A fractional calculus framework for open quantum dynamics: From Liouville to Lindblad to memory kernels

Area of Science:

• Quantum Dynamics
• Open Quantum Systems
• Fractional Calculus

Background:

• Open quantum systems display diverse dynamics, from unitary evolution to irreversible dissipation.
• The Gorini-Kossakowski-Sudarshan-Lindblad equation describes Markovian, completely positive and trace-preserving (CPTP) evolution.
• Many systems exhibit non-Markovian features like algebraic relaxation and coherence backflow, requiring advanced modeling.

Purpose of the Study:

• To develop a unified framework for modeling non-Markovian quantum dynamics using fractional calculus.
• To embed fractional master equations within existing open-system formalisms.
• To provide a computationally efficient and rigorous approach for simulating long-memory effects in quantum systems.

Main Methods:

• Developed a unified framework incorporating fractional master equations into open-system formalisms.
• Utilized Bochner-Phillips subordination to ensure a CPTP representation via averaging over Lindblad semigroups.
• Connected fractional dynamics to established non-Markovian approaches like Nakajima-Zwanzig kernels and hierarchical equations of motion.

Main Results:

• Fractional equations form a structured subclass of memory-kernel models, reducing to the Lindblad form at unit order.
• The framework admits a CPTP representation, ensuring physical consistency.
• Fractional dynamics provide a compact and efficient surrogate for long-memory effects in quantum systems.

Conclusions:

• Fractional calculus offers a rigorous and practical language for modeling non-Markovian quantum dynamics.
• The proposed framework is CPTP-preserving and computationally efficient.
• This approach is particularly valuable for simulating condensed-phase environments with significant long-time memory and dissipation in chemical and physical chemistry.