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Physical Review Letters

Semiclassical Real-Time Tunneling by Multiple Spawning of Classical Trajectories

Frank Großmann

Published Jul 31, 2000• 42 citations
Publisher

Abstract

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.

DOI
10.1103/physrevlett.85.903

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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 independent semiclassical treatment of barrier tunneling has been understood for a very long time. Several semiclassical approaches to time dependent tunneling through barriers have also been presented. These typically involve trajectories for which the position variable is a complex function of time. In this paper, a method is presented that uses only real valued trajectories, thus avoiding the complications that can arise when complex trajectories are employed. This is accomplished by expressing the time dependent wave packet as an integration over momentum. The action function in the exponent in this expression is expanded to second order in the momentum. The expansion is around the momentum, p0(*) , at which the derivative of the real part of the action is zero. The resulting Gaussian integral is then taken. The stationary phase approximation requires that the derivative of the full action is zero at the expansion point, and this leads to a complex initial momentum and complex tunneling trajectories. The "pseudo-stationary phase" approximation employed in this work results in real values for the initial momentum and real valued trajectories. The transmission probabilities obtained are found to be in good agreement with exact quantum results.
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.
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.
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.
Semiclassical initial value representation (IVR) formulas for the propagator have difficulty describing tunneling through barriers. A key reason is that these formulas do not automatically reduce, in the classical limit, to the version of the Van Vleck-Gutzwiller (VVG) propagator required to treat barrier tunneling, which involves trajectories that have complex initial conditions and that follow paths in complex time. In this work, a simple IVR expression, that has the correct tunneling form in the classical limit, is derived for the propagator in the case of one-dimensional barrier transmission. Similarly, an IVR formula, that reduces to the Generalized Gaussian Wave Packet Dynamics (GGWPD) expression [D. Huber, E. J. Heller, and R. Littlejohn, J. Chem. Phys. 89, 2003 (1988)] in the classical limit, is derived for the transmitted wave packet. Uniform semiclassical versions of the IVR formulas are presented and simplified expressions in terms of real trajectories and WKB penetration factors are described. Numerical tests show that the uniform IVR treatment gives good results for wave packet transmission through the Eckart and Gaussian barriers in all cases examined. In contrast, even when applied with the proper complex trajectories, the VVG and GGWPD treatments are inaccurate when the mean energy of the wave packet is near the classical transmission threshold. The IVR expressions for the propagator and wave packet are cast as contour integrals in the complex space of initial conditions and these are generalized to potentially allow treatment of a larger variety of systems. A steepest descent analysis of the contour integral formula for the wave packet in the present cases confirms its relationship to the GGWPD method, verifies its semiclassical validity, and explains results of numerical calculations.