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Semiclassical wave packet tunneling in real-time

Joachim Ankerhold, Saltzer, M

Published Oct 18, 2002• 12 citations
OpenAlex

Abstract

Quantum mechanical real-time tunneling through general scattering potentials is studied in the semiclassical limit. It is shown that the exact path integral of the real-time propagator is dominated in the long time sector by quasi-stationary fluctuations associated with caustics. This leads to an extended semiclassical propagation scheme for wave packet dynamics which accurately describes deep tunneling through static and, for the first time, driven barrier potentials.

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Quantum mechanical real-time tunneling through general scattering potentials is studied in the semiclassical limit. It is shown that the exact path integral of the real-time propagator is dominated in the long time sector by quasi-stationary fluctuations associated with caustics. This leads to an extended semiclassical propagation scheme for wave packet dynamics which accurately describes deep tunneling through static and, for the first time, driven barrier potentials.
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.
We systematically extend semiclassical real-time propagation methods based on Gaussian wave packets by using the composition property of the time-dependent quantum-mechanical Green function. For the first time, in an application to the problem of barrier tunneling, a semiclassical time-domain calculation of the transmission probability exhibits good agreement with exact quantum mechanical values at energies below the barrier top. Only two insertions of unity are needed for these results in a benchmark model of reactive scattering.
The time dependent density matrix of a system with potential barrier is studied using path integrals. The characterization of the initial state, which is assumed to be restricted to one side of the barrier, and the time evolution of the density matrix lead to a threefold path integral which is evaluated in the semiclassical limit. The semiclassical trajectories are found to move in the complex coordinate plane and barrier penetration only arises due to fluctuations. Both the form of the semiclassical paths and the relevant fluctuations change significantly as a function of temperature. The semiclassical analysis leads to a detailed picture of barrier penetration in the real time domain and the changeover from thermal activation to quantum tunneling. Deep tunneling is associated with quasizero modes in the fluctuation spectrum about the semiclassical orbits in the long time limit. The connection between this real time description of tunneling and the standard imaginary time instanton approach is established. Specific results are given for a double well potential and an Eckart barrier.
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.
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.