their heterogeneity-induced dispersion -and used in the tunnelling regime (below their cut-off frequency) could yield reflectionless tunnelling, i.e.…”

Tunnelling time of a gaussian wave packet through two potential barriers

“…Previous papers, both theoretical [1][2][3] and experimental [4,5], have shown very interesting behaviour of a quantum wave packet, occuring when it travels through a sequence of two potential barriers. The tunnelling time, evaluated as a simple phase time following the calculation of Hartman [6], turns out to be independent of the thickness a of the barriers (as in the case of the Hartman effect [7,8] for the tunnelling through one single barrier) and, moreover, results are also independent of the distance b between the two barriers (generalised Hartman effect).…”

Hartman effect and nonlocality in quantum networks

“…Since the stored energy in the evanescent field decreases exponentially within the barrier after a certain decay distances it becomes independent of the width of the barrier. The Hartman effect has been found in one dimensional barrier tunneling [18] as well as for cases beyond one dimension as in tunneling through mesoscopic rings in presence of Aharonov-Bohm flux [22]. In the current note we extend the study of phase times for branched networks of quantum wires.…”

“…The universality of 'phase time' distributions in random and chaotic systems has already been established earlier [17]. In the case of 'not too opaque' barriers, the tunneling time evaluated either as a simple 'phase time' [5] or calculated through the analysis of the wave packet behaviour [18] becomes independent of the barrier width. This phenomenon is termed as the Hartman effect [13,18,19].…”

“…In the case of 'not too opaque' barriers, the tunneling time evaluated either as a simple 'phase time' [5] or calculated through the analysis of the wave packet behaviour [18] becomes independent of the barrier width. This phenomenon is termed as the Hartman effect [13,18,19]. This implies that for sufficiently large barriers the effective velocity of the particle can become arbitrarily large, even larger than the light speed in the vacuum (superluminal effect).…”