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arXiv:1409.2889 · Canadian Journal of Physics

Time-dependent potential barriers and superarrivals

Hamid Karami, S.V. Mousavi

Published Feb 19, 2015• 6 citations
arXivHTMLPDF

Abstract

Scattering of a Gaussian wavepacket from rectangular potential barriers with increasing widths or heights is studied numerically. It is seen that during a certain time interval the time-evolving transmission probability increases compared to the corresponding unperturbed cases. In the literature this effect is known as superarrival in transmission probability. We present a trajectory-based explanation for this effect by using the concept of quantum potential energy and computing a selection of Bohmian trajectories. Relevant parameters in superarrivals are determined for the case that the barrier width increases linearly during the dispersion of the wavepacket. Nonlinearity in time perturbation is also considered.

DOI
10.1139/cjp-2014-0311

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Scattering of a Gaussian wavepacket from rectangular potential barriers with increasing widths or heights is studied numerically. It is seen that during a certain time interval the time-evolving transmission probability increases compared to the corresponding unperturbed cases. In the literature this effect is known as superarrival in transmission probability. We present a trajectory-based explanation for this effect by using the concept of quantum potential energy and computing a selection of Bohmian trajectories. Relevant parameters in superarrivals are determined for the case that the barrier width increases linearly during the dispersion of the wavepacket. Nonlinearity in time perturbation is also considered.
A curious effect is uncovered by calculating the time evolving probability of reflection of a Gaussian wave packet from a rectangular potential barrier while it is perturbed by reducing its height. A time interval is found during which this probability of reflection is larger (superarrivals) than in the unperturbed case. This nonclassical effect can be explained by requiring a wave function to act as a ``field'' through which an action, induced by the perturbation of the boundary condition, propagates at a speed depending on the rate of reducing the barrier height.
An analytical treatment of a propagating wave packet incident on a transient barrier reveals a counterintuitive quantum mechanical effect in which, for a particular time interval, the time-varying transmission probability {\it exceeds} (`superarrival') that for the free propagation of the wave packet. It is found that the speed with which the information about the barrier perturbation propagates across the wave packet can exceed the group velocity of the wave packet. An interesting implication of this effect regarding information transfer is analyzed by showing one-to-one correspondence between the strength of the barrier and the magnitude of `superarrival'.
Semiclassical approximations for tunneling processes usually involve complex trajectories or complex times. In this paper we use a previously derived approximation involving only real trajectories propagating in real time to describe the scattering of a Gaussian wavepacket by a finite square potential barrier. We show that the approximation describes both tunneling and interferences very accurately in the limit of small Plank's constant. We use these results to estimate the tunneling time of the wavepacket and find that, for high energies, the barrier slows down the wavepacket but that it speeds it up at energies comparable to the barrier height.
Semiclassical approximations for tunnelling processes usually involve complex trajectories or complex times. In this paper, we use a previously derived approximation involving only real trajectories propagating in real time to describe the scattering of a Gaussian wavepacket by a finite square potential barrier. We show that the approximation describes both tunnelling and interferences very accurately in the limit of small Planck’s constant. We use these results to estimate the tunnelling time of the wavepacket and find that, for high energies, the barrier slows down the wavepacket but that it speeds it up at energies comparable to the barrier height. PACS numbers: 03.65.Sq, 31.15.Gy (Some figures in this article are in colour only in the electronic version.) 1.
Ignoring thermal fluctuations of the environment, taking into account only its dissipative effects, free propagation of a Gaussian wavepacket is studied in the framework of the linear Caldirola-Kanai (CK) equation and non-linear equations, the Schrodinger-Langevin (SL) equation, known as Kostin equation, and the Schuch-Chung-Hartmann (SCH) equation. By a Gaussian ansatz for the probability density one obtains two equations, one for the evolution of the center of the wave packet which is just the classical Langeving equation and one for the evolution of the width of the wavepacket. This last equation has different forms in different approaches having only an analytic solution in the CK and SCH frameworks. Computations show that for a given friction, the width of the wavepacket increases with time in all approaches. In a given time, it reduces with friction in both CK and SL approaches revealing localization effects of dissipation, while has opposite behavior in the SCH approach. Furthermore, energy expectation value and its time-derivative is computed and compared in all approaches. It is shown that the rate of energy is given by the expectation value of momentum field in the SL framework. Finally, transmission of a Gaussian wavepacket from a rectangular barrier is numerically studied in the context of CK and compared to the non-dissipative case. The transmission decreases considerably with the dissipation.