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AIP conference proceedings

Quantal Cumulant Dynamics for Dissipative Systems

Yasuteru Shigeta, Theodore E. Simos, George Maroulis, George Psihoyios, Ch. Tsitouras

Published Jan 1, 2008• 1 citation
Publisher

Abstract

We develop a quantal cumulant dynamics method for the quantum tunneling in dissipative environment. Reduced equations of motion of classical and quantal cumulant variables without bath degrees of freedom are derived. We observed suppression of the tunneling that depends on the sign of a friction constant for an Ohmic approximation and on the magnitude of a bath frequency for a single bath mode approximation. A possible mechanism of the suppression is explored by analyzing an effective quantal potential of the tunneling path.

DOI
10.1063/1.2997313

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Physical Review A•Nov 1, 1996•25 citations
We develop a quantal cumulant dynamics method for the quantum tunneling in dissipative environment. Reduced equations of motion of classical and quantal cumulant variables without bath degrees of freedom are derived. We observed suppression of the tunneling that depends on the sign of a friction constant for an Ohmic approximation and on the magnitude of a bath frequency for a single bath mode approximation. A possible mechanism of the suppression is explored by analyzing an effective quantal potential of the tunneling path.
We develop a quantal cumulant dynamics method for the quantum tunneling in dissipative environment. Reduced equations of motion of classical and quantal cumulant variables without bath degrees of freedom are derived. We observed suppression of the tunneling that depends on the sign of a friction constant for an Ohmic approximation and on the magnitude of a bath frequency for a single bath mode approximation. A possible mechanism of the suppression is explored by analyzing an effective quantal potential of the tunneling path.
A wave equation formulation of the problem of quantum tunneling in a dissipative medium is developed by considering a many-body system in which the central particle is subject to an arbitrary force law, and at the same time is coupled to a bath of noninteracting harmonic oscillators. For the motion of the central particle it is possible to obtain an effective Lagrangian and Hamiltonian by eliminating the degrees of freedom of the oscillators. However both of these operators are nonlocal, and it is difficult to derive a wave equation for this motion. As an alternative method one can write a many-body Schrödinger equation for the whole system, and then eliminate the wave functions of all of the oscillators. This result is a many-channel Schrödinger equation for the motion of the central particle. By truncating this set of coupled equations, one can solve the problem for different force laws. In particular, in this work, the cases of dissipative tunneling, hopping, and quantum coherence are studied in detail. It is also shown how this approach can be generalized to multidimensional dissipative systems.
A derivation of the equations of motion for a mixed quantum-classical system in a dissipative environment is presented. The classical environment of the quantum subsystem is taken to be composed of two parts: the first part, termed the classical subsystem, comprises those degrees of freedom that are coupled directly to the quantum subsystem, whereas the second part is a large bath that is coupled only to the classical subsystem. Projection operator methods are used to eliminate the bath degrees of freedom yielding a dissipative equation of motion for the coupled quantum and classical subsystems.
The dynamics of a mixed quantum-classical system, in which the classical subsystem interacts with a dissipative bath, is investigated. This description of the dynamics will be appropriate if the details of the bath dynamics are unimportant but its presence plays an important role in the dissipation of energy to the environment. In this dynamical description, which can be simulated employing an ensemble of stochastic surface-hopping trajectories, the strength of the dissipation is controlled by a friction coefficient. We show that if decoherence, whose effects are controlled by the bath friction, is sufficiently rapid, the equation of motion can be reduced to a master equation. Thus, decoherence and the validity of master equation models may be explored as a function of bath friction. We use this framework to study the mechanism of decoherence in a simple model nonadiabatic chemical reaction.
We study the quantal dynamics of a harmonically driven quartic double well in the presence of dissipation. A master equation for the reduced density operator is derived in the Floquet representation. In the classical limit, this system corresponds to a Duffing oscillator with Ohmic damping. We present numerical results for the transient time evolution and for the stationary sate. The influence of the weak dissipation on interference effects in the context of driven tunnelling is discussed on the basis of these results. We find that the coherent suppression of tunnelling can be stabilized by reservoir-induced noise for a suitably chosen temperature.
Applying a technique developed recently [L. H. Yu and C. P. Sun, Phys. Rev. A 49, 592 (1994); L. H. Yu, Phys. Lett. 202, 167 (1995)] for a harmonic oscillator coupled to a bath of harmonic oscillators, we present an exact solution for the tunneling problem in an Ohmic dissipative system with inverted harmonic potential. The result shows that while the dissipation tends to suppress the tunneling, the Brownian motion tends to enhance the tunneling. Whether the tunneling rate increases or not would then depend on the initial conditions. We give a specific formula to calculate the tunneling probability determined by various parameters and the initial conditions. \textcopyright{} 1996 The American Physical Society.