Time dependence of the probability density in the transient regime for tunneling

Abstract

An exact analytical solution to the time-dependent Schrödinger equation with cutoff wave initial conditions is used to investigate the fast tunneling response of a rectangular potential barrier. We find that just across the tunneling region, the probability density exhibits at short times a transient behavior that may be characterized by a peak t p and a width ⌬t. We show that t p provides the earliest tunneling response of the system and that the top-barrier S-matrix poles play an important role in the process. As a function of the barrier width, t p exhibits two regimes. Along the first regime, t p remains almost a constant; as the barrier width increases, a second regime appears where t p grows linearly with the barrier width.M (x,Ϯk;t) and M (x,k n ;t) are defined as M ͑ x,q;t ͒ϭ 1 2 e (imx 2 /2បt) e y q 2 erfc͑ y q ͒, ͑3͒where the argument y q is given by y q ϵe Ϫi/4 ͩ m 2បt ͪ 1/2 ͫ xϪ បq m t ͬ .

Wavepacket basis for time-dependent processes and its application to relaxation in resonant electronic transport

“…The time-dependence in tunnelling has been addressed using various wavepackets by many authors in the past, e.g. [22,23,24,25,26,27]. The distinguishing contribution of the present treatment is the correct many-electron occupation corresponding to finite bias and low temperatures and hence its direct consequences for the current-voltage characteristics.…”

Delay time and tunneling transient phenomena

“…[6] a cutoff wave initial state has, in addition to tunneling components, momentum components that go above the barrier height. One sees from Eq.…”

“…On the theoretical side the above achievements have stimulated work on the issue of time-dependent tunneling. In particular, one finds a number of works that deal with the solution to the time-dependent Schrödinger's equation for cutoff wave initial states [3,4,5,6]. One interesting feature of these approaches is that at asymptotically long times the time-dependent solution goes into the well known stationary solution.…”

Equivalence between the real-time Feynman histories and the quantum-shutter approaches for the “passage time” in tunneling

“…There are a number of approaches to this problem [2]. In this paper, an unexpected close relationship is found between a real time Feynman path integral approach [3,4,5] and the quantum shutter approach [6,7], which are, at first sight, unlikely to be related.…”

“…Since initially there is no particle along the tunneling region, detecting the particle at the barrier edge at time t should provide a relevant time scale of the tunneling process. In recent work, two of the authors [7,13] analyzed the time evolution of the probability density |Ψ(d, t)| 2 for a rectangular potential barrier using the above formalism. There, it was found that the probability density at the right barrier edge x = d, exhibits at short times a transient structure that they named time-domain resonance.…”