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Physical review. B, Condensed matter

Quantum dynamics of a dissipative particle in a constant field: Exact path-integral approach

Xin-Qi Li, Zhao-Bin Su, Fuzhou Gong

Published Jun 15, 1994• 1 citation
Publisher

Abstract

We show that the dynamics of a dissipative quantum particle in a constant electric field can be solved exactly by applying the Feynman-Vernon influence-functional theory. We derive the drift velocity and its autocorrelation function as well as the fluctuation-dissipation theorem. For such a simple linear dissipative system, it is interesting to find that the drift velocity always depends on the field linearly, and the fluctuation is not influenced by the field. We also find, for the Ohmic dissipation, a ${\mathit{t}}^{\mathrm{\ensuremath{-}}2}$ power law long-time tail for the drift-velocity autocorrelation function at zero temperature.

DOI
10.1103/physrevb.49.16277

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We show that the dynamics of a dissipative quantum particle in a constant electric field can be solved exactly by applying the Feynman-Vernon influence-functional theory. We derive the drift velocity and its autocorrelation function as well as the fluctuation-dissipation theorem. For such a simple linear dissipative system, it is interesting to find that the drift velocity always depends on the field linearly, and the fluctuation is not influenced by the field. We also find, for the Ohmic dissipation, a t\ensuremath−2{\mathit{t}}^{\mathrm{\ensuremath{-}}2}t\ensuremath−2 power law long-time tail for the drift-velocity autocorrelation function at zero temperature.
This chapter demonstrates the equivalence between Feynman–Vernon influence functional approach and Keldysh diagram technique developed to describe non-equilibrium phenomena in quantum many-body systems with interactions. Path integrals appearing in both quantum mechanics and quantum field theory may be viewed as a formal representation of the exact infinite series of Feynman diagrams. In other words, path integral and diagram techniques are formally equivalent to each other. The chapter discusses a single degree of freedom, introducing interactions with a bath of harmonic oscillators. In a way, to employ the Caldeira–Leggett model of a bath of harmonic oscillators can be considered just as a physical trick helping to derive the influence functional in the case of linear Ohmic damping. Gaussian influence functionals are exact provided the environment consists of an arbitrary number of harmonic oscillators coupled linearly to the system of interest q .
In this thesis the path integral formalism is applied to the calculation of the dynamics of dissipative quantum systems. The time evolution of a system of bilinearly coupled bosonic modes is treated using the real-time path integral technique in coherent-state representation. This method is applied to a damped harmonic oscillator within the Caldeira-Leggett model. In order to get the stationary trajectories the corresponding Lagrangian function is diagonalized and then the path integrals are evaluated by means of the stationary-phase method. The time evolution of the reduced density matrix in the basis of coherent states is given in simple analytic form for weak system-bath coupling, i.e. the so-called rotating-wave terms can be evaluated exactly but the non-rotating-wave terms only in a perturbative manner. The validity range of the rotating-wave approximation is discussed from the viewpoint of spectral equations. In addition, it is shown that systems without initial system-bath correlations can exhibit initial jumps in the population dynamics even for rather weak dissipation. Only with initial correlations the classical trajectories for the system coordinate can be recovered. The path integral formalism in a combined phase-space and coherent-state representation is applied to the problem of curve-crossing dynamics. The system of interest is described by two coupled one-dimensional harmonic potential energy surfaces interacting with a heat bath. The mapping approach is used to rewrite the Lagrangian function of the electronic part of the system. Using the Feynman-Vernon influence-functional method the bath is eliminated whereas the non-Gaussian part of the path integral is treated using the perturbation theory in the small coordinate shift between potential energy surfaces. The vibrational and the population dynamics is considered in a lowest order of the perturbation. The dynamics of a Gaussian wave packet is analyzed along a one-dimensional reaction coordinate. Also the damping rate of coherence in the electronic part of the relevant system is evaluated within the ordinary and variational perturbation theory. The analytic expressions for the rate functions are obtained in the low and high temperature regimes.
A nonperturbative technique is developed to solve for the quantum dynamics of an electron interacting strongly with its environment. Real-time path-integral methods are used to solve numerically a quantum dissipation problem that simulates nonequilibrium electron transport in solids. A linearly coupled electron-phonon interaction is used to model the dissipation. This method is nonperturbative in both the electron-phonon coupling and the external electric field, making it useful for electron-transport calculations in cases of both high fields and large phonon-scattering rates. Monte Carlo sampling over the electron paths is done using the Feynman-Vernon influence functional as a weight. Importance-sampling methods are developed for the numerical solution of the quantum propagation. Results of the calculations are shown which include the motion of the electron, the wave-packet spread, and temperature effects.
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.
We use Feynman's influence-functional theory to study the quantum dynamics of a heavy particle moving in a free Fermi gas with arbitrary average velocity. A semiclassical expansion yields a nonlinear Langevin equation with the exact friction coefficient as derived in an earlier publication. The fluctuations around a steady state far from equilibrium are due to a nonclassical state-dependent noise term and can be described by a diffusion constant. In the limit of zero average velocity, the Einstein relation is fulfilled for arbitrary temperatures. For finite velocities the diffusion around the steady state is different in longitudinal and transverse directions and can be expressed in terms of the transport cross section and a ``diffusion'' cross section. In the case where the frictional force exhibits a maximum as a function of velocity and thus an unstable branch for v>vc{\mathit{v}}_{\mathit{c}}vc​, the longitudinal diffusion constant diverges on approaching vc{\mathit{v}}_{\mathit{c}}vc​ from below. Numerical results for the noise spectrum and the temperature and velocity dependence of the diffusion constants are presented for simple repulsive interaction potentials in one and three dimensions.
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.