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The Journal of Chemical Physics

Hydrodynamic analysis of dynamical tunneling

Dmytro Babyuk, Róbert E. Wyatt, John H. Frederick

Published Sep 18, 2003• 15 citations
Publisher

Abstract

The process of dynamical tunneling between local mode states is analyzed within the Bohmian approach to quantum mechanics. A number of quantum trajectories are followed for both a coupled Morse potential and the Barbanis potential. The superposition of nearly degenerate eigenstates leads to the formation of numerous permanent nodes concentrated near the potential valley. Around these nodes, vortices are evident on the flux map. Quantum trajectories can be attracted by a vortex and their velocity and kinetic energy will be significantly changed as they spiral around the vortex. However, trajectories starting remotely from vortices reflect more realistically the transition between a pair of localized states. These quantum trajectories can go through one or more regions where the potential energy is higher than the total energy. Hence, barrier tunneling accompanies dynamical tunneling.

DOI
10.1063/1.1605385

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•Jun 15, 2012•0 citations
The process of dynamical tunneling between local mode states is analyzed within the Bohmian approach to quantum mechanics. A number of quantum trajectories are followed for both a coupled Morse potential and the Barbanis potential. The superposition of nearly degenerate eigenstates leads to the formation of numerous permanent nodes concentrated near the potential valley. Around these nodes, vortices are evident on the flux map. Quantum trajectories can be attracted by a vortex and their velocity and kinetic energy will be significantly changed as they spiral around the vortex. However, trajectories starting remotely from vortices reflect more realistically the transition between a pair of localized states. These quantum trajectories can go through one or more regions where the potential energy is higher than the total energy. Hence, barrier tunneling accompanies dynamical tunneling.
Vortices are known to play a key role in the dynamics of the quantum trajectories defined within the framework of the de Broglie-Bohm formalism of quantum mechanics. It has been rigourously proved that the motion of a vortex in the associated velocity field can induce chaos in these trajectories, and numerical studies have explored the rich variety of behaviors that due to their influence can be observed. In this paper, we go one step further and show how the theory of dynamical systems can be used to construct a general and systematic classification of such dynamical behaviors. This should contribute to establish some firm grounds on which the studies on the intrinsic stochasticity of Bohm's quantum trajectories can be based. An application to the two dimensional isotropic harmonic oscillator is presented as an illustration.
Vortices are known to play a key role in many important processes in physics and chemistry. Here, we study vortices in connection with the quantum trajectories that can be defined in the framework provided by the de Broglie–Bohm formalism of quantum mechanics. In a previous work, it was shown that the presence of a single moving vortex is enough to induce chaos in these trajectories. Here, this situation is explored in more detail by discussing the relationship between Lyapunov exponents and the parameters characterizing the vortex dynamics. We also consider the issue when more than one vortex exists. In this case, the interaction among them can annihilate or create pairs of vortices with opposite vorticity. This phenomenon is analyzed from a dynamical point of view, showing how the size of the regular regions in phase space grows, as vortices disappear.
The hydrodynamic analogy for quantum mechanics is developed in a quantum phase-space representation. The key technical step in this development is the formulation of a phase-space flux (or current) density expression. Lagrangian fluid trajectories in phase space are explicitly found as solutions to a set of first-order ordinary differential equations which are formulated without making semiclassical approximation to the quantum dynamics. These fluid trajectories may be used to construct sharp structures in quantum phase space in the same way that one uses classical trajectories in classical mechanics. While these sharp structures do not themselves represent states, their properties ``control'' the dynamics of states in a manner familiar from the theory of first-order flows. The formalism reduces to Liouville dynamics in the classical limit, \ensuremath{\Elzxh}\ensuremath{\rightarrow}0. In addition to several analytic applications to the pure- and mixed-state dynamics, the approach is numerically implemented for tunneling dynamics in a double-well problem. A new picture for quantum tunneling emerges where the tunneling transport occurs when density spirals outward from one well along real-valued fluid trajectories and then into the other well. The tunneling is found to occur through localized portals, or ``flux gates'' in the phase space. We advocate the phase-space hydrodynamic model as a general tool to understand classical quantum correspondence.
In this paper we compute quantum trajectories arising from Bohm’s causal description of quantum mechanics. Our computational methodology is based upon a finite-element moving least-squares method (MWLS) presented recently by Wyatt and co-workers [Lopreore and Wyatt, Phys. Rev. Lett. 82, 5190 (1999)]. This method treats the “particles” in the quantum Hamilton–Jacobi equation as Lagrangian fluid elements that carry the phase, S, and density, ρ, required to reconstruct the quantum wave function. Here, we compare results obtained via the MWLS procedure to exact results obtained either analytically or by numerical solution of the time-dependent Schrödinger equation. Two systems are considered: first, dynamics in a harmonic well and second, tunneling dynamics in a double well potential. In the case of tunneling in the double well potential, the quantum potential acts to lower the barrier, separating the right- and left-hand sides of the well, permitting trajectories to pass from one side to another. However, as probability density passes from one side to the other, the effective barrier begins to rise and eventually will segregate trajectories in one side from the other. We note that the MWLS trajectories exhibited long time stability in the purely harmonic cases. However, this stability was not evident in the barrier crossing dynamics. Comparisons to exact trajectories obtained via wave packet calculations indicate that the MWLS trajectories tend to underestimate the effects of constructive and destructive interference effects.
The quantum hydrodynamic approach is based on the de Broglie-Bohm formulation of quantum mechanics. The resulting fluid-like equations of motion describe the flow of probability and an accurate solution to these equations is equivalent to solving the time-dependent Schroedinger equation. Furthermore, the hydrodynamic approach provides new insight into the mechanisms as well as an alternative computational approach for treating tunneling phenomena. New concepts include well-defined 'quantum trajectories', 'quantum potential', and 'quantum force' all of which have classical analogues. The quantum potential and its associated force give rise to all quantum mechanical effects such as zero point energy, tunneling, and interference. A new numerical approach called the Iterative Finite Difference Method (IFDM) will be discussed. The IFDM is used to solve the set of non-linear coupled hydrodynamic equations. It is 2nd-order accurate in both space and time and exhibits exponential convergence with respect to the iteration count. The stability and computational efficiency of the IFDM is significantly improved by using a 'smart' Eulerian grid which has the same computational advantages as a Lagrangian or Arbitrary Lagrangian Eulerian (ALE) grid. The IFDM is also capable of treating anharmonic potentials. Example calculations using the IFDM will be presented which include: a one-dimensional Gaussian wave packet tunneling through an Eckart barrier, a one-dimensional bound-state Morse oscillator, and a two-dimensional (2D) model collinear reaction using an anharmonic potential energy surface. Approximate treatments of the quantum hydrodynamic equations will also be discussed which could allow scaling of the calculations to hundreds of degrees of freedom which is important for treating tunneling phenomena in condensed phase systems.