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arXiv:2102.09484 · Physical review. B./Physical review. B

Nonequilibrium open quantum systems with multiple bosonic and fermionic environments: A hierarchical equations of motion approach

Jakob Bätge, Yaling Ke, Christoph Kaspar, Michael Thoss

Published Jun 8, 2021• 30 citations
arXivHTMLPDF

Abstract

We present a hierarchical equations of motion approach, which allows a numerically exact simulation of nonequilibrium transport in general open quantum systems involving multiple macroscopic bosonic and fermionic environments. The performance of the method is demonstrated for a model of a nanosystem, which involves interacting electronic and vibrational degrees of freedom and is coupled to fermionic and bosonic baths. The results show the intricate interplay of electronic and vibrational degrees of freedom in this nonequilibrium transport scenario for both voltage and thermally driven transport processes. Furthermore, the use of importance criteria to improve the efficiency of the method is discussed.

DOI
10.1103/physrevb.103.235413

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We present a hierarchical equations of motion approach, which allows a numerically exact simulation of nonequilibrium transport in general open quantum systems involving multiple macroscopic bosonic and fermionic environments. The performance of the method is demonstrated for a model of a nanosystem, which involves interacting electronic and vibrational degrees of freedom and is coupled to fermionic and bosonic baths. The results show the intricate interplay of electronic and vibrational degrees of freedom in this nonequilibrium transport scenario for both voltage and thermally driven transport processes. Furthermore, the use of importance criteria to improve the efficiency of the method is discussed.
We present a hierarchical quantum master equation (HQME) approach, which allows a numerically exact simulation of nonequilibrium transport in general open quantum systems involving multiple bosonic and fermionic environments. The performance of the method is demonstrated for a model of a nanosystem, which involves interacting electronic and vibrational degrees of freedom and is coupled to fermionic and bosonic baths. The results show the intricate interplay of electronic and vibrational degrees of freedom in this nonequilibrium transport scenario for both voltage and thermally driven transport processes. Furthermore, the use of importance criteria to improve the efficiency of the method is discussed.
We extend the twin-space formulation of the hierarchical equations of motion approach in combination with the matrix product state representation [R. Borrelli, J. Chem. Phys. 150, 234102 (2019)] to nonequilibrium scenarios where the open quantum system is coupled to a hybrid fermionic and bosonic environment. The key ideas used in the extension are a reformulation of the hierarchical equations of motion for the auxiliary density matrices into a time-dependent Schrödinger-like equation for an augmented multi-dimensional wave function as well as a tensor decomposition into a product of low-rank matrices. The new approach facilitates accurate simulations of non-equilibrium quantum dynamics in larger and more complex open quantum systems. The performance of the method is demonstrated for a model of a molecular junction exhibiting current-induced mode-selective vibrational excitation.
We extend a standard stochastic theory to study open quantum systems coupled to a generic quantum environment. We exemplify the general framework by studying a two-level quantum system coupled bilinearly to the three fundamental classes of non-interacting particles: bosons, fermions, and spins. In this unified stochastic approach, the generalized stochastic Liouville equation (SLE) formally captures the exact quantum dissipations when noise variables with appropriate statistics for different bath models are applied. Anharmonic effects of a non-Gaussian bath are precisely encoded in the bath multi-time correlation functions that noise variables have to satisfy. Starting from the SLE, we devise a family of generalized hierarchical equations by averaging out the noise variables and expand bath multi-time correlation functions in a complete basis of orthonormal functions. The general hierarchical equations constitute systems of linear equations that provide numerically exact simulations of quantum dynamics. For bosonic bath models, our general hierarchical equation of motion reduces exactly to an extended version of hierarchical equation of motion which allows efficient simulation for arbitrary spectral densities and temperature regimes. Similar efficiency and flexibility can be achieved for the fermionic bath models within our formalism. The spin bath models can be simulated with two complementary approaches in the present formalism. (I) They can be viewed as an example of non-Gaussian bath models and be directly handled with the general hierarchical equation approach given their multi-time correlation functions. (II) Alternatively, each bath spin can be first mapped onto a pair of fermions and be treated as fermionic environments within the present formalism.
Abstract The hierarchical equations of motion (HEOM) approach can describe the reduced dynamics of a system simultaneously coupled to multiple bosonic and fermionic environments. The complexity of exactly describing the system-environment interaction with the HEOM method usually results in time-consuming calculations and a large memory cost. Here, we introduce an open-source software package called HierarchicalEOM.jl: a Julia framework integrating the HEOM approach. HierarchicalEOM.jl features a collection of methods to compute bosonic and fermionic spectra, stationary states, and the full dynamics in the extended space of all auxiliary density operators (ADOs). The required handling of the ADOs multi-indexes is achieved through a user-friendly interface. We exemplify the functionalities of the package by analyzing a single impurity Anderson model, and an ultra-strongly coupled charge-cavity system interacting with bosonic and fermionic reservoirs. HierarchicalEOM.jl achieves a significant speedup with respect to the corresponding method in the Quantum Toolbox in Python (QuTiP), upon which this package is founded.
Investigations of strongly correlated quantum impurity systems (QIS), which exhibit diversified novel and intriguing quantum phenomena, have become a highly concerning subject in recent years. The hierarchical equations of motion (HEOM) method is one of the most popular numerical methods to characterize QIS linearly coupled to the environment. This review provides a comprehensive account of a formally rigorous and numerical convergent HEOM method, including a modeling description of the QIS and an overview of the fermionic HEOM formalism. Moreover, a variety of spectrum decomposition schemes and hierarchal terminators have been proposed and developed, which significantly improve the accuracy and efficiency of the HEOM method, especially in cryogenic temperature regimes. The practicality and usefulness of the HEOM method to tackle strongly correlated issues are exemplified by numerical simulations for the characterization of nonequilibrium quantum transport and strongly correlated Kondo states as well as the investigation of nonequilibrium quantum thermodynamics.
In this review we give a comprehensive account of a hierarchical equations of motion (HEOM) approach to the characterization of stationary and dynamic properties of open quantum systems. This approach is rooted at the Feynman-Vernon influence functional path integral formalism, but much more implementable numerically and operationally for the study of various complex molecular dynamics and quantum transport in strongly correlated electronic systems. By construction, HEOM resolves nonperturbatively the combined effects of many-particle interaction, system-bath coupling,and non-Markovian memory. Finally the practicality of HEOM to address physical and chemical problems is exemplified with a model simulation of coherent two-dimensional spectroscopy signals of a biological light-harvesting system and a time-dependent quantum transport system involving dynamic Kondo transition.級聯方程已成為研究量子開放系統的穩態性質和動力學過程的重要方法。本文旨在系統綜述量子耗散和量子輸運的級聯方程組方法的建立、發展以及在理論、算法和應用方面的一些最新進展。級聯方程形式理論的建立以影響泛函路徑積分為基礎,并具有數值上的高效性和應用上的靈活性,可用于研究分子體系的復雜動力學過程以及強關聯電子體系中的量子輸運。其級聯耦合結構以非微擾的方式揭示了多體相互作用、體系-環境耦合、非馬爾可夫記憶等的綜合效應。作為應用示例,我們采用級聯方程模擬了生物光富集體系的二維相干動力學光譜以及含時電子輸運過程中的動態近藤效應。