Simulating Non-Markovian Dynamics in Open Quantum Systems
Meng Xu, Vasilii Vadimov, J. T. Stockburger, J. Ankerhold
TL;DR
The paper presents quantum dissipation with minimally extended state space (QD-MESS) as a unifying framework that connects a broad class of non-Markovian open-quantum-system methods. By embedding a finite set of effective reservoir modes and exploiting Gaussian bath statistics, it shows how hierarchical (HEOM), Lindblad-pseudomode, Brownian-motion, stochastic Liouville–von Neumann, HOPS, thermofield, and chain-mapping approaches are interrelated through equivalent time-local formulations. It discusses topologies (star vs chain), Green's-function representations of bath spectra, and practical decompositions of $S_eta()$ to achieve efficient simulations across parameter regimes, including strong coupling, structured reservoirs, and low temperatures. The work underscores that embedding methods offer broad applicability, numerical stability, and computational efficiency, while also outlining current limitations and directions, such as tensor networks, machine learning, and environment engineering, to advance non-Markovian quantum dynamics simulations in complex systems.
Abstract
Recent advances in quantum technologies and related experiments have created a need for highly accurate, versatile, and computationally efficient simulation techniques for the dynamics of open quantum systems. Long-lived correlation effects (non-Markovianity), system-environment hybridization, and the necessity for accuracy beyond the Born-Markov approximation form particular challenges. Approaches to meet these challenges have been introduced, originating from different fields, such as hierarchical equations of motion, Lindblad-pseudomode formulas, chain-mapping approaches, quantum Brownian motion master equations, stochastic unravelings, and refined quantum master equations. This diversity, while indicative of the field's relevance, has inadvertently led to a fragmentation that hinders cohesive advances and their effective cross-community application to current problems for complex systems. How are different approaches related to each other? What are their strengths and limitations? Here we give a systematic overview and concise discussion addressing these questions. We make use of a unified framework which very conveniently allows to link different schemes and, this way, may also catalyze further progress. In line with the state of the art, this framework is formulated not in a fully reduced space of the system but in an extended state space which in a minimal fashion includes effective reservoir modes. This in turn offers a comprehensive understanding of existing methods, elucidating their physical interpretations, interconnections, and applicability.