The study of an extended hierarchy equation of motion in the spin-boson model: The cutoff function of the sub-Ohmic spectral density

Duan, Chenru; Wang, Qianlong; Tang, Zhoufei; Wu, Juan

doi:10.1063/1.4997669

Cited by 25 publications

(17 citation statements)

References 41 publications

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“…136 To obtain a numerically converged solution as the thermal equilibrium state, it is important to express the noise correlation function in terms of the decaying functions as c j exp(−ζ j t) or c j exp(−ζ j ± iω j ), where c j , ζ j , and ω j characterize the noise correlation for the jth hierarchical elements, to maintain the positivity of the HEOM formalism. [137][138][139][140][141][142][143][144][145][146][147][148] Although the HEOM can be constructed for an arbitrary SDF in the form of a Fourier expansion 87 or in terms of special functions, [149][150][151][152][153][154][155][156] the HEOM that are derived from a finite set of non-timedecaying functions do not converge in time and thus may not describe phenomena toward the thermal equilibrium state.…”

Section: F the Bathentanglement States And The Positivity Of The Heommentioning

confidence: 99%

“…Alternatively, we can extend the HEOM for arbitrary SDFs using Fourier, 89 Gauss-Legendre, or Chebyshevquadrature spectral decompositions (eHEOM). [149][150][151][152][153][154][155][156] This approach allowed the investigation of a spin-boson system coupled to a bath with sub-Ohmic SDF at zero temperature. [153][154][155] An extension of the HEOM for a combination of exponential/non-exponential bath correlation functions has also been presented.…”

Section: A Heom For Arbitrary Spectral Distribution Functionsmentioning

confidence: 99%

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Numerically “exact” approach to open quantum dynamics: The hierarchical equations of motion (HEOM)

Tanimura

2020

The Journal of Chemical Physics

408

293

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An open quantum system refers to a system that is further coupled to a bath system consisting of surrounding radiation fields, atoms, molecules, or proteins. The bath system is typically modeled by an infinite number of harmonic oscillators. This system-bath model can describe the time-irreversible dynamics through which the system evolves toward a thermal equilibrium state at finite temperature. In nuclear magnetic resonance and atomic spectroscopy, dynamics can be studied easily by using simple quantum master equations under the assumption that the system-bath interaction is weak (perturbative approximation) and the bath fluctuations are very fast (Markovian approximation). However, such approximations cannot be applied in chemical physics and biochemical physics problems, where environmental materials are complex and strongly coupled with environments. The hierarchical equations of motion (HEOM) can describe numerically "exact" dynamics of a reduced system under nonperturbative and non-Markovian system-bath interactions, which has been verified on the basis of exact analytical solutions (non-Markovian tests) with any desired numerical accuracy. The HEOM theory has been used to treat systems of practical interest, in particular to account for various linear and nonlinear spectra in molecular and solid state materials, to evaluate charge and exciton transfer rates in biological systems, to simulate resonant tunneling and quantum ratchet processes in nanodevices, and to explore quantum entanglement states in quantum information theories. This article, presents an overview of the HEOM theory, focusing on its theoretical background and applications, to help further the development of the study of open quantum dynamics.

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Section: F the Bathentanglement States And The Positivity Of The Heommentioning

confidence: 99%

Section: A Heom For Arbitrary Spectral Distribution Functionsmentioning

confidence: 99%

Numerically “exact” approach to open quantum dynamics: The hierarchical equations of motion (HEOM)

Tanimura

2020

The Journal of Chemical Physics

408

293

View full text Add to dashboard Cite

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“…Much like our number of surrogate modes, the number of exponentials needed for an adequate fit is one of the main factors determining complexity of HEOM simulations, the other being the tier at which the hierarchy needs to be truncated. This number should be the same for the two methods if one requires the same accuracy and uses efficient expansion techniques for the correlation function [115,116] (these overcome the well-known problem of the more traditional Matsubara-frequency expansion [117,118], which at low temperatures needs a very large number of exponential terms in order to converge). HEOM scales with the complexity of the spectral density in a similar manner as our method and can likewise account implicitly for temperature through the approximate correlation function.…”

