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The Journal of Physical Chemistry Letters

Tunneling Dynamics Using Classical-like Trajectories with an Effective Quantum Force

Huaqing Li, Jens Aage Poulsen, Gunnar Nyman

Published Aug 20, 2013• 8 citations
Publisher

Abstract

We develop an effective quantum force to propagate trajectories in Wigner phase space. In this way, the quantum entanglement between trajectories is represented by the effective quantum force, which is related to the variance of the momentum distribution. Reaction probabilities are obtained for three model potentials. For a cubic potential, the reaction probabilities are found to continuously grow in time long after the classical reaction probabilities have become stationary. This growth is due to tunneling and is in good agreement with numerically exact quantum results. The variance of the momentum distribution is updated at each time step of the trajectories, which is the key to the increase of the reaction probabilities at longer times and allows a nice interpretation of tunneling.

DOI
10.1021/jz4012564

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In this Letter, we present a new method for simulating quantum processes in the context of classical molecular dynamics simulations. The approach is based on solving numerically the quantum Liouville equation in the Wigner representation using ensembles of classical trajectories. Quantum effects arise in this formulation as a breakdown of the statistical independence of the ensemble. New interaction forces between ensemble members are derived, which require the trajectory ensemble representing the state to evolve as an entangled, unified whole. The method is applied to the simulation of quantum tunneling in a one-dimensional model system, yielding excellent agreement with exact quantum calculations.
The quantum trajectory method (QTM) for time-dependent wave packet dynamics involves integration of the de Broglie–Bohm hydrodynamic equations for the evolving probability fluid [C. Lopreore and R. E. Wyatt, Phys. Rev. Lett. 82, 5190 (1999)]. The equations of motion for discretized elements of the probability fluid (particles) are integrated in the Lagrangian, moving with the fluid, picture. These fluid elements move under the influence of both the usual potential energy function and the quantum potential, which involves the curvature of the quantum amplitude. The quantum potential and the quantum force are evaluated using a moving weighted least squares algorithm. As a demonstration of applicability, the QTM is applied to a model collinear reaction with an activation barrier. The reaction probabilities are in good agreement with exact quantum results, even for a relatively small number of particles in the ensemble. The QTM accurately describes tunneling using only real valued trajectories. In addition to the reaction probability, plots are presented to show the probability density and the flux distributions at several time steps during the reaction.
We study a wavepacket tunneling in one-dimensional periodically driven double-well system using entangled trajectory molecular dynamics method. The tunneling dynamics dependents on the amplitude and frequency of the driven force are present. Both resonant and nonresonant tunneling process are enhanced by the driven force when the system is chaotic under classical dynamics. We give entangled trajectory in phase space compared to corresponding classical trajectory with same initial state to visually show quantum tunneling process. The average values of quantum tunneling probability after long time evolution have been shown in the parameter spaces, the effect of resonance and chaos on the tunneling dynamics are present. The relation between chaos and the uncertainly product is discussed in the end. © 2016 Wiley Periodicals, Inc.
Quantum tunneling, as a quintessential quantum phenomenon, has been investigated in detail both theoretically and experimentally. Still, the physical picture of the tunneling process is not intuitive, leading to some confusion and paradoxes. In this paper, we have tried to gain insight into quantum tunneling by a phase-space approach. For this purpose, we scrutinize the evolution of the Wigner distribution during tunneling and derive the energy and momentum spectra by integrating over a segment of phase space. In this way, some of the difficulties and paradoxes in tunneling probability, energy conservation, and tunneling time are given a clearer interpretation. Negative probabilities in the Wigner distribution play a key role in the tunneling process, and the origin of negative probabilities clearly indicates that the volatility of matter underlies the various exotic phenomena involved.
We demonstrate a different method of simulation of nonstationary quantum processes, considering the tunneling of two interacting identical particles, represented by wave packets. The used method of quantum molecular dynamics (WMD) is based on the Wigner representation of quantum mechanics. In the context of this method ensembles of classical trajectories are used to solve quantum Wigner-Liouville equation. These classical trajectories obey Hamiltonian-like equations, where the effective potential consists of the usual classical term and the quantum term, which depends on the Wigner function and its derivatives. The quantum term is calculated using local distribution of trajectories in phase space, therefore, classical trajectories are not independent, contrary to classical molecular dynamics. The developed WMD method takes into account the influence of exchange and interaction between particles. The role of direct and exchange interactions in tunneling is analyzed. The tunneling times for interacting particles are calculated.