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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Generalized quantum Langevin equations from the forward-backward path integral

Kazuhiro Tsusaka

Published May 1, 1999• 6 citations
Publisher

Abstract

Kleinert and Shabanov [H. Kleinert and S. V. Shabanov, Phys. Lett. A 200, 224 (1995)] have derived the quantum Langevin equations from the Feynman-Vernon forward-backward path integral for a density matrix of a quantum system in a thermal oscillator bath. However, their derivation is confined to an Ohmic case. In this paper we derive the generalized quantum Langevin equations from the forward-backward path integral, by extending the Kleinert-Shabanov method to a general case.

DOI
10.1103/physreve.59.4931

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Kleinert and Shabanov [H. Kleinert and S. V. Shabanov, Phys. Lett. A 200, 224 (1995)] have derived the quantum Langevin equations from the Feynman-Vernon forward-backward path integral for a density matrix of a quantum system in a thermal oscillator bath. However, their derivation is confined to an Ohmic case. In this paper we derive the generalized quantum Langevin equations from the forward-backward path integral, by extending the Kleinert-Shabanov method to a general case.
In a previous paper a new derivation of the Langevin equations for a system coupled with a heat reservoir was given based on the Feynman path-integral representation of the transition probability, i.e. the diagonal part of a reduced density matrix of the system. The derived Langevin equations are of the form of the Lagrangian equations of motion. In this article the temporal change of the reduced density matrix as a whole, or more precisely of the Wigner phase-space distribution, is treated, and the Langevin equations equivalent to those mentioned above but in the form of the canonical equation of motion are derived in the second order perturbational approximation with respect to interactions between the system and the reservoir.
We establish a direct connection between the Feynman-Vernon path integral formalism for open quantum systems and the Wiener path integral used in classical stochastic dynamics. By considering a generalized influence functional in the strong decoherence limit, we demonstrate that integrating over the quantum coherence length leads to a derivation of stochastic Langevin dynamics. Specifically, we show that the quantum Feynman measure transforms into the stochastic Wiener measure. Applying this framework to the Wigner function representation, we show that the system follows a stochastic path interpretable via classical probability theory. Finally, we address the inverse problem: constructing an equivalent quantum influence functional from a given classical Langevin equation.
A new derivation of the Langevin equation for a system in contact with a heat reservoir in equilibrium is given based on the concept of path probability inherent in the Feynman approach to quantum mechanics. The derived Langevin equation is a generalization of the Magalinski type, i.e., it is a second order differential equation and contains a shift term, a retarded damping term, a destruction term and a random force term in addition to the usual external potential force term. It is a semi-classical equation in the sense that it is a classical equation but contains a quantal response function of the Kubo formula type and the second moment of the random force is given by a quantal symmetrized correlation function. The fluctuation-dissipation theorem of the second kind is exactly valid between the response and the correlation functions.
We have presented a simple approach to quantum theory of Brownian motion and barrier crossing dynamics. Based on an initial coherent state representation of bath oscillators and an equilibrium canonical distribution of quantum-mechanical mean values of their co-ordinates and momenta we have derived a c number generalized quantum Langevin equation. The approach allows us to implement the method of classical non-Markovian Brownian motion to realize an exact generalized non-Markovian quantum Kramers' equation. The equation is valid for arbitrary temperature and friction. We have solved this equation in the spatial diffusion-limited regime to derive quantum Kramers' rate of barrier crossing and analyze its variation as a function of the temperature and friction. While almost all the earlier theories rest on quasiprobability distribution functions (e.g., Wigner function) and path integral methods, the present work is based on true probability distribution functions and is independent of path integral techniques. The theory is a natural extension of the classical theory to quantum domain and provides a unified description of thermally activated processes and tunneling.
Abstract We establish a formal connection between the Feynman–Vernon path integral formalism for open quantum systems and the Wiener path integral used in classical stochastic dynamics, further exploring its applications. By considering a generalized influence functional in the strong decoherence limit, we demonstrate that integrating over the quantum coherence length leads to stochastic Langevin dynamics. Specifically, under our model, we show that the quantum Feynman differential transforms into an underdamped multiplicative non-Markovian Langevin dynamics. Applying this framework to the Wigner function representation, we show that the system follows a stochastic path interpretable via classical probability theory, allowing for the investigation of decoherence through classical dynamics. Finally, we address the inverse problem: constructing an equivalent quantum influence functional from a given classical Langevin equation.