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Quantum Mechanics with Dissipation: Influence Functional Theory

Andrei D. Zaikin, Dmitry S. Golubev

Published May 24, 2019• 0 citations
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Abstract

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 .

DOI
10.1201/9780429298233-1

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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.
We study the Caldeira-Leggett model where a quantum Brownian particle interacts with an environment or a bath consisting of a collection of harmonic oscillators in the path-integral formalism. Compared to the contours that the paths take in the conventional Schwinger-Keldysh formalism, the paths in our study are deformed in the complex time plane as suggested by the recent study by C. Aron, G. Biroli, and L. F. Cugliandolo [SciPost Phys. 4, 008 (2018)10.21468/SciPostPhys.4.1.008]. This is done to investigate the connection between the symmetry properties in the Schwinger-Keldysh action and the equilibrium or nonequilibrium nature of the dynamics in an open quantum system. We derive the influence functional explicitly in this setting, which captures the effect of the coupling to the bath. We show that in equilibrium the action and the influence functional are invariant under a set of transformations of path-integral variables. The fluctuation-dissipation relation is obtained as a consequence of this symmetry. When the system is driven by an external time-dependent protocol, the symmetry is broken. From the terms that break the symmetry, we derive a quantum Jarzynski-like equality for a quantum mechanical worklike quantity given as a function of fluctuating quantum trajectory. In the classical limit, the transformations becomes those used in the functional integral formalism of the classical stochastic thermodynamics to derive the classical fluctuation theorem.
The effective dynamics of a system interacting with a bath or environment is presented in two ways, (1) the (LGKS) replacement of the von Neuman equation for the density matrix and (2) the Feynman-Vernon path-integral derivation, by integrating out the bath degree of freedom, to arrive at a system's density matrix. In this paper, I connect the two methods by deriving a Lindbladian in a mechanical example, a point particle interacting with a bath of harmonic oscillators, previously considered by Feynman and Vernon (FV) and expounded on later by Caldeira and Leggett (CL). But the (FV)/(CL) results only in non-Markov effect, memory terms from the bath interaction. To derive a Lindbladian, I changed the interaction term they considered to take into account the point particle interacting with the bath harmonic oscillators to something more realistic. From the resulting path-integral expression of the system's propagator for the density matrix, the Lindbladian and non-Markov terms are read for this simple problem. I also point out the causes of these terms, the Markov Lindbladian from the very local interaction of the point particle with the classical solutions of the harmonic oscillator and the non-Markov term from the global interaction of the point particle with the fluctuation of the classical solutions.
I review the generating function for quantum-statistical mechanics, known as the Feynman-Vernon influence functional, the decoherence functional, or the Schwinger-Keldysh path integral. I describe a probability-conserving iεi\varepsiloniε prescription from a path-integral implementation of Lindblad evolution. I also explain how to generalize the formalism to accommodate out-of-time-ordered correlators (OTOCs), leading to a Larkin-Ovchinnikov path integral. My goal is to provide step-by-step calculations of path integrals associated to the harmonic oscillator.
The change of the von Neumann entropy of a set of harmonic oscillators initially in thermal equilibrium and interacting linearly with an externally driven quantum system is computed by adapting the Feynman–Vernon influence functional formalism. This quantum entropy production has the form of the expectation value of three functionals of the forward and backward paths describing the system history in the Feynman–Vernon theory. In the classical limit of Kramers–Langevin dynamics (Caldeira–Leggett model) these functionals combine to three terms, where the first is the entropy production functional of stochastic thermodynamics, the classical work done by the system on the environment in units of k B T, and the second and the third other functionals which have no analogue in stochastic thermodynamics.