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

Journal Article · Sat Feb 28 00:00:00 EST 2026 · The Journal of Chemical Physics

DOI:https://doi.org/10.1063/5.0312309· OSTI ID:3023751

Peng, Bo; Zhang, Yu

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.

• Research Organization: Pacific Northwest National Laboratory (PNNL), Richland, WA (United States)
• Sponsoring Organization: USDOE
• Grant/Contract Number: AC05-76RL01830
• OSTI ID: 3023751
• Report Number(s): PNNL-SA-217990
• Journal Information: The Journal of Chemical Physics, Journal Name: The Journal of Chemical Physics Journal Issue: 8 Vol. 164
• Country of Publication: United States
• Language: English