The Hartman effect has been investigated in various ways by extending the system not only for a single barrier but also for double [8, 9] and multiple barrier [10, 11] structures. Olkhovsky, Recami, and Salesi came out with the idea that for non-resonant tunneling through two potential barriers, the tunneling time (which is a phase time) is
independent not only of the barrier width but also of the barrier separation [8]. The approximations introduced in this reference to obtain the unknown coefficients, led these authors to unphysical results like the generalized Hartman effect. This has been called the generalized Hartman effect (GHE). The two-barrier problem can be solved without approximations, see for example, in the work of Pereyra [12]. An experiment to check this effect was performed by check details Longhi et learn more al. [10]
where optical pulses of 1,550 nm wavelength were transmitted through a double-barrier system of Bragg gratings. In this reference, non-conclusive and inappropriately presented results for five different values of the gratings separation were reported. Most of the theoretical conclusions were based on questionable formulas and unnecessarily involved calculations. For example, Equation 2 (used in Equations 3 and 4) of [8] is not the actual transmission coefficient through a double Bragg grating. A criticism on the mathematical rigor on GHE is also given by S. Kudaka and S. Matsumoto [13, 14]. It is easy to check from a straightforward calculation, or from the precise and general formulas published in [7] as quoted below, that the phase time for a double barrier (DB) with separation L has the structure SCH 900776 cost (1) with T 2 and T the double- and single-barrier transmission coefficients, respectively, k the wave number, ω the frequency and A i , A r , F, and G simple functions of the potential parameters (P. Pereyra and Flucloronide H. P. Simanjuntak, unpublished work). Despite this clear dependence on L, involved and contradictory
arguments lead to establish that τ becomes independent of L[8, 10, 11]. In the following we will consistently use a for the separation between barriers. For the inference of a generalized Hartman effect to be meaningful for multi-barriers, double superlattices (SLs) or double Bragg gratings (BG), one would of course need to keep the physical parameters [like the energy (wavelength) of the particle (wave)] fixed as the length between barriers is increased. The tunneling and transmission times behavior should be taken with care when one tries to find a Hartman effect due to barrier separation in multi-barrier systems [8, 11] since, in general, the density of resonance energies grows rapidly as the separation increases. It is well known that the non-resonant gaps in the band structure of a SL or a BG become resonating when these systems are divided and separated; and the separation is increasingly varied. This was already recognized in [15] (for double SL) and in [10] (for double BG).