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Physical Review A

Generalized Feynman-Vernon approach to dissipative quantum systems

C. Morais Smith, A. O. Caldeira

Published Oct 1, 1987• 60 citations
Publisher

Abstract

In this paper the so-called Feynman-Vernon influence functional theory has been generalized in order to include the possibility of dealing with initial conditions of the system plus environment other than the factorizable one. It has been shown that the new influence functional can now be written in terms of paths which represent the coupling of quantum-statistical to quantum-dynamical effects. Once the thermal paths are integrated out of this new influence functional, a generalization of the original Feynman-Vernon expression is obtained.

DOI
10.1103/physreva.36.3509

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In this paper the so-called Feynman-Vernon influence functional theory has been generalized in order to include the possibility of dealing with initial conditions of the system plus environment other than the factorizable one. It has been shown that the new influence functional can now be written in terms of paths which represent the coupling of quantum-statistical to quantum-dynamical effects. Once the thermal paths are integrated out of this new influence functional, a generalization of the original Feynman-Vernon expression is obtained.
The Feynman-Vernon influence functional is a powerful result, based on path integrals, for describing the reduced state of a system interacting with a Gaussian environment. Here, the authors derive in detail a canonical (i.e., operatorial) form of the influence functional for fermionic environments, and show how it can be used to compute the dynamics under arbitrary parity-symmetry initial conditions as well as to derive generalized versions of the Lindblad master equation and the hierarchical equations of motion.
The problem of metastability due to quantum and thermal fluctuations in a weakly damped system is treated within Feynman's influence-functional theory. It is shown that quite generally the nonequilibrium density matrix may be written as a single path integral. For a cubic metastable potential it has the form of a Feynman propagator with a complex Lagrangian containing second-order derivatives in time and terms noninvariant under time reversal. Dissipative tunneling and thermal activation emerge together as different solutions of the associated Lagrange equations. The low-temperature correction to the exponent of the tunneling rate behaves as Tn+1{T}^{n+1}Tn+1 for a spectrum of bath excitations vanishing as ${\ensuremath{\omega}}^{n}$. It is shown, moreover, how this result may be understood in terms of environment-assisted tunneling. Thus the low-temperature behavior of the tunneling rate provides a method to measure the dissipation mechanism in the quantum regime.
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.
Abstract We present a derivation of the Feynman–Vernon approach to open quantum systems in the language of super-operators. We show that this gives a new and more direct derivation of the generating function of energy changes in a bath, or baths. As found previously, this generating function is given by a Feynman–Vernon-like influence functional, with only time shifts in the kernels coupling the forward and backward paths. We further show that the new approach extends to an-harmonic and possible non-equilibrium baths, provided that the interactions are bi-linear, and that the baths do not interact between themselves. Such baths are characterized by non-trivial cumulants. Every non-zero cumulant of certain environment correlation functions is thus a kernel in a higher-order term in the Feynman–Vernon action.
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 .
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.