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Physical Review A

Non-Markovian dynamics without using a quantum trajectory

Chengjun Wu, Yang Li, Mingyi Zhu, Hong Mei Guo

Published May 17, 2011• 3 citations
Publisher

Abstract

Open quantum systems interacting with structured environments is important and manifests non-Markovian behavior, which was conventionally studied using a quantum trajectory stochastic method. In this paper, by dividing the effects of the environment into two parts, we propose a deterministic method without using a quantum trajectory. This method is more efficient and accurate than the stochastic method in most Markovian and non-Markovian cases. We also extend this method to the generalized Lindblad master equation.

DOI
10.1103/physreva.83.052116

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Open quantum systems interacting with structured environments is important and manifests non-Markovian behavior, which was conventionally studied using a quantum trajectory stochastic method. In this paper, by dividing the effects of the environment into two parts, we propose a deterministic method without using a quantum trajectory. This method is more efficient and accurate than the stochastic method in most Markovian and non-Markovian cases. We also extend this method to the generalized Lindblad master equation.
Open quantum system interacting with structured environment is important and manifests non- Markovian behavior, which was conventionally studied using quantum trajectory stochastic method. In this paper, by dividing the effects of the environment into two parts, we propose a deterministic method without using quantum trajectory. This method is more efficient and accurate than stochastic method in most Markovian and non-Markovian cases. We also extend this method to the generalized Lindblad master equation.
General open quantum systems display memory features, their master equations are non-Markovian. We show that the subclass of Gaussian non-Markovian open system dynamics is tractable in a depth similar to the Markovian class. The structure of master equations exhibits a transparent generalization of the Lindblad structure. We find and parametrize the class of stochastic Schrödinger equations that unravel a given master equation, such a class was previously known for Markovian systems only. We show that particular non-Markovian unravelings known in the literature are special cases of our class.
The Monte Carlo wave function method or the quantum-trajectory-jump approach is a powerful tool to study dissipative dynamics governed by the Markovian master equation, in particular for high-dimensional systems and when it is difficult to simulate directly. We extend this method to the non-Markovian case described by the generalized Lindblad master equation. Two examples to illustrate the method are presented and discussed. The results show that the method can correctly reproduce the dissipative dynamics for the system. The difference between this method and the traditional Markovian jump approach and the computational efficiency of this method is also discussed.
A large class of non-Markovian quantum processes in open systems can be formulated through time-local master equations which are not in Lindblad form. It is shown that such processes can be embedded in a Markovian dynamics which involves a time-dependent Lindblad generator on an extended state space. If the state space of the open system is given by some Hilbert space H\mathcal{H}H, the extended state space is the triple Hilbert space $\mathcal{H}\ensuremath{\bigotimes}{\mathbb{C}}^{3}$ which is obtained by combining the open system with a three-state system. This embedding is used to derive an unraveling for non-Markovian time evolution by means of a stochastic process in the extended state space. The process is defined through a stochastic Schr\"odinger equation which generates genuine quantum trajectories for the state vector conditioned on a continuous monitoring of an environment. The construction leads to a continuous measurement interpretation for non-Markovian dynamics within the framework of the theory of quantum measurement.
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.
For the efficient simulation of open quantum systems, we often use quantum jump trajectories given by pure states that evolve stochastically to unravel the dynamics of the underlying master equation. In the Markovian regime, when the dynamics is described by a Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation, this procedure is known as Monte Carlo wave function approach. However, beyond ultraweak system-bath coupling, the dynamics of the system is not described by an equation of GKSL type, but rather by the Redfield equation, which can be brought into pseudo-Lindblad form. Here, negative dissipation strengths prohibit the conventional approach. To overcome this problem, we propose a pseudo-Lindblad quantum trajectory (PLQT) unraveling. It does not require an effective extension of the state space, like other approaches, except for the addition of a single classical bit. We test the PLQT for the eternal non-Markovian master equation for a single qubit and an interacting Fermi-Hubbard chain coupled to a thermal bath and discuss its computational effort compared to solving the full master equation.