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arXiv:2511.13038 · The Journal of Chemical Physics

A Fractional Calculus Framework for Open Quantum Dynamics: From Liouville to Lindblad to Memory Kernels

Bo Peng, Yu Zhang

quant-phSubmitted Nov 17, 2025• Announced Nov 18, 2025• 3 citations
v1 · Nov 17, 2025v2 · Dec 17, 2025
arXivHTMLPDF

Abstract

Open quantum systems exhibit dynamics ranging from purely unitary evolution to irreversible dissipative relaxation. The Gorini--Kossakowski--Sudarshan--Lindblad (GKSL) equation uniquely characterizes Markovian dynamics that are completely positive and trace-preserving (CPTP), yet many physical systems display non-Markovian features such as algebraic relaxation and coherence backflow beyond the reach of semigroup evolution. Fractional calculus provides a natural framework for describing such long-memory behavior through power-law temporal kernels introduced by fractional time derivatives. Here we establish a unified hierarchy that embeds fractional quantum master equations within the broader landscape of open system dynamics. The fractional master equation forms a structured subclass of memory-kernel models, reducing to the GKSL form at unit fractional order. Through Bochner--Phillips subordination, fractional evolution is expressed as an average over Lindblad semigroups weighted by a power-law waiting-time distribution. This construction ensures physical consistency, explains the algebraic origin of long-time decay, and bridges unitary, Markovian, and structured non-Markovian regimes. The resulting framework positions fractional calculus as a rigorous and unifying language for quantum dynamics with intrinsic memory, enabling new directions for theoretical analysis and quantum simulation.

DOI
10.1063/5.0312309

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Open quantum systems exhibit dynamics ranging from purely unitary evolution to irreversible dissipative relaxation. The Gorini--Kossakowski--Sudarshan--Lindblad (GKSL) equation uniquely characterizes Markovian dynamics that are completely positive and trace-preserving (CPTP), yet many physical systems display non-Markovian features such as algebraic relaxation and coherence backflow beyond the reach of semigroup evolution. Fractional calculus provides a natural framework for describing such long-memory behavior through power-law temporal kernels introduced by fractional time derivatives. Here we establish a unified hierarchy that embeds fractional quantum master equations within the broader landscape of open system dynamics. The fractional master equation forms a structured subclass of memory-kernel models, reducing to the GKSL form at unit fractional order. Through Bochner--Phillips subordination, fractional evolution is expressed as an average over Lindblad semigroups weighted by a power-law waiting-time distribution. This construction ensures physical consistency, explains the algebraic origin of long-time decay, and bridges unitary, Markovian, and structured non-Markovian regimes. The resulting framework positions fractional calculus as a rigorous and unifying language for quantum dynamics with intrinsic memory, enabling new directions for theoretical analysis and quantum simulation.
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In this paper, we proposed the exactly solvable model of non-Markovian dynamics of open quantum systems. This model describes open quantum systems with memory and periodic sequence of kicks by environment. To describe these systems, the Lindblad equation for quantum observable is generalized by taking into account power-law fading memory. Dynamics of open quantum systems with power-law memory are considered. The proposed generalized Lindblad equations describe non-Markovian quantum dynamics. The quantum dynamics with power-law memory are described by using integrations and differentiation of non-integer orders, as well as fractional calculus. An example of a quantum oscillator with linear friction and power-law memory is considered. In this paper, discrete-time quantum maps with memory, which are derived from generalized Lindblad equations without any approximations, are suggested. These maps exactly correspond to the generalized Lindblad equations, which are fractional differential equations with the Caputo derivatives of non-integer orders and periodic sequence of kicks that are represented by the Dirac delta-functions. The solution of these equations for coordinates and momenta are derived. The solutions of the generalized Lindblad equations for coordinate and momentum operators are obtained for open quantum systems with memory and kicks. Using these solutions, linear and nonlinear quantum discrete-time maps are derived.
In this paper, we propose a fractional time extension of the Quan tum Master Equation. We introduce a Caputo-type fractional derivative in time as an extension of the exponential decay of the Lindblad framework through the incorporation of fractional derivatives into the Lindblad framework. We show that the analytical and numerical results of our analytical and numerical models, demonstrate that fractional dynamics produces long-memory coherence decay naturally and provides an interpretable and flexible model of non-Markovianity.
This preprint presents a self-contained and mathematically explicit reconstruction of the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation starting from exact microscopic unitary dynamics through the Nakajima-Zwanzig projection operator formalism. Beginning with the exact non-local generalized master equation, we systematically analyze the hierarchy of approximations required for the emergence of a Markovian quantum dynamical semigroup. The Born (weak-coupling), Markov (time-localization), and secular approximations are derived and interpreted as successive information-reduction steps, each associated with clearly identifiable mathematical assumptions and physical consequences. Particular attention is devoted to the role of secularization in transforming the Redfield generator into a completely positive GKSL structure. To provide a concrete worked example, the formalism is applied to a damped two-level system coupled to a bosonic thermal reservoir, where the Redfield and GKSL equations are derived explicitly and compared at the microscopic level. Beyond the derivation itself, the work develops a systematic map of validity limits and failure regimes. The breakdown of the Born, Markov, secular, factorization, and stationarity assumptions is analyzed in terms of environmental correlation times, spectral-density structure, non-Markovian memory effects, information backflow, and loss of semigroup behavior. The resulting framework clarifies the precise operational domain in which GKSL dynamics can be regarded as a physically reliable effective description and provides a rigorous foundation for applications involving entropy-production theorems, quantum thermodynamics, and open-system information theory.
This monograph delivers a rigorous, non-perturbative closed-form analytical solution to the generalized non-Markovian Lindblad-Nakajima-Zwanzig master equation governing open quantum-classical systems coupled to arbitrary non-equilibrium thermal baths. By lifting the dissipative Liouville-von Neumann evolution onto an extended non-Hermitian supersymmetric (SUSY) fibre bundle equipped with a Cartan-Darboux connection, the environmental memory kernel is mapped onto the characteristic flow of an exceptional point foliation. We prove two foundational theorems: (1) Global Closed-Form Integrability, reducing non-local integro-differential dynamics to exact meromorphic Wilson holonomy loops with O(N^3) deterministic complexity, and (2) Strict Complete Positivity and Trace Preservation (CPTP) across all finite and asymptotic temporal regimes, definitively overcoming the unphysical negative-probability breakdown during informational backflow intervals. The formulation includes a production-certified, zero-heap embedded ISO C99 deterministic kernel executing in 1.85 microseconds on standard bare-metal controllers, establishing a new foundation for decoherence-resilient autonomous cybernetic computing.
The control of open-system dynamics provides a powerful mechanism for using quantum information to process sequential tasks. While quantum reservoir computing typically relies on Markovian dissipation to process sequential data, the computational role of non-Markovian memory effects remains largely unexplored. We introduce a framework for quantum reservoir computing where non-Markovianity is explicitly modeled and regulated using fractional derivatives. By employing fractional time subordination, we generate tunable, heavy-tailed relaxation dynamics that govern the information backflow between the system and its environment. Non-Markovian information backflow regulates the overall temporal retention of the system, maximizing short-term linear memory capacity. Within an optimal operating regime, fractional non-Markovianity redistributes the reservoir memory towards recent inputs, improving both linear memory capacity and nonlinear prediction accuracy at short delays, at the expense of long-range temporal retention. System-environment interaction proves being not merely a requirement for quantum reservoir computing, but an active resource that embeds memory retention directly into the quantum evolution.