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arXiv:1907.02671 · Journal of Physics A Mathematical and Theoretical

An operator derivation of the Feynman–Vernon theory, with applications to the generating function of bath energy changes and to an-harmonic baths

Erik M. Aurell, Ryoichi Kawai, Ketan Goyal

Published May 12, 2020• 21 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1751-8121/ab9274

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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 .

•May 24, 2019•0 citations
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.
The thermodynamics of a quantum system interacting with an environment that can be assimilated to a harmonic oscillator bath has been extensively investigated theoretically. In recent experiments, the system under study however does not interact directly with the bath, but though a cavity or a transmission line. The influence on the system from the bath is therefore seen through an intermediate system, which modifies the characteristics of this influence. Here we first show that this problem is elegantly solved by a transform, which we call the Vernon transform, mapping influence action kernels on influence action kernels. We also show that the Vernon transform takes a particularly simple form in the Fourier domain, though it then must be interpreted with some care. Second, leveraging results in quantum thermodynamics we show how the Vernon transform can also be used to compute the generating function of energy changes in the environment. We work out the example of a system interacting with two baths of the Caldeira-Leggett type, each of them seen through a cavity.
FINDING: The most substantive mathematical content is in the Feynman-Vernon model of a moving thermal environment (arXiv:1803.10300), which generalizes the influence functional to non-stationary, moving baths — the YouTube results are pedagogical noise with no derivable equations. | MATH: The Feynman-Vernon influence functional F[x,x′]=exp⁡[−1ℏ∫t0tdt′∫t0t′ds (x(t′)−x′(t′))(η(t′−s)(x(s)−x′(s))+iν(t′−s)(x(s)+x′(s)))] \mathcal{F}[x,x'] = \exp\left[-\frac{1}{\hbar}\int_{t_0}^{t} dt' \int_{t_0}^{t'} ds \, (x(t')-x'(t')) \left( \eta(t'-s) (x(s)-x'(s)) + i \nu(t'-s) (x(s)+x'(s)) \right) \right] F[x,x′]=exp[−ℏ1​∫t0​t​dt′∫t0​t′​ds(x(t′)−x′(t′))(η(t′−s)(x(s)−x′(s))+iν(t′−s)(x(s)+x′(s)))], where η(t)=∫0∞dωπJ(ω)cos⁡(ωt) \eta(t) = \int_0^\infty \frac{d\omega}{\pi} J(\omega) \cos(\omega t) η(t)=∫0∞​πdω​J(ω)cos(ωt) (dissipation kernel) and ν(t)=∫0∞dωπJ(ω)coth⁡(ℏω/2kBT)sin⁡(ωt) \nu(t) = \int_0^\infty \frac{d\omega}{\pi} J(\omega) \coth(\hbar\omega/2k_B T) \sin(\omega t) ν(t)=∫0∞​πdω​J(ω)coth(ℏω/2kB​T)sin(ωt) (noise kernel). For Lorentzian spectral density J(ω)=2γω02ωω02+ω2 J(\omega) = \frac{2\gamma \omega_0^2 \omega}{\omega_0^2 + \omega^2} J(ω)=ω02​+ω22γω02​ω​, the decoherence rate scales as Γ∝γkBT/ℏ \Gamma \propto \gamma k_B T / \hbar Γ∝γkB​T/ℏ in the high-T limit (linear in T), and as \( \Gamma \propto \gamma (k_B T/\hbar)^2 / Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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.
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.
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.
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 .