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The Journal of Chemical Physics

On the dynamics of coupled Bohmian and phase-space variables: A new hybrid quantum-classical approach

Irène Burghardt, Gérard Parlant

Published Feb 10, 2004• 47 citations
Publisher

Abstract

A new approach to the coupling of quantum and classical dynamics is developed, by combining a hydrodynamic, Bohmian description for the quantum subsystem with a Liouville-space description for the classical subsystem. To this end, partial hydrodynamic moments are introduced, the dynamics of which is determined by a hierarchy of equations derived from the quantum Liouville equation. We focus on pure states (wave functions) and introduce a trajectory representation in a hybrid hydrodynamic-Liouvillian phase space. The interleaved trajectory dynamics is guided by a new type of quantum force. For illustration, we consider a pair of bilinearly coupled harmonic oscillators, for which the method is exact.

DOI
10.1063/1.1647059

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A new approach to the coupling of quantum and classical dynamics is developed, by combining a hydrodynamic, Bohmian description for the quantum subsystem with a Liouville-space description for the classical subsystem. To this end, partial hydrodynamic moments are introduced, the dynamics of which is determined by a hierarchy of equations derived from the quantum Liouville equation. We focus on pure states (wave functions) and introduce a trajectory representation in a hybrid hydrodynamic-Liouvillian phase space. The interleaved trajectory dynamics is guided by a new type of quantum force. For illustration, we consider a pair of bilinearly coupled harmonic oscillators, for which the method is exact.
The theoretical framework of the mixed quantum-classical description given by Burghardt and Parlant [J. Chem. Phys. 120, 3055 (2004)] is detailed. A representation in terms of partial hydrodynamic moments is developed, the dynamics of which is determined by a hierarchy of equations derived from the quantum Liouville equation. Exact equations of motion are obtained, whose quantum-classical approximants are associated with a fluid-dynamical trajectory representation which couples classical variables to quantum hydrodynamic variables. The latter evolve under a generalized hydrodynamic force which also depends upon the classical phase-space variables. The hydrodynamic moment description is shown to be closely connected to mixed quantum-classical phase-space methods.
The hybrid quantum-classical approach of Burghardt and Parlant [Burghardt, I.; Parlant, G. J. Chem. Phys. 2004, 120, 3055], referred to here as the quantum-classical moment (QCM) approach, is demonstrated for the dynamics of a quantum double well coupled to a classical harmonic coordinate. The approach combines the quantum hydrodynamic and classical Liouvillian representations by the construction of a particular type of moments (that is, partial hydrodynamic moments) whose evolution is determined by a hierarchy of coupled equations. For pure states, which are at the center of the present study, this hierarchy terminates at the first order. In the Lagrangian picture, the deterministic trajectories result in dynamics which is Hamiltonian in the classical subspace, while the projection onto the quantum subspace evolves under a generalized hydrodynamic force. Importantly, this force also depends upon the classical (Q, P) variables. The present application demonstrates the tunneling dynamics in both the Eulerian and Lagrangian representations. The method is exact if the classical subspace is harmonic, as is the case for the systems studied here.
Abstract The formulation of quantum hydrodynamics for mixed states provides a complement, in a quantum‐statistical context, to its pure‐state “Bohmian mechanics” analogue. We summarize our recent work in this area, focusing on two key aspects. We first describe the formulation of a generalized hydrodynamic force, which includes the effects of thermal fluctuations and “quantum fluctuations” and thus determines the time evolution under the influence of dissipation. Second, we introduce a mixed hydrodynamic–Liouville space representation in terms of partial moments, as applied in the context of coupled electronic states and mixed quantum‐classical dynamics. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004
A novel time-dependent hybrid quantum/classical propagation scheme based on Bohmian quantum trajectories is presented. The quantum subsystem is described by a wave packet depending on the quantum variables x and, via the total potential energy of the system, parametrically on the classical trajectories X(t). The wave packet is used to calculate de Broglie–Bohm quantum trajectories x(t) which are used to calculate the force acting on the classical variables. Quantum corrections of the classical equation of motion are also included. The method is applied to a simple case of two coupled oscillators. Comparison between exact quantum and approximate results demonstrates that these MQCB (Mixed Quantum/Classical Bohmian) trajectories provide a good description of the energy exchange between the two oscillators.
A hybrid method is proposed to propagate system-bath quantum dynamics that use both basis functions and coupled quantum trajectories. In it, the bath is represented with an ensemble of Bohmian trajectories while the system degrees of freedom are accounted through reduced density matrices. By retaining the Hilbert space structure for the system, the method is able to capture interference processes that are challenging to describe in Bohmian dynamics due to singularities that these processes introduce in the quantum potential. By adopting quantum trajectories to represent the bath, the method beats the exponential scaling of the computational cost with the bath size. This combination makes the method suitable for large-scale ground and excited state fully quantum molecular dynamics simulations. Equations of motion for the quantum trajectories and reduced density matrices are derived from the Schrodinger equation and a computational algorithm to solve these equations is proposed. Through computations in two-dimensional model systems, the method is shown to offer an accurate description of subsystem observables and of quantum decoherence, which is difficult to obtain when the quantum nature of the bath is ignored. The scaling of the method is demonstrated using a model with 21 degrees of freedom. The limit of independent trajectories is recovered when the mass of bath degrees of freedom is much larger than the one of the system, in agreement with mixed quantum-classical descriptions.
A combined method for analyzing quantum dynamical equations which uses the Bohmian mechanics and the quantum phase space representation is proposed. It is based on a presentation of the wave function in phase space in a polar form. The combined method is very effective for comparison of classical and quantum dynamics for polynomial potentials of order no higher than second. Application for the harmonic oscillator and quartic double well potential is performed. Quantum fluxes and particle energy are derived by means of the proposed method.