Non-Markovian Quantum Processes with Memory Kernels Extension of the Lindblad Formalism via the Generalized Nakajima-Zwanzig EquationLast & I think for a bit of time, formalized conclusion https://zenodo.org/records/19517607 Description This repository contains the complete theoretical foundations of a (futur) rigorous extension to the Lindblad master equation for describing non-Markovian quantum processes. The work introduces a general mathematical framework based on memory kernels
K(t,s) to capture the effects of a structured environment with finite memory, going beyond the Markovian approximation. It is accompanied by an interactive, web-based version of this work, featuring a fully functional simulator, dynamic graphs, a 3D Bloch sphere visualization, and unit tests, available at: ((...)) Core Contributions 1. Axiomatic Definition of
KCP A rigorous definition of the space of memory kernels that preserve the fundamental properties of quantum states: complete positivity and trace conservation. Necessary and sufficient conditions for a kernel to be admissible are established and proven. Axiom A1 (Hermiticity):
K(t,s)=K∗(s,t) Axiom A2 (
L2-integrability):
∫0∞∫0∞∣K(t,s)∣2dsdt<∞ Axiom A3 (Complete Positivity):
σmin(K^(ω))≥0, ∀ω 2. Characterization of the Instantaneous Generator
L(t) A formal link between the memory kernel
K(t,s) and the instantaneous generator
L(t) is established, with explicit formulas for different kernel types:
L(t)=L+∫0tK(τ)dτ−K(0) 3. Exactly Solvable Examples Detailed analysis of four canonical memory kernels: Exponential (Ornstein-Uhlenbeck process):
ρ11(t)=ρ11(∞)+[ρ11(0)−ρ11(∞)]e−γt Oscillatory (Cavity QED, damped Rabi oscillations):
K(t,s)=ΩR2cos[ΩR(t−s)]e−κ∣t−s∣ Compact Support (finite memory) Power-Law (
1/f noise, non-ergodicity):
K(t)∼1/t1−α 4. Quantitative Measures of Non-Markovianity Definition of three new, operationally significant measures to quantify the degree of memory in a quantum process: Measure Definition Interpretation
ML T1∫0T∣L˙(t)∣dt Based on the temporal variation of
L(t).
ML=0 iff Markovian.
MK inf{Ki}∥K−∑iKi⊗Ki∗∥ Distance to a simple Kraus representation. $\Deg(K)$
limt→∞ln(1/t)ln∣K(t)∣ Asymptotic exponent classifying the memory length (0 for Markovian, 1 for strongly non-Markovian). 5. Comprehensive Validation Comparison with experimental data from NMR, superconducting cavities, and the FMO complex. Proof-of-concept numerical simulations using an adaptive step-size Runge-Kutta 4 (RK45) integrator, integrated into the interactive version. Interactive Simulator (Online Version) The interactive version of this work, located at ((...)), includes a powerful falsification toolbox: Real-time RK45 Simulator with 5 selectable kernel types. Dynamic 2D plots for population
ρ11(t), information flow
σ(t), and comparison with Lindblad. 3D Bloch sphere visualization of the qubit state. Integrated unit tests for trace conservation and complete positivity. Metrics display (
ML,
MK, $\Deg(K)$,
τmem, RK4 error). Comparison with experimental datasets (NMR, FMO, etc.). Export options for PDF, LaTeX, Markdown, and JSON data. Repository Contents CORE-IMMUABLE - extension-non-markovienne.md (8.6 MB): The comprehensive, publication-ready manuscript in Markdown format. It contains all theoretical derivations, proofs, and references. This is the permanent, citable record of the work.Online interactive version Citation If you use this work, please cite it as: DESVAUX, G. J. Y. (2023). Non-Markovian Extension. Zenodo. https://doi.org/10.5281/zenodo.19102132 Keywords Open quantum systems, Non-Markovianity, Memory kernels, Lindblad equation, Nakajima-Zwanzig equation, Complete positivity, Kraus representation, Quantum simulator, Runge-Kutta,
1/f noise Additional Information Languages: The core manuscript is in English. Titles are also available in French ("Extension non-markovienne") and Chinese ("非马尔可夫扩展"). NON-DISCLOSURE AGREEMENT Academic Collaboration Disclosing Party Receiving Party Legal Name: Hope 'n Mind SASU Name: Representative: Guillaume Desvaux Title / Position: Role: President-Founder Institution / Affiliation: Registration: RCS Brest, SIREN 938 261 310 Institutional Address: Address: 10 Rue Amiral Vallon, 29200 Brest, France Contact Information: This Non-Disclosure Agreement (the “Agreement”) is effective as of the date of the last signature below. 1. Purpose The Receiving Party is being provided with certain Confidential Information solely for the following purpose: Permitted Purpose: Any use of the Confidential Information outside the Permitted Purpose is strictly prohibited unless the Disclosing Party gives prior written consent. 2. 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Research in open quantum system dynamics has received growing interest in recent years, and its ongoing investigation has uncovered many intriguing physics departing from closed systems. A particularly useful application of the open quantum system theory is to simulate quantum dynamical processes in condensed phase materials. As the quantum degree of freedom is constantly under the influence of its thermal environment, the accurate description of the dynamics often requires non-Markovian time evolution at finite temperature. Such calculation is usually quite challenging on classical computers as extensive memory storage is required. In this work, we focus on a quantum system linearly coupled to its harmonic bath and present a path integral based quantum algorithm that time-evolves the reduced density matrix under finite temperature non-Markovian dynamics. To treat the nonunitary time evolution, the Sz.-Nagy dilation scheme is used for the conversion to unitary dynamics that can be implementable on gate-based quantum computers. The modified Hadamard test is then used to retrieve all the information of the reduced density matrix. Complexity analysis shows that the memory requirement has exponential reduction on the quantum machine whereas the runtime complexity stays roughly the same as for the classical computer. It points to the possibility of using this algorithm to simulate multilevel and multisite non-Markovian quantum dynamics that are beyond the reach of classical computers. The algorithm makes no ad hoc assumptions and extends naturally beyond Markovian and weak coupling regimes. We validated the algorithm on the quantum simulator with the spin-boson model and demonstrated its excellent agreement with the classical computer benchmark. Published by the American Physical Society 2024
