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Physical Review A

Transient and asymptotic effects in tunneling

S. Brouard, Juan Gonzalo Muga

Published Oct 1, 1996• 52 citations
Publisher

Abstract

The time dependence, from the transient to the stationary regime, of a quantum state initially prepared as a cutoff plane wave impinging on a one-dimensional square barrier is analyzed in detail using a decomposition based on a discrete set of critical points in the complex momentum plane. The motion and shape of the main wave front, the forerunners, and asymptotic (large time) features of the wave are discussed by means of the exact solution and in terms of velocities and characteristic times associated with the partial contributions of the critical points. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.54.3055

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arXiv:quant-ph/9602017•Physical Review A•Dec 1, 1996•22 citations
The time dependence, from the transient to the stationary regime, of a quantum state initially prepared as a cutoff plane wave impinging on a one-dimensional square barrier is analyzed in detail using a decomposition based on a discrete set of critical points in the complex momentum plane. The motion and shape of the main wave front, the forerunners, and asymptotic (large time) features of the wave are discussed by means of the exact solution and in terms of velocities and characteristic times associated with the partial contributions of the critical points. \textcopyright{} 1996 The American Physical Society.
An exact analytic solution of the time-dependent Schrödinger equation with cutoff wave initial conditions, that goes into the stationary solution at asymptotically long times, is used to investigate the transient behavior in the time evolution of the probability density along the internal and transmitted regions of a potential barrier. The approach involves the complex poles and residues (resonant states) of the outgoing Green's function of the problem. The solution exhibits absence of propagation along the classically forbidden internal region of the potential, and along the transmitted region it splits into two propagating structures. One of them is a transient effect describing the earliest response of the system to the initial condition whereas the other structure evolves towards the stationary solution.
Analytic solutions to the time-dependent Schr\"odinger equation for cutoff wave initial conditions are used to investigate the time evolution of the transmitted probability density for tunneling. For a broad range of values of the potential barrier opacity $\ensuremath{\alpha},$ we find that the probability density exhibits two evolving structures. One refers to the propagation of a forerunner related to a time domain resonance [Phys. Rev. A 64, 0121907 (2001)], while the other consists of a semiclassical propagating wave front. We find a regime where the forerunners are absent, corresponding to positive time delays, and show that this regime is characterized by opacities $\ensuremath{\alpha}<{\ensuremath{\alpha}}_{c}.$ The critical opacity ${\ensuremath{\alpha}}_{c}$ is derived from the analytical expression for the delay time, which reflects a link between transient effects in tunneling and the delay time.
Abstract. The time evolution of plane waves in the presence of a 1-dimensional square quantum barrier is considered. Comparison is made between the cases of an infinite and a cut-off (shutter) initial plane wave. The difference is relevant when the results are applied to the analysis of the tunnelling regime. This work is focused on the analytical calculation of the time-evolved solution and highlights the contribution of the resonant (Gamow) states. PACS numbers: 11.10.Ef, 11.10.Lm, 04.60Plane wave tunnelling in a square barrier 2 1.
Time dependence for barrier penetration is considered in the phase space. An asymptotic phase-space propagator for nonrelativistic scattering on a one-dimensional barrier is constructed. The propagator has a form universal for various initial state preparations and local potential barriers. It is manifestly causal and includes time-lag effects and quantum spreading. Specific features of quantum dynamics which disappear in the standard semiclassical approximation are revealed. The propagator may be applied to calculation of the final momentum and coordinate distributions, for particles transmitted through or reflected from the potential barrier, as well as for elucidating the tunneling time problem. \textcopyright{} 1996 The American Physical Society.