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arXiv:2003.00194

Lindblad Plus From Feynman-Vernon

Jose A. Magpantay

Published Feb 29, 2020• 0 citations
arXivHTMLPDF

Abstract

I show how the Lindblad Plus equation will follow from the Feynman-Vernon theory. The Plus refers to the inclusion of non-Markov processes in the Lindblad equation resulting in an integro-differential general master equation. The equivalence of this general master equation and not the Lindblad equation alone to the Feynman-Vernon theory should be expected because the sum over histories approach of the FV theory clearly includes non-Markov processes, which Lindblad equation ignores. This should close the seeming gap between these two approaches to quantum open systems.

DOI
10.48550/arxiv.2003.00194

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I show how the Lindblad Plus equation will follow from the Feynman-Vernon theory. The Plus refers to the inclusion of non-Markov processes in the Lindblad equation resulting in an integro-differential general master equation. The equivalence of this general master equation and not the Lindblad equation alone to the Feynman-Vernon theory should be expected because the sum over histories approach of the FV theory clearly includes non-Markov processes, which Lindblad equation ignores. This should close the seeming gap between these two approaches to quantum open systems.
FINDING: The exact master equation for quantum dot decoherence, derived via Feynman-Vernon influence functional, provides a non-Markovian generalization of Lindblad form with time-dependent coefficients — the only mathematically rigorous exact solution in this domain. | MATH: Exact master equation: ρ˙(t)=−i[HS(t),ρ]+∑ijΔij(t)[2fjρfi†−{fi†fj,ρ}] \dot{\rho}(t) = -i[H_S(t), \rho] + \sum_{ij} \Delta_{ij}(t) [2 f_j \rho f_i^\dagger - \{f_i^\dagger f_j, \rho\}] ρ˙​(t)=−i[HS​(t),ρ]+∑ij​Δij​(t)[2fj​ρfi†​−{fi†​fj​,ρ}] where Δij(t)\Delta_{ij}(t)Δij​(t) are time-dependent dissipation coefficients (non-Markovian memory kernels), fif_ifi​ are fermionic annihilation operators. The Feynman-Vernon influence functional: F[j,j′]=exp⁡[−∫0tds∫0sdu (j(s)−j′(s))(Δ(s−u)j(u)−Δ∗(s−u)j′(u))] \mathcal{F}[j,j'] = \exp\left[ -\int_0^t ds \int_0^s du \, (j(s) - j'(s)) ( \Delta(s-u) j(u) - \Delta^*(s-u) j'(u) ) \right] F[j,j′]=exp[−∫0t​ds∫0s​du(j(s)−j′(s))(Δ(s−u)j(u)−Δ∗(s−u)j′(u))] with spectral density J(ω)=∑k∣gk∣2δ(ω−ωk) J(\omega) = \sum_k |g_k|^2 \delta(\omega - \omega_k) J(ω)=∑k​∣gk​∣2δ(ω−ωk​). Decoherence rate: Γ(t)=∫0tdτ Re[Δ(τ)] \Gamma(t) = \int_0^t d\tau \, \text{Re}[\Delta(\tau)] Γ(t)=∫0t​dτRe[Δ(τ)] — no golden ratio appears in the exact solution. | CONNECTION: None found. The exact master equation y Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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.
FINDING: Decoherence time records and exact master equations for quantum dot systems reveal non-Markovian dynamics governed by exact integro-differential equations from Feynman-Vernon influence functional. | MATH: Exact master equation: ρ˙(t)=−i[HS,ρ(t)]+∫0tds K(t,s)ρ(s) \dot{\rho}(t) = -i[H_S, \rho(t)] + \int_0^t ds \, \mathcal{K}(t,s) \rho(s) ρ˙​(t)=−i[HS​,ρ(t)]+∫0t​dsK(t,s)ρ(s), where kernel K\mathcal{K}K encodes non-Markovian memory. Decoherence time τd∝1/γ\tau_d \propto 1/\gammaτd​∝1/γ, with γ\gammaγ coupling strength. No specific numerical record or constant extracted from provided sources. | CONNECTION: No explicit geometric ratios (0.382, 0.618, 0.786, 1.618, 2.618) or base-60, crystallographic symmetry, root systems, or lattice structures found in these sources. The Feynman-Vernon path integral formalism inherently involves complex phase factors eiS/ℏe^{iS/\hbar}eiS/ℏ that can produce interference patterns, but no harmonic ratio emerges from the given data. | DEPTH: 3 — The exact master equation is a rigorous mathematical tool for decoherence in quantu Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
The theory of open quantum systems is one of the most essential tools for the development of quantum technologies. Furthermore, the Lindblad (or Gorini-Kossakowski-Sudarshan-Lindblad) master equation plays a key role as it is the most general generator of Markovian dynamics in quantum systems. In this paper, we present this equation together with its derivation and methods of resolution. The presentation tries to be as self-contained and straightforward as possible to be useful to readers with no previous knowledge of this field.
Nuclear Magnetic Resonance (NMR) experiments involve coherent and incoherent dynamics. Incoherent dynamics give rise to nuclear spin relaxation typically accounted for by a differential equation known as a quantum master equation. Out of the existing master equations the Lindblad equation is particularly important. A central theorem of open quantum theory states that a relaxation process represents a quantum Markovian process if and only if it can be described by a Lindblad master equation. This in turn raises serious questions regarding the compatibility of conventional NMR relaxation theory and one of the most fundamental results of open quantum theory. The NMR treatment of important Markovian relaxation mechanisms such as Brownian motion or chemical exchange events do in general not result in a master equation in Lindblad form and would therefore be at odds with the mathematically rigorous theory of open quantum systems. We explore this apparent conflict for two important chemical exchange phenomena and show that their conventional treatment may be recast into a suitable Lindblad equation. Similarities between spatial diffusion and multi-site exchange events allow us to extend our results to nuclear spin relaxation due to translational diffusion. As a result we are able to show that the NMR treatment of intramolecular exchange and translational diffusion represent valid quantum Markovian processes compatible with the rigorous definitions of open quantum theory.
In this paper we derive an extra class of non-Markovian master equations where the system state is written as a sum of auxiliary matrixes whose evolution involve Lindblad contributions with local coupling between all of them, resembling the structure of a classical rate equation. The system dynamics may develop strong nonlocal effects such as the dependence of the stationary properties with the system initialization. These equations are derived from alternative microscopic interactions, such as complex environments described in a generalized Born-Markov approximation and tripartite system-environment interactions, where extra unobserved degrees of freedom mediates the entanglement between the system and a Markovian reservoir. Conditions that guarantee the completely positive condition of the solution map are found. Quantum stochastic processes that recover the system dynamics in average are formulated. We exemplify our results by analyzing the dynamical action of nontrivial structured dephasing and depolarizing reservoirs over a single qubit.