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Physical Review A

Exact dynamics of a quantum dissipative system in a constant external field

Chang-Pu Sun, Li-Hua Yu

Published Mar 1, 1995• 87 citations
Publisher

Abstract

We study the quantum dynamics of the simplest dissipative system, a particle moving in a constant external field and interacting with a bath of harmonic oscillators with Ohmic spectral density. Applying the main idea and methods developed in our recent work [L. H. Yu and C. P. Sun, Phys. Rev. A 49, 592 (1994)] to this system, we obtain the simple and exact solutions for the coordinate operator of the system in the Heisenberg picture and the wave function of the composite system of the system and the bath in the Schr\"odinger picture. An effective Hamiltonian for the dissipative system is explicitly derived from these solutions. The meaning of the wave function described by this effective Hamiltonian is clarified by analyzing the effect of the Brownian motion. In particular, the general effective Hamiltonian for an arbitrary potential is directly derived with this method for the case when the Brownian motion can be ignored. Using this effective Hamiltonian, we show an interesting result that the dissipation suppresses the wave-packet spreading.

DOI
10.1103/physreva.51.1845

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We study the quantum dynamics of the simplest dissipative system, a particle moving in a constant external field and interacting with a bath of harmonic oscillators with Ohmic spectral density. Applying the main idea and methods developed in our recent work [L. H. Yu and C. P. Sun, Phys. Rev. A 49, 592 (1994)] to this system, we obtain the simple and exact solutions for the coordinate operator of the system in the Heisenberg picture and the wave function of the composite system of the system and the bath in the Schr"odinger picture. An effective Hamiltonian for the dissipative system is explicitly derived from these solutions. The meaning of the wave function described by this effective Hamiltonian is clarified by analyzing the effect of the Brownian motion. In particular, the general effective Hamiltonian for an arbitrary potential is directly derived with this method for the case when the Brownian motion can be ignored. Using this effective Hamiltonian, we show an interesting result that the dissipation suppresses the wave-packet spreading.
The quantum dynamics of a simplest dissipative system, a particle moving in a constant external field , is exactly studied by taking into account its interaction with a bath of Ohmic spectral density. We apply the main idea and methods developed in our recent work [1] to quantum dissipative system with constant external field. Quantizing the dissipative system we obtain the simple and exact solutions for the coordinate operator of the system in Heisenberg picture and the wave function of the composite system of system and bath in Schroedinger picture. An effective Hamiltonian for the dissipative system is explicitly derived from these solutions with Heisenberg picture method and thereby the meaning of the wavefunction governed by it is clarified by analyzing the effect of the Brownian motion. Especially, the general effective Hamiltonian for the case with arbitrary potential is directly derived with this method for the case when the Brownian motion can be ignored. Using this effective Hamiltonian, we show an interesting fact that the dissipation suppresses the wave packet spreading.
A microscopic approach treating the quantum dissipation process presented by Sun and Yu (Phys. Rev. A49 (1994) 592; A51 (1995) 1845) is invoked to construct the wavefunction of the composite system—the model for a harmonic oscillator interacting with a many-oscillator bath under the rotating wave approximation. It shows the back-action of the system on the bath. In particular, the dynamic evolution of the wavefunction for the composite system maintains a factorized form in its wavefunction. In the limited temperature, the reduced density matrix for the system is also calculated to clarify the influence of Brownian motion on the system.
For a dissipative system with Ohmic friction, we obtain a simple and exact solution for the wave function of the system plus the bath. It is described by the direct product in two independent Hilbert spaces. One of them is described by an effective Hamiltonian, the other represents the effect of the bath, i.e., the Brownian motion, thus clarifying the structure of the wave function of the system whose energy is dissipated by its interaction with the bath. No path-integral technology is needed in this treatment. The derivation of the Weisskopf-Wigner linewidth theory follows easily.
In this paper, a microscopic approach treating the quantum dissipation process presented by Yu and Sun [Phys. Rev. A 49, 592 (1994); 51, 1845 (1995)] is developed to analyze the wave function structure for the dynamic evolution of a typical dissipative system---a single-mode boson soaked in a bath of many bosons. The wave function of the total system is explicitly obtained as a product of two components of the system and bath in a coherent state representation. It not only describes the influence of the bath on the variable of the system through the Brownian motion, but also manifests the back-action of the system on the bath and the mutual interaction among the bosons of bath. Due to the back-action, the total wave function can only be partially factorizable even if the Brownian motion can be completely ignored in certain cases, e.g., weak coupling and large detuning. The semiclassical implication of the back-action, the mutual interaction, and the Brownian motion in the present model are also discussed by considering the wave packet evolution of the dissipative system in the coordinate representation.
Exact quantum master equation for a driven Brownian oscillator system is constructed via a Wigner phase-space Gaussian wave packet approach. The interplay between external field and dissipation leads to this system an effective field correction that enhances the polarization. This cooperative property is resulted from an effective bath response to the external field applied on the system. It is important in the low-frequency driving and intermediate bath memory region. We demonstrate this non-Markovian effect on the linear response and nonlinear dynamics and analyze the results together with analytical asymptotic expressions.
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.