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Markovian Embeddings of Non-Markovian Quantum Systems: Coupled Stochastic and Quantum Master Equations for Non-Markovian Quantum Systems

Hendra I. Nurdin

TL;DR

The paper tackles non-Markovian quantum dynamics, where standard GKSL descriptions fail due to memory effects. It introduces Markovian embeddings that couple a non-Markovian principal system to auxiliary degrees of freedom and compound baths driven by quantum white noise, enabling a tractable description via coupled stochastic and quantum master equations. The authors derive explicit forms of stochastic and deterministic evolutions—an operator-valued stochastic master equation (SME) and a corresponding quantum master equation (QME)—for general embedding architectures, including direct-coupling, cascaded, and multi-bath setups, with the reduced dynamics recovering GKSL behavior when appropriate. This framework supports open-loop and feedback control and provides scalable reduced models for numerical simulation, offering deeper insight into the structure of continuous-time non-Markovian quantum dynamics.

Abstract

Quantum Markov models are employed ubiquitously in quantum physics and in quantum information theory due to their relative simplicity and analytical tractability. In particular, these models are known to give accurate approximations for a wide range of quantum optical and mesoscopic systems. However, in general, the validity of the Markov approximation entails assumptions regarding properties of the system of interest and its environment, which may not be satisfied or accurate in arbitrary physical systems. Therefore, developing useful modelling tools for general non-Markovian quantum systems for which the Markov approximation is inappropriate or deficient is an undertaking of significant importance. This work considers non-Markovian principal quantum systems that can be embedded in a larger Markovian quantum system with one or more compound baths consisting of an auxiliary quantum system and a quantum white noise field, and derives a set of coupled stochastic and quantum master equations for embedded non-Markovian quantum systems. The case of a purely Hamiltonian coupling between the principal and auxiliary systems as a closed system without coupling to white noises is included as a special case. The results are expected to be of interest for (open-loop and feedback) control of continuous-time non-Markovian systems and studying reduced models for numerical simulation of such systems. They may also shed more light on the general structure of continuous-time non-Markovian quantum systems.

Markovian Embeddings of Non-Markovian Quantum Systems: Coupled Stochastic and Quantum Master Equations for Non-Markovian Quantum Systems

TL;DR

The paper tackles non-Markovian quantum dynamics, where standard GKSL descriptions fail due to memory effects. It introduces Markovian embeddings that couple a non-Markovian principal system to auxiliary degrees of freedom and compound baths driven by quantum white noise, enabling a tractable description via coupled stochastic and quantum master equations. The authors derive explicit forms of stochastic and deterministic evolutions—an operator-valued stochastic master equation (SME) and a corresponding quantum master equation (QME)—for general embedding architectures, including direct-coupling, cascaded, and multi-bath setups, with the reduced dynamics recovering GKSL behavior when appropriate. This framework supports open-loop and feedback control and provides scalable reduced models for numerical simulation, offering deeper insight into the structure of continuous-time non-Markovian quantum dynamics.

Abstract

Quantum Markov models are employed ubiquitously in quantum physics and in quantum information theory due to their relative simplicity and analytical tractability. In particular, these models are known to give accurate approximations for a wide range of quantum optical and mesoscopic systems. However, in general, the validity of the Markov approximation entails assumptions regarding properties of the system of interest and its environment, which may not be satisfied or accurate in arbitrary physical systems. Therefore, developing useful modelling tools for general non-Markovian quantum systems for which the Markov approximation is inappropriate or deficient is an undertaking of significant importance. This work considers non-Markovian principal quantum systems that can be embedded in a larger Markovian quantum system with one or more compound baths consisting of an auxiliary quantum system and a quantum white noise field, and derives a set of coupled stochastic and quantum master equations for embedded non-Markovian quantum systems. The case of a purely Hamiltonian coupling between the principal and auxiliary systems as a closed system without coupling to white noises is included as a special case. The results are expected to be of interest for (open-loop and feedback) control of continuous-time non-Markovian systems and studying reduced models for numerical simulation of such systems. They may also shed more light on the general structure of continuous-time non-Markovian quantum systems.
Paper Structure (8 sections, 23 equations, 3 figures)

This paper contains 8 sections, 23 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Mathematical preliminaries
  3. Markovian embedding of non-Markovian quantum systems
  4. Direct coupling embedding
  5. Cascaded embedding
  6. General Markovian embedding
  7. Coupled non-Markovian quantum stochastic and quantum master equations
  8. Conclusion

Figures (3)

  • Figure 1: Embedding by direct coupling. A principal quantum is directly coupled to the auxiliary quantum system in a compound bath consisting of the auxiliary coupled to a single traveling quantum field.
  • Figure 2: Markovian embedding via cascading. A principal quantum I/O system is cascaded to a compound bath consisting of an auxiliary quantum I/O system that is coupled to a single traveling quantum field. The output field of the compound bath becomes the input to the principal system.
  • Figure 3: A general Markovian embedding. The principal quantum system is coupled to $M$ compound baths. Each compound bath labeled $l$ consists of an auxiliary system that is coupled to one or more quantum fields. The principal system is coupled to each compound bath through a direct interaction term (thin bidirectional arrows) and instantaneous feedback interconnections by mediated by output quantum fields (thick one directional arrows).

Theorems & Definitions (1)

  • Remark 1