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

TL;DR

This paper introduces a fractional calculus-based framework for modeling open quantum systems with non-Markovian dynamics, unifying various approaches and enabling efficient simulation of long-memory effects.

Contribution

It develops a unified fractional master equation framework that generalizes Lindblad dynamics and connects to established non-Markovian models, offering analytical and computational advantages.

Findings

Fractional equations form a structured subclass of memory-kernel models.

The framework reduces to Lindblad form at unit order.

It admits a CPTP representation as an average over Lindblad semigroups.

Abstract

Open quantum systems exhibit dynamics ranging from unitary evolution to irreversible dissipation. While the Gorini--Kossakowski--Sudarshan--Lindblad (GKSL) equation uniquely characterizes Markovian CPTP evolution, many physical platforms display non-Markovian features such as algebraic relaxation and coherence backflow. Fractional calculus provides a natural way to model such long-memory behavior through power-law temporal kernels introduced by fractional time derivatives. Here we develop a unified framework that embeds fractional master equations within the broader hierarchy of open-system formalisms. The fractional equation forms a structured subclass of memory-kernel models, reduces to the Lindblad form at unit order, and, through Bochner--Phillips subordination, admits a CPTP representation as an average over Lindblad semigroups. Its resolvent structure further connects fractional dynamics to established non-Markovian approaches, including Nakajima--Zwanzig kernels and hierarchical equations of motion, providing a compact surrogate for long-memory effects. This formulation positions fractional calculus as a rigorous and practical language for quantum dynamics with intrinsic memory, supporting both analytical insight and efficient quantum simulation.