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arXiv:2010.02755 · Europhysics Letters (EPL)

General(ized) Hartman effect

Mohammad Hasan, Bhabani Prasad Mandal

Published Jan 1, 2021• 5 citations
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Abstract

Abstract In this letter we prove explicitly that if the Hartman effect exists for an arbitrary “unit cell” potential, then it also exists for a periodic system constructed using the same “unit cell” potential repeatedly. We further show that if the Hartman effect exists, the tunneling time in the limiting case of a sufficiently thick “unit cell” potential is the same as that of its periodic system. This is true for any arbitrary value of the intervening gap between the consecutive “unit cell” of the periodic system. Thus, the generalized Hartman effect always occurs for a general potential constructed using multiple copies of single potential which shows Hartman effect.

DOI
10.1209/0295-5075/133/20001

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Abstract In this letter we prove explicitly that if the Hartman effect exists for an arbitrary “unit cell” potential, then it also exists for a periodic system constructed using the same “unit cell” potential repeatedly. We further show that if the Hartman effect exists, the tunneling time in the limiting case of a sufficiently thick “unit cell” potential is the same as that of its periodic system. This is true for any arbitrary value of the intervening gap between the consecutive “unit cell” of the periodic system. Thus, the generalized Hartman effect always occurs for a general potential constructed using multiple copies of single potential which shows Hartman effect.
Recent papers suggest that tunneling wave packets traverse the region of allowed propagation between two potential barriers with superluminal group velocity and in a time independent of barrier separation. This phenomenon has been termed the "generalized Hartman effect" and extended to multiple barriers. Here we show that this delay time is not a transit time but a cavity lifetime. It does not imply superluminal velocity. Reported experimental verifications of this effect are reinterpreted.
With reference to a particle tunneling through two successive barriers, it seems to have been generally accepted that the tunneling time does not depend on the separation distance between the barriers. This phenomenon has been called the {\it generalized Hartman effect}. In this letter, we point out a lack of mathematical rigour in the reasoning by which this effect was deduced about ten years ago. A mathematically rigorous treatment shows us that the tunneling time does indeed depend on the length of the free space between the barriers.
In a recent review paper [Phys. Rpeorts 214 (1992) 339] we proposed, within conventional quantum mechanics, new definitions fot the subarrier tunnelling and reflection times. Aims of the present paper are: (i) presenting and analysing the results of various numerical calculations (based on our equations) on the penetration and return times $\\left \\langle tau_P_e_n \\right \\rangle $ , $\\left \\langle tau_R_e_t\\right \\rangle $ during tunnelling inside a rectangular potential barrier, for various penetration depths xfx_fxf​ ; (ii) putting forth and discussing suitable definitions. Besides of the mean values, also of the variances (or dispersions)$D\\tau_T\\ $ and $ D\\tau_R\\ $ for the tima durations of trasmission and reflection processes; (iii) mentioning that our definition $\\left \\langle tau_T \\right \\rangle $ for the average transmission time results to constituite an improvemente of the ordinary dwell-time $ \\bar tau^D^W\\ $ formula: (iv) commenting upon some recent criticism by C.R. Leavens, on the basis of our new numerical results. We stress that our numerical evaluations confirm that our approach implied, and implies, the existence of the Hartman effect: an effect that in these days (due to the theoretical connections between tunnelling and evanescent-wave propagation) is receiving - at Cologne, Berkeley, Florence and Wien - indirect, but quite interesting, experimental verifications. At last we briefly analyze some other definitions of tunnelling times.