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fractional-time deformation of quantum coherence in open systems: a non-markovian framework beyond lindblad dynamics

Taylan Demir

TL;DR

This work introduces a fractional-time quantum master equation by replacing the standard time derivative in the Lindblad form with a Caputo derivative, thereby embedding memory effects without explicit kernels. The authors establish well-posedness and physical consistency, show that solutions involve Mittag-Leffler functions, and demonstrate non-exponential, long-tail coherence decay. Through a two-level system with amplitude damping, they show how the fractional order $α$ continuously tunes memory strength, transitioning from Markovian to strongly non-Markovian dynamics. The framework offers a simple, physically interpretable phenomenological approach to non-Markovian open quantum dynamics and suggests avenues for connecting the fractional order to environmental structure and experimental data.

Abstract

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.

fractional-time deformation of quantum coherence in open systems: a non-markovian framework beyond lindblad dynamics

TL;DR

This work introduces a fractional-time quantum master equation by replacing the standard time derivative in the Lindblad form with a Caputo derivative, thereby embedding memory effects without explicit kernels. The authors establish well-posedness and physical consistency, show that solutions involve Mittag-Leffler functions, and demonstrate non-exponential, long-tail coherence decay. Through a two-level system with amplitude damping, they show how the fractional order continuously tunes memory strength, transitioning from Markovian to strongly non-Markovian dynamics. The framework offers a simple, physically interpretable phenomenological approach to non-Markovian open quantum dynamics and suggests avenues for connecting the fractional order to environmental structure and experimental data.

Abstract

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.

Paper Structure

This paper contains 30 sections, 17 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Mathematical Preliminaries
  3. Density Operators and Quantum States
  4. Completely Positive Maps
  5. Fractional Derivatives in Time
  6. Trace Preservation and Physical Consistency
  7. Fractional-Time Quantum Master Equation
  8. Classical Lindblad master equation
  9. Fractional-time generalization
  10. Well-posedness of the fractional master equation
  11. Physical admissibility conditions
  12. Quantum Coherence and Memory Effects
  13. Quantum coherence measures
  14. Fractional order and decoherence rate
  15. Comparison with exponential Markovian decay
  16. ...and 15 more sections

Figures (2)

  • Figure 1: Fractional-time decoherence dynamics (2D): time evolution of the $\ell_1$-norm coherence $C_{\ell_1}(t)$ for different fractional orders $\alpha$. Decreasing $\alpha$ delays coherence loss, reflecting the strengthening of memory effects in the fractional-time evolution.
  • Figure 2: Fractional-time quantum coherence landscape (3D): coherence level $C_{\ell_1}$ as a function of time and fractional order $\alpha$. Smaller values of $\alpha$ correspond to more persistent coherence and non-exponential long-tail relaxation.