Quantization of Time in Dynamic Barrier Tunnelling

TL;DR

This paper introduces a novel time quantization approach in dynamic barrier tunnelling, leading to a finite energy spectrum and new insights into inelastic scattering channels during electron transit.

Contribution

It presents an exact temporal wave-function matching method that discretizes time, resulting in a finite energy spectrum and a new understanding of inelastic scattering in tunnelling.

Findings

Derived absorption and emission times T_+ and T_- for different frequency regimes.

Showed T_+= T_- in low frequency and T_+< T_- in high frequency regimes.

Provided a finite system of inelastic scattering channels during tunnelling.

Abstract

In the B\"uttiker-Landauer perturbation approach to electron tunnelling, through a time-modulated rectilinear potential barrier, the Tien-Gordon identity was invoked, together with its infinite energy spectrum. Here, an exact treatment is presented which is based on the temporal wave-function matching procedure, that led to a finite energy spectrum. In seeking the condition governing the time evolution of the tunnelling process, the Euler formula provided the crucial ingredient for time quantization, which discretised the continuous time in the oscillating barrier potential and energy harmonic equations. As a result, a finite system of inelastic scattering channels was created. When an electron entered the elastic channel, it was scattered, instantaneously, into finite neighbouring energy-level scattering channels, by absorption (emission) of photon energy from (to) the oscillating field, during the transit period across the dynamic barrier. The absorption and emission times of barrier traversal, T_+ and T_-, respectively, were derived for the low and high frequency regimes of the barrier oscillations. Calculations revealed that in the low (high) frequency situation T_+= T_- (T_+< T_-).