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Journal of Physics B Atomic Molecular and Optical Physics

Non-Markovian effects in optical Bloch dynamics: geometry and power spectra

Octavian Postavaru

Published Jun 9, 2026• 1 citation
Publisher

Abstract

Abstract Fractional-order extensions of quantum dynamical equations provide an effective framework for incorporating memory and non-Markovian effects into the description of open quantum systems. In this work we investigate a laser-driven two-level system governed by a fractional generalization of the optical Bloch equations based on Caputo derivatives. The resulting evolution is analyzed in phase space, where temporal memory manifests itself through systematic geometric modifications of the relaxation trajectories. In particular, decreasing the fractional order produces a rotation and contraction of the phase-space spirals, together with a displacement of their effective centers, revealing a consistent reorganization of the dissipative oscillatory dynamics. We further connect this time-domain behavior to experimentally accessible frequency-domain signatures by constructing the incoherent resonance-fluorescence power spectrum from the stationary dipole correlation function. The fractional spectra recover the classical three-peak (Mollow-type) structure in the limit α → 1 , while for α < 1 they exhibit a redistribution of spectral weight accompanied by modified peak localization, sideband separation, and intensity ratios. These results demonstrate how fractional dynamics links the geometry of relaxation to measurable spectral morphology in non-Markovian quantum systems.

DOI
10.1088/1361-6455/ae7ad0

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Abstract Fractional-order extensions of quantum dynamical equations provide an effective framework for incorporating memory and non-Markovian effects into the description of open quantum systems. In this work we investigate a laser-driven two-level system governed by a fractional generalization of the optical Bloch equations based on Caputo derivatives. The resulting evolution is analyzed in phase space, where temporal memory manifests itself through systematic geometric modifications of the relaxation trajectories. In particular, decreasing the fractional order produces a rotation and contraction of the phase-space spirals, together with a displacement of their effective centers, revealing a consistent reorganization of the dissipative oscillatory dynamics. We further connect this time-domain behavior to experimentally accessible frequency-domain signatures by constructing the incoherent resonance-fluorescence power spectrum from the stationary dipole correlation function. The fractional spectra recover the classical three-peak (Mollow-type) structure in the limit α → 1 , while for α < 1 they exhibit a redistribution of spectral weight accompanied by modified peak localization, sideband separation, and intensity ratios. These results demonstrate how fractional dynamics links the geometry of relaxation to measurable spectral morphology in non-Markovian quantum systems.
Classical Maxwell–Bloch models are often inadequate for describing electromagnetic wave propagation in complex optical media where memory effects, anomalous relaxation processes, and long-range electromagnetic interactions are important. To address these limitations, this work develops a fractional nonlocal Maxwell–Bloch framework that simultaneously incorporates Caputo fractional temporal memory and nonlocal electromagnetic coupling. The principal novelty lies in the unified treatment of temporal memory and spatial nonlocality within the Maxwell–Bloch framework, together with the corresponding analysis of their effects on optical-wave dynamics. The proposed model captures hereditary polarization dynamics and spatially distributed electromagnetic interactions that are not represented in the standard local integer-order formulation. The analytical part establishes the well-posedness structure of the governing system, including existence, boundedness, and stability properties of solutions in appropriate functional spaces. A spectral stability analysis is further employed to characterize fractional instability thresholds and transitions in the resulting wave propagation. Numerical simulations demonstrate the emergence of localized fractional optical structures, memory-dependent pulse propagation, nonlinear wave interactions, and complex dynamical regimes. A systematic parametric study further shows that variations in the fractional order significantly modify the electric-field profile, energy evolution, and pulse amplitude. The results demonstrate that the combined effects of fractional temporal memory and spatial nonlocality can substantially alter the propagation and stability of electromagnetic waves compared with the classical Maxwell–Bloch limit. The framework therefore provides a unified mathematical and computational basis for investigating memory-dependent and spatially nonlocal optical dynamics.
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.
We present a new approach for investigating the Markovian to non-Markovian transition in quantum aggregates strongly coupled to a vibrational bath through the analysis of linear absorption spectra. Utilizing hierarchical algebraic equations in the frequency domain, we elucidate how these spectra can effectively reveal transitions between Markovian and non-Markovian regimes, driven by the complex interplay of dissipation, aggregate–bath coupling, and intra-aggregate dipole–dipole interactions. Our results demonstrate that reduced dissipation induces spectral peak splitting, signaling the emergence of bath-induced non-Markovian effects. The spectral peak splitting can also be driven by enhanced dipole–dipole interactions, although the underlying mechanism differs from that of dissipation-induced splitting. In addition, with an increase in aggregate–bath coupling strength, initially symmetric or asymmetric peaks with varying spectral amplitudes may merge under weak dipole–dipole interactions, whereas strong dipole–dipole interactions are more likely to cause peak splitting. Moreover, we find that spectral features serve as highly sensitive indicators for distinguishing the geometric structures of aggregates while also unveiling the critical role that geometry plays in shaping non-Markovian behavior. This study not only deepens our understanding of the Markovian to non-Markovian transition but also provides a robust framework for optimizing and controlling quantum systems.
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.
Abstract Harnessing non-Markovian effects has emerged as a resource for quantum control, where a structured environment can act as a quantum memory. We investigate the quench dynamics from specific initial states to equilibrium steady states in strongly correlated quantum dot systems. The distance between quantum states is quantified using the Bures metric, which endows the space of reduced density matrices with a Riemannian geometric structure. Using the numerically exact hierarchical equations of motion method, we demonstrate a quantum Mpemba effect (QME) arising from non-Markovianity. Unlike previous studies centered on non-interacting limits, we reveal a distinct non-Markovian QME uniquely driven by the initial-state-dependent formation of the many-body Kondo resonance. Furthermore, by tracking the dynamical evolution in the generalized Bloch parameter space, we uncover an intrinsic QME. This trajectory-dependent analysis provides a rigorous geometric perspective on the circuitous relaxation pathways driven by Kondo correlations. Our results underscore the critical role of memory effects in quantum quench dynamics and suggest new pathways for controlling anomalous relaxation in open quantum systems.
We define non-Markovian quantum dynamics as evolution in which the current state depends on all past states, and completely characterize its structure under the assumptions of complete positivity and non-signalling. The resulting continuous-time dynamics is an integro-differential equation that augments the Gorini-Kossakowski-Sudarshan-Lindblad equation with a memory integral, and is capable of describing the quantum state of systems exposed to noise with any integrable power spectral density with no further approximations. We then establish a formalism to evaluate multi-time correlations of measurement outcomes in this general setting, obviating the need for a regression theorem. As an application, we derive the emission spectrum of a driven two-level system coupled to a non-Markovian bath: the familiar Mollow triplet acquires a frequency-dependent linewidth that encodes the memory of the bath. Our work provides a rigorous yet transparent description of the quantum state of non-Markovian systems, opening the door for state estimation and state-based quantum control beyond the Markovian regime.