The reduced description of the quantum dynamic processes in the condensed phase environment leads to the equation of motion with a memory kernel. Such a memory effect, termed non-Markovianity, presents more complex dynamics compared to its memoryless or Markovian counterpart, and many chemical systems have been demonstrated through numerical simulations to exhibit non-Markovian quantum dynamics. Explicitly how the memory impacts the dynamic process remains largely unexplored. In this work, we focus on ways to separate the non-Markovian contributions from the dynamics and study the non-Markovian effects. Specifically, we developed a rigorous procedure for mapping the exact non-Markovian quantum propagator to the Lindblad form. Consequently, it allows us to extract the negative decay rate from the Lindbladian that is the signature of the non-Markovianity. By including or excluding the negative rate in the time evolution, we can decisively pinpoint the influence of non-Markovianity on the system's properties such as coherence, entanglement, and equilibrium state distribution. The understanding of such memory effects on the dynamic process suggests the possibility of leveraging non-Markovianity for quantum control.
The Lindblad equation determines the time evolution of the density operator of open quantum systems. While valid for any system size, its use is, in practice, restricted to prototype/surrogate models with the aim of tackling specific aspects of the overall quantum complexity of a multi-atomic system. Path integral molecular dynamics (PIMD) instead provides static and dynamical quantum statistical averages of physical observables for systems in equilibrium composed of up to thousands of atoms over timescales up to nanoseconds, under the condition that short-time quantum coherence is not relevant for the properties of interest. PIMD relies on the well-established technique of molecular dynamics (MD) with its associated classical trajectories. However, it cannot describe a direct time evolution of a system and its convergence to a stationary state in situations out of equilibrium. In this work, we analyze the link between the Lindblad equation and PIMD; specifically, we will discuss how PIMD can actually be used to calculate the time evolution of ensemble-averaged physical observables and their convergence to a stationary state for situations out of equilibrium, bypassing the need of explicitly solving the Lindblad equation. Yet, at the same time, the Lindblad equation and PIMD are linked to one another through a formal relation of equivalence, which provides an argument for the consistency of PIMD results, namely the positivity of the density operator at any time. A numerical study of a prototype system, which is of interest in chemical physics, will be used to showcase the method.
We introduce a new approach for calculating quantum time-correlation functions and time-dependent expectation values in many-body thermal systems; both electronically adiabatic and non-adiabatic cases can be treated. Our approach uses a path integral simulation to sample an initial thermal density matrix; subsequent evolution of this density matrix is equivalent to solution of the time-dependent Schrödinger equation, which we perform using a linear expansion of Gaussian wavepacket basis functions which evolve according to simple classical-like trajectories. Overall, this methodology represents a formally exact approach for calculating time-dependent quantum properties; by introducing approximations into both the imaginary-time and real-time propagations, this approach can be adapted for complex many-particle systems interacting through arbitrary potentials. We demonstrate this method for the spin Boson model, where we find good agreement with numerically exact calculations. We also discuss future directions of improvement for our approach with a view to improving accuracy and efficiency.
Memory effects play a key role in the dynamics of strongly correlated systems driven out of equilibrium. In this paper, we explore the nature of memory in the nonequilibrium Anderson impurity model. The Nakajima-Zwanzig-Mori formalism is used to derive an exact generalized quantum master equation for the reduced density matrix of the interacting quantum dot, which includes a non-Markovian memory kernel. A real-time path integral formulation is developed in which all diagrams are stochastically sampled in order to numerically evaluate the memory kernel. We explore the effects of temperature down to the Kondo regime, as well as the role of source-drain-bias voltage and bandwidth on the memory. Typically, the memory decays on time scales significantly shorter than the dynamics of the reduced density matrix itself, yet under certain conditions, it develops a low magnitude but long-ranged tail. In addition, we address the conditions required for the existence, uniqueness, and stability of a steady state.