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arXiv:2108.05755

Pseudomode description of general open quantum system dynamics: non-perturbative master equation for the spin-boson model

Graeme Pleasance, Francesco Petruccione

Published Aug 12, 2021• 2 citations
arXivHTMLPDF

Abstract

We outline a non-perturbative approach for simulating the behavior of open quantum systems interacting with a bosonic environment defined by a generalized spectral density function. The method is based on replacing the environment by a set of damped harmonic oscillators - the pseudomodes - thereby forming an enlarged open system whose dynamics is governed by a Markovian master equation. Each pseudomode is connected to one of the poles of the spectral density when analytically continued to the lower-half complex frequency plane. Here, we extend previous results to a completely generic class of open system models, and discuss how our framework can be used as a powerful and versatile tool for analyzing non-Markovian open system dynamics. The effectiveness of the method is demonstrated on the spin-boson model by accurately benchmarking its predictions against numerically exact results.

DOI
10.48550/arxiv.2108.05755

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We outline a non-perturbative approach for simulating the behavior of open quantum systems interacting with a bosonic environment defined by a generalized spectral density function. The method is based on replacing the environment by a set of damped harmonic oscillators - the pseudomodes - thereby forming an enlarged open system whose dynamics is governed by a Markovian master equation. Each pseudomode is connected to one of the poles of the spectral density when analytically continued to the lower-half complex frequency plane. Here, we extend previous results to a completely generic class of open system models, and discuss how our framework can be used as a powerful and versatile tool for analyzing non-Markovian open system dynamics. The effectiveness of the method is demonstrated on the spin-boson model by accurately benchmarking its predictions against numerically exact results.
We develop an exact framework for describing the non-Markovian dynamics of an open quantum system interacting with an environment modeled by a generalized spectral density function.The approach relies on mapping the initial system onto an auxiliary configuration, comprising the original open system coupled to a small number of discrete modes, which in turn are each coupled to an independent Markovian reservoir.Based on the connection between the discrete modes and the poles of the spectral density function, we show how expanding the system using the discrete modes allows for the full inclusion non-Markovian effects within an enlarged open system whose dynamics is governed by an exact Lindblad master equation.Initially we apply this result to obtain a generalization of the pseudomode method [B.M. Garraway, Phys. Rev. A. 55, 2290 (1997)] in cases where the spectral density function has a Lorentzian structure.For many other types of spectral density function, we extend our proof to show that an open system dynamics may be modeled physically using discrete modes which admit a non-Hermitian coupling to the system and for such cases determine the equivalent master equation to no longer be of Lindblad form.For applications involving two discrete modes, we demonstrate how to convert between pathological and Lindblad forms of the master equation using the techniques of the pseudomode method.
Using the pseudomode model to simulate open quantum systems is important in quantum physics. Here we report a systematic and efficient pseudomode model based on dissipative, noninteracting harmonic oscillators, capable of accurately simulating any type of environmental spectrum. The dynamical evolution is reduced to a Lindblad master equation with only one dissipation term. A simple procedure to determine the parameters of the pseudomodes is proposed. We evaluate its accuracy by some spin-boson models under different environmental spectra. Our findings consistently exhibit high accuracy. Finally, we extend this pseudomode method to larger systems with more complicated interactions. Our method may have potential applications in quantum information.
The pseudomode framework provides an exact description of the dynamics of an open quantum system coupled to a non-Markovian environment. Using this framework, the influence of the environment on the system is studied in an equivalent model, where the open system is coupled to a finite number of unphysical pseudomodes that follow a time-local master equation. Building on the insight that this master equation does not need to conserve the hermiticity of the pseudomode state, we here ask for the most general conditions on the master equation that guarantee the correct reproduction of the system's original dynamics. We demonstrate that our generalized approach decreases the number of pseudomodes that are required to model, for example, underdamped environments at finite temperature. We also provide an unraveling of the master equation into quantum jump trajectories of non-Hermitian states, which further facilitates the utilization of the pseudomode technique for numerical calculations by enabling the use of easily parallelizable Monte Carlo simulations. Finally, we show that pseudomodes, despite their unphysical nature, provide a natural picture in which physical processes, such as the creation of system-bath correlations or the exchange of heat, can be studied. Hence, our results pave the way for future investigations of the system-environment interaction leading to a better understanding of open quantum systems far from the Markovian weak-coupling limit. Published by the American Physical Society 2024
Embedding non-Markovian open quantum dynamics into an enlarged Markovian space offers a powerful route to nonperturbative simulations, where the dynamics of the extended space can be governed by multiple distinct Markovian equations. We show that these distinct embeddings arise from different unravelings of Gaussian bath self-energies, generating a family of deterministic, time-local equations for the extended system. Using the Brownian-oscillator spectral density as an illustrative example, we clarify the relationships among existing approaches, including the Hierarchical Equations of Motion (HEOM) and the Lindblad--pseudomode formalism, and demonstrate how this framework enables numerically stable and efficient simulations. This work provides both a transparent theoretical foundation for embedding techniques and a flexible platform for developing new methods to simulate non-Markovian quantum dynamics.
The pseudomode framework provides an efficient description of open quantum systems in non-Markovian and nonperturbative coupling regimes by utilizing simple Markovian Lindblad master equations that incorporate a set of damped bosonic pseudomodes. Developing an efficient numerical scheme to solve these equations accurately is highly desirable. In this work, we propose a modified multiple Davydov D1 density operator ansatz and combine it with the Dirac-Frenkel time-dependent variational principle to simulate the dynamics governed by these master equations. We apply this variational method to the spin-boson model, both with and without a periodic external driving field, and compute both single- and two-time expectation values in non-Markovian, strong-coupling regimes. The accuracy of our variational approach is validated by comparing its predictions with results obtained from the small matrix path integral method. We demonstrate that the variational and path-integral approaches are in excellent agreement for both single- and two-time expectations across nondriven and driven cases, spanning both low- and high-temperature regimes. Our results establish the reliability and broad applicability of the pseudomode-based variational approach for calculating two-time correlation functions in open quantum systems.
We identify the conditions that guarantee equivalence of the reduced dynamics of an open quantum system (OQS) for two different types of environments-one a continuous bosonic environment leading to a unitary system-environment evolution and the other a discrete-mode bosonic environment resulting in a system-mode (nonunitary) Lindbladian evolution. Assuming initial Gaussian states for the environments, we prove that the two OQS dynamics are equivalent if both the expectation values and two-time correlation functions of the environmental interaction operators are the same at all times for the two configurations. Since the numerical and analytical description of a discrete-mode environment undergoing a Lindbladian evolution is significantly more efficient than that of a continuous bosonic environment in a unitary evolution, our result represents a powerful, nonperturbative tool to describe complex and possibly highly non-Markovian dynamics. As a special application, we recover and generalize the well-known pseudomodes approach to open-system dynamics.