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Physical Review A

Semiclassical computations of time-dependent tunneling

M. Kira, Ilkka Juhani Tittonen, W. K. Lai, Stig Stenholm

Published Apr 1, 1995• 23 citations
Publisher

Abstract

In this paper we consider the time evolution of wave-packet tunneling through a potential barrier. Using a path-summation approach, we can describe the quantum evolution in terms of particle trajectories both in the allowed and forbidden regions. Near the barrier edges, where the potential changes rapidly, the wave aspects dominate. These make the trajectories branch into infinite families of paths, which have to be included in the summation. Combining these features, we treat correctly the complementary aspects of the full quantum process, and this allows us to obtain a computational method that gives accurate numerical results with considerably less expenditure of computer time than a direct integration of the Schr\"odinger equation. We test the method on a simple rectangular potential by comparing the calculations with both conventional wave scattering results and computations from the Schr\"odinger equation. A general potential can be approximated by a staircase, which allows the application of our method. A simple adiabatic extension is shown to work excellently for the case of a periodically modulated barrier.

DOI
10.1103/physreva.51.2826

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International Journal of Quantum Chemistry•Mar 14, 1992•9 citations
In this paper we consider the time evolution of wave-packet tunneling through a potential barrier. Using a path-summation approach, we can describe the quantum evolution in terms of particle trajectories both in the allowed and forbidden regions. Near the barrier edges, where the potential changes rapidly, the wave aspects dominate. These make the trajectories branch into infinite families of paths, which have to be included in the summation. Combining these features, we treat correctly the complementary aspects of the full quantum process, and this allows us to obtain a computational method that gives accurate numerical results with considerably less expenditure of computer time than a direct integration of the Schr"odinger equation. We test the method on a simple rectangular potential by comparing the calculations with both conventional wave scattering results and computations from the Schr"odinger equation. A general potential can be approximated by a staircase, which allows the application of our method. A simple adiabatic extension is shown to work excellently for the case of a periodically modulated barrier.
We study to what extent propagation of wave packets through potential barriers can be described semiclassically in real time. By analyzing the semiclassical approximation to the exact path-integral representation of the quantum-mechanical time evolution operator, we show that in the long-time sector it is dominated by quasiclassical fluctuations associated with caustics. Contrary to the description of tunneling in imaginary times, simple tunneling paths even for complex energies do not exist. Our findings lead to an extended semiclassical propagation scheme for wave-packet dynamics which is numerical efficient and accurately describes deep incoherent tunneling through static and externally driven barrier potentials.
There has been a surprising degree of uncertainty about how, or even whether, the standard semiclassical theory of Van Vleck and Gutzwiller (VVG) can be used to treat the time-dependent dynamics of tunneling through barriers. At issue is how semiclassical theory describes the time evolution of tunneling. This uncertainty has hindered development of practical techniques for semiclassical calculations of multidimensional systems. It is shown here that the VVG theory can indeed be applied to barrier tunneling, but generally requires consideration of classical trajectories with both complex initial conditions and complex time paths, even for systems with one degree of freedom. The present approach is verified by applying it to the classically forbidden propagation of a wave packet through a one-dimensional barrier---a very basic problem that cannot be treated by the VVG theory without the modifications described here.
We present a time-dependent semiclassical method based on quantum trajectories. Quantum-mechanical effects are described via the quantum potential computed from the wave function density approximated as a linear combination of Gaussian fitting functions. The number of the fitting functions determines the accuracy of the approximate quantum potential (AQP). One Gaussian fit reproduces time-evolution of a Gaussian wave packet in a parabolic potential. The limit of the large number of fitting Gaussians and trajectories gives the full quantum-mechanical result. The method is systematically improvable from classical to fully quantum. The fitting procedure is implemented as a gradient minimization. We also compare AQP method to the widely used semiclassical propagator of Herman and Kluk by computing energy-resolved transmission probabilities for the Eckart barrier from the wave packet time-correlation functions. We find the results obtained with the Herman–Kluk propagator to be essentially equivalent to those of AQP method with a one-Gaussian density fit for several barrier widths.
Tunneling through a one-dimensional Eckart barrier is investigated using a recently developed propagation scheme based on semiclassical wave packets. This version of the time-dependent discrete variable representation method yields linear equations for the parameters, is fully adaptive, and does not require a frozen ansatz in order to approximate the exact solution of the Schrödinger equation accurately. We rely on an analytical result to derive a new algorithm to spawn a second family of semiclassical wave packets after the tunneling has occurred. Numerical results for a benchmark problem demonstrate the accuracy of the new method.
A variety of different theoretical procedures have been used to determine tunneling times. These include (1) phase time; (2) the Stevens procedure; (3) Larmor times; (4) a complex “time”; (5) dwell time; (6) the Büttikker-Landauer time; (7) Feynman path-integrals; (8) scattering theory; (9) a stochastic formulation; (10) oscillatory perturbation of the barrier; (11) kinetic time; and (12) the semiclassical solution. Interest in tunneling times is not purely pedagogical, since correction for changes in image potential during transit is required in determining the conductance of several new high-speed semiconductor devices. A new procedure to evaluate transit time is presented, which assumes that energy fluctuations keep a tunneling particle above the barrier while transiting the classically forbidden region. For large rectangular barriers, the most probable fluctuations minimize the product of their magnitude and the transit time. This results in the semiclassical solution. For the case of very small barriers, there is a separate solution in which the transit time is indeterminate but bounded. The new solution is compared with 12 different results by others. Numerical values, limiting forms, and interpretations are presented for these various tunneling times. Difficulties in using time domain numerical solutions to determine the transit time for a wave packet will also be described. © 1992 John Wiley & Sons, Inc.