Section: Impactmentioning

confidence: 99%

Optimized auxiliary oscillators for the simulation of general open quantum systems

Mascherpa,

Smirne,

Somoza

et al. 2019

Preprint

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A method for the systematic construction of few-body damped harmonic oscillator networks accurately reproducing the effect of general bosonic environments in open quantum systems is presented. Under the sole assumptions of a Gaussian environment and regardless of the system coupled to it, an algorithm to determine the parameters of an equivalent set of interacting damped oscillators obeying a Markovian quantum master equation is introduced. By choosing a suitable coupling to the system and minimizing an appropriate distance between the two-time correlation function of this effective bath and that of the target environment, the error induced in the reduced dynamics of the system is brought under rigorous control. The interactions among the effective modes provide remarkable flexibility in replicating non-Markovian effects on the system even with a small number of oscillators, and the resulting Lindblad equation for the system and the modes may therefore be integrated at a very reasonable computational cost using standard methods for Markovian problems, even in strongly non-perturbative coupling regimes and at arbitrary temperatures including zero. We apply the method to an exactly solvable problem in order to demonstrate its accuracy, and present two studies based on current research in the context of coherent transport in biological aggregates and organic photovoltaics as more realistic examples of its use and potential; performance and versatility are highlighted, and theoretical and numerical advantages over existing methods, as well as possible future improvements, are discussed.

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“…While HEOM is, in principle, exact for systems interacting with arbitrary harmonic baths, practically, it has been mostly restricted to simulating the case of baths described by a Drude spectral densities. Attempts have been made to develop efficient HEOM-based algorithms that are applicable to general spectral densities [8,[10][11][12][13]. In case the solvent is atomistically defined and anharmonic effects are important, the IF does not have a closed-form expression.…”

Section: Introductionmentioning

confidence: 99%

A Multisite Decomposition of the Tensor Network Path Integrals

Bose,

Walters

2021

Preprint

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Tensor network decompositions of path integrals for simulating open quantum systems have recently been proven to be useful. In this work, we extend the tensor network path integral (TNPI) framework to efficiently simulate extended systems coupled with local vibrational and phononic modes. The Feynman-Vernon influence functional is a very popular approach used to account for the effect of a bath on the dynamics of the system. In order to facilitate the incorporation of the influence functional into a multisite framework (MS-TNPI), we combine a matrix product state decomposition of the reduced density tensor of the system along the sites with a corresponding tensor network representation of the time axis to construct an efficient 2D tensor network. The 2D MS-TNPI network, when finally contracted, yields the time-dependent reduced density tensor of the extended system as a matrix product state. The decomposition and algorithm presented are independent of the nature of the system Hamiltonian. We also outline an iteration scheme to take the simulation beyond the non-Markovian memory length introduced by the dissipative baths. Applications to spin chains coupled to local harmonic baths is presented; we consider interactions defined by the Ising, XXZ and the Heisenberg models. We demonstrate that the presence of dissipative environments can often dissipate the entanglement between the sites as measured by the bond dimension of the reduced density matrix product state. The MS-TNPI method would be useful for studying a variety of extended quantum systems coupled with vibrational baths or phononic modes.

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The study of an extended hierarchy equation of motion in the spin-boson model: The cutoff function of the sub-Ohmic spectral density

Cited by 25 publications

References 41 publications

Numerically “exact” approach to open quantum dynamics: The hierarchical equations of motion (HEOM)

Numerically “exact” approach to open quantum dynamics: The hierarchical equations of motion (HEOM)

Optimized auxiliary oscillators for the simulation of general open quantum systems

A Multisite Decomposition of the Tensor Network Path Integrals

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