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Zeitschrift für Naturforschung A

Tunneling Through the Opaque Inverted Parabola

G.-J. Ni, W. Wonneberger

Published Mar 1, 1983• 0 citations
Publisher

Abstract

Abstract The inverted parabola remains an exactly solvable tunneling barrier when imaginary parts are added to the potential. They represent the effect of particle absorption. In the limit of strong absorption (“opaque barrier”) convenient limiting forms for the tunneling probability are derived using asymptotic expansions of the exact scattering solution. An application to Zener tunneling is made and the result is compared to the case of tunneling under viscous dissipation. Significant differences are found between these two approaches to tunneling under dissipation.

DOI
10.1515/zna-1983-0301

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Abstract The inverted parabola remains an exactly solvable tunneling barrier when imaginary parts are added to the potential. They represent the effect of particle absorption. In the limit of strong absorption (“opaque barrier”) convenient limiting forms for the tunneling probability are derived using asymptotic expansions of the exact scattering solution. An application to Zener tunneling is made and the result is compared to the case of tunneling under viscous dissipation. Significant differences are found between these two approaches to tunneling under dissipation.
The quantum tunneling rate through an inverted parabola potential in an oscillating external field is studied. The tunneling rate is calculated within quasi-classical accuracy as a function of the frequency of the field. In this calculation, the complex-time path integral method, which has been previously applied to the case of tunneling through static potentials, is used after being appropriately extended for application to the case of tunneling through explicitly time-dependent potentials. Since the Feynman kernel for the case considered in the present paper can be exactly obtained, we are able to calculate the tunneling rate over a considerably wider range of frequencies of the oscillating field than in the case that this quantity is calculated through a perturbative treatment of the external field, as done in the previous studies. The present analysis reveals that the perturbative analysis for the present problem is insufficient even in the region where it has been considered to be appropriate. From the numerical calculation of the quasi-classical tunneling rate, we find that, as the frequency of the external field increases, all of the tunneling rates are initially enhanced by a significant amount, but beyond a certain value, all tunneling rates become rapidly decaying functions of this frequency. This fact may imply that the consideration beyond quasi-classical accuracy is needed in the high frequency regime. Tunneling processes that do not conserve energy are also considered. They cannot be neglected in this time-dependent problem, and we find that tunneling rates become large when tunneling particles acquire energy during tunneling.
Abstract Quantum tunneling through potential barriers plays a central role in diverse physical phenomena ranging from nuclear fusion and radioactive decay to semiconductor devices and superconducting technologies. In this work, we calculate tunneling probabilities and wave functions for an arbitrary number of potential barriers with diverse functional forms, including square, Gaussian, parabolic, and Lorentzian profiles, using a general numerical framework. The numerical results are first validated by comparison with exact analytical solutions for double, triple and quadruple-square barrier systems, demonstrating excellent agreement. The results reveal that for a system with N barriers, a single resonance peak splits into ( N − 1 ) distinct peaks, which eventually merge into an energy band as N becomes large, in a manner fully consistent with the existing literature. A key additional result is the incorporation of absorption effects through an imaginary component of the potential, providing a realistic description of particle trapping and leading to a progressive suppression of resonant transmission. For sufficiently strong absorption, the resonant structures are completely washed out. Further, if the barriers become non-uniform or disordered, the regular resonance pattern is also observed to be completely washed out. These findings demonstrate the broad applicability of the approach to complex potential landscapes and offer insights relevant to both fundamental quantum mechanics and the design of next-generation quantum devices.
The problem of inter-band tunneling in a semiconductor (Zener breakdown) in a nonstationary and homogeneous electric field is solved exactly. Using the exact analytical solution, the approximation based on classical trajectories is studied. A new mechanism of enhanced tunneling through static non-one-dimensional barriers is proposed in addition to well known normal tunneling solely described by a trajectory in imaginary time. Under certain conditions on the barrier shape and the particle energy, the probability of enhanced tunneling is not exponentially small even for non-transparent barriers, in contrast to the case of normal tunneling.
In the case of tunneling of relativistic particles, differently from the nonrelativistic case, a limit of "transparent" barrier can also lead to an apparent "superluminal" behavior when considering the phase time. In this limit, the restricting condition of "opaque" barrier of the nonrelativistic case is avoided, nevertheless, the very thin width of a single barrier to obtain this ?transparent? limit can result in a problem itself, for probing the effect. A combination of two successive transparent barriers can show an apparent "superluminal" behavior along a macroscopic arbitrary distance "L". Two solutions for energy E above and below the potential square barrier V are found, for both solutions there the apparent superluminal behavior is possible above a threshold of free travelling group velocity (energy) and dependent on the ratio barriers length free path as function of the ratio group velocity - speed of light.
The quantum tunneling rate of a macroscopic particle in one-dimensional potentials subject to oscillating external fields is considered using the complex time path integral method and the time-dependent Schrödinger equation. A simple calculation scheme for obtaining the tunneling rate within quasi-classical accuracy is described and applied to two systems. The one is the tunneling motion through an inverted parabola potential and the other is the escaping motion from a cubic potential well. Both systems are subject to small oscillating fields. It is found that both of the rates are exponentially enhanced due to the imaginary time motion of the tunneling particle under oscillating potential barriers, and that time scale is identical to the Büttiker-Landauer traversal time.