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

TL;DR

The paper integrates fractional calculus into open quantum dynamics by embedding memory at the generator level through Caputo derivatives and Bochner–Phillips subordination. It shows that fractional master equations are a structured, CPTP subclass that interpolates Liouville, Lindblad, and non-Markovian memory kernels, while remaining connected to Nakajima–Zwanzig, HEOM, and influence-functional approaches. The authors provide both analytic results (equivalence to convolution forms, Mittag–Leffler solutions, and subordination representations) and practical benchmarks against a solvable spin–boson model, including predictive rules for α and λ from spectral densities and a plateau-normalized extension for super-Ohmic baths. They also offer two quantum-simulation paths: QSP/QSVT-based polynomial approximations and fractional quantum trajectories via subordination sampling, enabling scalable simulation of memory effects on future hardware. Overall, the framework supplies a rigorous, flexible, and tractable language for long-memory quantum dynamics with direct routes to inference and quantum algorithms."

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.

Figures (8)

• Figure 1: Schematic hierarchy of quantum dynamical regimes unified by fractional subordination. The parameter $\alpha$ is the memory exponent of the renewal process governing the operational time of the dynamics:$\alpha=1$ yields Markovian Lindblad semigroups with exponential waiting times, while $0<\alpha<1$ generates long-tailed waiting statistics and non-Markovian evolution with algebraically decaying memory. The parameter $\lambda$ controls the dissipative strength of the underlying GKSL generator. Fractional or subordinated Lindblad dynamics ($0<\alpha<1$) therefore provide a CPTP bridge connecting unitary Liouville evolution ($\alpha=1,\lambda=0$), Markovian semigroups ($\alpha=1,\lambda>0$), and general memory--kernel quantum master equations ($\alpha\neq 1$).
• Figure 2: General subordination replaces physical time by a random operational time. Fractional (inverse–stable) laws yield the fractional subordination, while other waiting-time distributions lead to non-fractional NZ kernels with short- or multi-scale memory (not necessarily power-law).
• Figure 3: Algebraic connections between the fractional Lindblad equation and mainstream non--Markovian approaches. Fractional dynamics arise as (i) a resolvent-level deformation of the NZ memory-kernel equation, (ii) a coarse-grained surrogate for HEOM self-energy structures, (iii) a subordinated Markovian process corresponding to Lévy-distributed operational times, and (iv) a compact representation of long-memory features also encoded in influence-functional methods such as QUAPI and MCTDH.
• Figure 4: Short-time ($t \le 0.2/ \omega_c$, left panels) and long-time ($t \ge 5/\omega_c$, right panels) asymptotic propagation of the dephasing functional $Q(t)$ and corresponding coherence amplitude $|\langle\sigma_+(t)\rangle|$ for three different bath exponents $\chi=0.5$, $1.0$, and $1.5$. Solid lines denote numerical integrations of Eq. \ref{['eq:Q-general']}; open circles show asymptotic expressions (Table. \ref{['tab:Q-short-long']}) with optimized prefactors, consistent with the asymptotic analysis in Refs. Leggett1987Grifoni1998Weiss2012deVega2017.
• Figure 5: Comparison of exact coherence from Figure \ref{['fig:Q_Sigma_exact']} with Markovian and non-Markovian fits. The asymptotic propagation of the coherence amplitude $|\langle\sigma_+(t)\rangle|$ for three different bath exponents $\chi=0.5$, $1.0$, and $1.5$ are computed by constant $\gamma$ (Markovian, left panel) and time-local $\gamma(t) = \tfrac{1}{2}\dot{Q}(t)$ (non-Markovian, right panel), respectively. The constant rate $\gamma$ is obtained by fitting $\log|\langle\sigma_+(t)\rangle|$ to a linear decay in the intermediate-time regime $t\in [2/\omega_c, 60/\omega_c]$ (blue shade, left panel), while solid lines are obtained by numerically integrating Eq. \ref{['eq:Q-general']}. The parameters shown here ($\gamma$ and $\gamma(t)$) belong to Markovian and time-local models only and are not related to the fractional parameters $(\alpha,\lambda)$ introduced in Figure \ref{['fig:frac_general']}.

Theorems & Definitions (8)

• Definition 1: Liouville Dynamics
• Definition 2: Lindblad/GKSL Dynamics
• Definition 3: Memory-Kernel Quantum Master Equation
• Definition 4: Fractional Master Equation
• Remark 1: Physical meaning of the fractional order $\alpha$.
• Remark 2: Fractional Liouville and Fractional Lindblad
• Proposition 1: Caputo--Volterra--Convolution equivalence
• Remark 3: Three kernels and their roles