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The Journal of Physical Chemistry B

Quantum Dissipation in the Hydrodynamic Moment Hierarchy: A Semiclassical Truncation Strategy

Jeremy B. Maddox, Eric R. Bittner

Published Jul 3, 2002• 20 citations
Publisher

Abstract

In this paper we formulate the hydrodynamic moment and cumulant expansions of the density matrix for a quantum system in which the time evolution is described by an infinite hierarchy of coupled hydrodynamic moment equations. For the case where the system interacts with a complex environment, we develop a semiclassical closure that truncates the hierarchy. Our numerical approach for solving the truncated system of equations is intimately related to the de Broglie−Bohm interpretation of quantum mechanics, in that the time evolution of each cumulant is constructed over a discrete ensemble of hydrodynamic paths. We discuss the advantages and disadvantages of this scheme and apply the method to a model problem for a particle escaping from a metastable potential well while in contact with a thermal bath.

DOI
10.1021/jp020845s

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In this paper we formulate the hydrodynamic moment and cumulant expansions of the density matrix for a quantum system in which the time evolution is described by an infinite hierarchy of coupled hydrodynamic moment equations. For the case where the system interacts with a complex environment, we develop a semiclassical closure that truncates the hierarchy. Our numerical approach for solving the truncated system of equations is intimately related to the de Broglie−Bohm interpretation of quantum mechanics, in that the time evolution of each cumulant is constructed over a discrete ensemble of hydrodynamic paths. We discuss the advantages and disadvantages of this scheme and apply the method to a model problem for a particle escaping from a metastable potential well while in contact with a thermal bath.
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In recent years trajectory based methodologies have become increasingly popular for evaluating the time evolution of quantum systems. A revival of the de Broglie Bohm interpretation of quantum mechanics has spawned several such techniques for examining quantum dynamics from a hydrodynamic perspective. Using techniques similar to those found in computational fluid dynamics one can construct the wavefunction of a quantum system at any time from the trajectories of a discrete ensemble of hydrodynamic fluid elements (Bohm particles) which evolve according to non-classical equations of motion. Until very recently these schemes have been limited to closed systems. In this paper we present our methodology for including the effects of a thermal environment into the hydrodynamic formulation of quantum dynamics. We derive hydrodynamic equations of motion from the Caldeira-Leggett master equation for the reduced density matrix and give a brief overview of our computational scheme which incorporates an adaptive Lagrangian mesh. Our applications focus upon the dissipative dynamics of open unbounded quantum systems. Using both the Wigner phase space representation and the linear entropy we probe the breakdown of the Markov approximation of the bath dynamics at low temperatures. We suggest a criteria for rationalizing the validity of the Markov approximation in open unbound systems and discuss decoherence, energy relaxation, and quantum/classical correspondence in the context of the Bohmian paths.
In recent years trajectory based methodologies have become increasingly popular for evaluating the time evolution of quantum systems. A revival of the de Broglie--Bohm interpretation of quantum mechanics has spawned several such techniques for examining quantum dynamics from a hydrodynamic perspective. Using techniques similar to those found in computational fluid dynamics one can construct the wave function of a quantum system at any time from the trajectories of a discrete ensemble of hydrodynamic fluid elements (Bohm particles) which evolve according to nonclassical equations of motion. Until very recently these schemes have been limited to conservative systems. In this paper, we present our methodology for including the effects of a thermal environment into the hydrodynamic formulation of quantum dynamics. We derive hydrodynamic equations of motion from the Caldeira-Leggett master equation for the reduced density matrix and give a brief overview of our computational scheme that incorporates an adaptive Lagrangian mesh. Our applications focus upon the dissipative dynamics of open unbounded quantum systems. Using both the Wigner phase space representation and the linear entropy, we probe the breakdown of the Markov approximation of the bath dynamics at low temperatures. We suggest a criteria for rationalizing the validity of the Markov approximation in open unbound systems and discuss decoherence, energy relaxation, and quantum/classical correspondence in the context of the Bohmian paths.
The hydrodynamic formulation of mixed quantum states involves a hierarchy of coupled equations of motion for the momentum moments of the Wigner function. In this work a closure scheme for the hierarchy is developed. The closure scheme uses information contained in the lower known moments to expand the Wigner phase-space distribution function in a Gauss-Hermite orthonormal basis. The higher moment required to terminate the hierarchy is then easily obtained from the reconstructed approximate Wigner function by a straightforward integration over the momentum space. Application of the moment closure scheme is demonstrated for the dissipative and nondissipative dynamics of two different systems: (i) double-well potential, (ii) periodic potential.
Quantum trajectories, originating from the de Broglie–Bohm hydrodynamic description of quantum mechanics, are used to construct time-correlation functions in an initial value representation. The formulation is fully quantum mechanical and the resulting equations for the correlation functions are similar in form to their semiclassical analogs but do not require the computation of the stability or monodromy matrix or conjugate points. We then move to a local trajectory description by evolving the cumulants of the wave function along each individual path. The resulting equations of motion are an infinite hierarchy, which we truncate at a given order. We show that time-correlation functions computed using these approximate quantum trajectories can be used to accurately compute the eigenvalue spectrum for various potential systems.