Spin-Boson Model Dynamics: Accurate Fourth-Order Correction and Non-Markovianity Analysis.

Researchers analytically derive a fourth-order time-convolutionless generator for the spin-boson model, valid for non-weak system-environment coupling and odd spectral densities. Results demonstrate corrections to dynamics in semiconductor double-dot systems and reveal that second-order master equations often overestimate non-Markovian behaviour, offering improved accuracy over numerical methods.

Understanding the behaviour of quantum systems interacting with their surroundings is fundamental to advancements in diverse fields, from materials science to quantum computing. These interactions often lead to complex, non-Markovian dynamics, requiring increasingly sophisticated theoretical tools for accurate description. Researchers from the Optics and Quantum Information Group at The Institute of Mathematical Sciences, Prem Kumar, K. P. Athulya, and Sibasish Ghosh, address this challenge in their work, “Asymptotic TCL4 Generator for the Spin-Boson Model: Analytical Derivation and Benchmarking”. They present a detailed analytical derivation of a fourth-order time-convolutionless (TCL) generator, a mathematical tool used to approximate the evolution of open quantum systems, for the widely studied spin-boson model. This model describes a two-level quantum system, such as an electron spin, interacting with a surrounding environment, and their results offer improved accuracy over existing approximations, particularly when the interaction between system and environment is not weak. The team rigorously validates their analytical approach against both specialised calculations and numerically exact methods, revealing limitations in commonly used approximations and providing a more reliable framework for modelling complex quantum dynamics.

Researchers have developed a fourth-order time-convolutionless (TCL) generator to model the spin-boson model, a fundamental construct in open quantum systems. This development addresses limitations inherent in describing strong interactions between a quantum system and its environment, and improves the precision of dynamical calculations. The spin-boson model specifically describes a two-level quantum system, often referred to as a qubit, interacting with a harmonic bath representing the environment. The TCL generator provides a mathematical framework to calculate how the system evolves over time.

Existing second-order TCL master equations, used to describe the system’s evolution, frequently overestimate the degree of non-Markovian behaviour. Non-Markovianity describes the memory effects within a system, where its future state is dependent on its past history, rather than solely on its present conditions. This overestimation highlights the necessity of employing higher-order approximations, such as the newly developed fourth-order generator, to accurately model open quantum systems exhibiting significant non-Markovian characteristics.

Rigorous benchmarking exercises confirm the accuracy of the fourth-order TCL generator, validating its ability to capture the dynamics of the spin-boson model even in regimes where the simplifying assumption of weak coupling between the system and environment breaks down. These comparisons involved specialised analytical calculations, performed using the Ohmic spectral density – a common description of the environmental bath – with a Drude cutoff, which introduces a high-frequency limit to the bath’s influence. Further validation was achieved through comparison with the numerically exact Hierarchical Equations of Motion (HEOM) technique, a computationally intensive method used as a gold standard for these types of calculations.

To promote reproducibility and facilitate further investigation, the researchers have made a comprehensive suite of computational files publicly available. These include Mathematica notebooks (.nb and .wl) and Python scripts (.py), detailing the derivation and evaluation of the generator. Key files such as TCLIntegrandCalcs.wl, TCL4GeneratorCalc.wl, and TCLVsHEOMFidelity.wl document the mathematical processes, while tcl_vs_heom.py enables direct comparison of the TCL method with the HEOM technique.

Calculations utilising this new generator reveal corrections to the predicted dynamics of systems such as semiconductor double quantum dots. These corrections become physically significant in specific parameter regimes, demonstrating the importance of employing accurate modelling techniques. This detailed analysis provides valuable insights into the behaviour of open quantum systems and validates the efficacy of the developed methodology.

The study extends previous calculations of the Breuer-Petruccione (BLP) measure, a metric used to quantify non-Markovianity, by maximizing it over a larger ensemble of states represented on the Bloch sphere. The Bloch sphere provides a geometrical representation of a qubit’s state, allowing for a comprehensive assessment of non-Markovian dynamics and confirming the robustness of the observed features. This broader analysis provides a more complete understanding of the system’s behaviour.