Reduced hierarchical equations of motion in real and imaginary time: Correlated initial states and thermodynamic quantities

Tanimura, Yoshitaka

doi:10.1063/1.4890441

Cited by 195 publications

(257 citation statements)

References 97 publications

(215 reference statements)

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“…(16) and (21), they define the hierarchal elements for the correlated initial equilibrium state. 108 To demonstrate this point, we consider the initial states of the density operators, obtained by setting t = 0 in Eq. (14),…”

Section: Reduced Hierarchal Equations Of Motion From Correlated Imentioning

confidence: 99%

“…IV. 108 In practice, the former approach is simpler, because it requires the real-time HEOM only. This approach has been used to set the correlated initial conditions of the HEOM derived from factorized initial conditions that are identical to those used with the present HEOM.…”

Section: Real-time Quantum Hierarchal Fokker-planck Equationsmentioning

confidence: 99%

“…While only the first element W (p, q; t) ≡ W (0) 0,0, ...,0 (p, q; t) has a physical meaning and the other elements W (n) j 1 , ..., j K (p, q; t) are initially introduced to avoid the explicit treatment of the inherent memory effects, it turns out, however, that these elements allow us to take into account the system-bath coherence, 8 entanglement, 88,116,117 and expectation values that include the bath operators as ⟨Ĥ I ⟩ ≡ −⟨q  α jx j ⟩. 108 The HEOM consist of an infinite number of equations, but they can be evaluated with the desired accuracy by depicting the asymptotic behavior of the hierarchal elements for different K and using this to determine whether or not there are sufficiently many members in the hierarchy. Essentially, the error introduced by the truncation to be negligibly small when K is sufficiently large.…”

Section: Real-time Quantum Hierarchal Fokker-planck Equationsmentioning

confidence: 99%

“…118,119 By applying the methodology developed in Ref. 108, we can derive the QHFP equations in imaginary time. This allows us to calculate the thermal equilibrium distribution W eq (p, q) at inverse temperature β .…”

Section: Imaginary-time Quantum Hierarchal Fokker-planck Equationsmentioning

confidence: 99%

“…37,38 Recently, the author derived the HEOM not only in real time but also in imaginary time, which represents an inverse temperature, starting from correlated initial conditions for a system described by discretized energy states. 108 Reduction of these HEOM to a system represented in Wigner space is not straightforward, because they involve derivatives with respect to the position and momentum that require a careful treatment with regards to the order of time slices in the path integral formalism. In this paper, we present the derivation of real-and imaginary-time HEOM in Wigner space and demonstrate the validity of these equations.…”

Section: Introductionmentioning

confidence: 99%

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Real-time and imaginary-time quantum hierarchal Fokker-Planck equations

Tanimura

2015

The Journal of Chemical Physics

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We consider a quantum mechanical system represented in phase space (referred to hereafter as "Wigner space"), coupled to a harmonic oscillator bath. We derive quantum hierarchal Fokker-Planck (QHFP) equations not only in real time but also in imaginary time, which represents an inverse temperature. This is an extension of a previous work, in which we studied a spin-boson system, to a Brownian system. It is shown that the QHFP in real time obtained from a correlated thermal equilibrium state of the total system possesses the same form as those obtained from a factorized initial state. A modified terminator for the hierarchal equations of motion is introduced to treat the non-Markovian case more efficiently. Using the imaginary-time QHFP, numerous thermodynamic quantities, including the free energy, entropy, internal energy, heat capacity, and susceptibility, can be evaluated for any potential. These equations allow us to treat non-Markovian, non-perturbative system-bath interactions at finite temperature. Through numerical integration of the real-time QHFP for a harmonic system, we obtain the equilibrium distributions, the auto-correlation function, and the first-and second-order response functions. These results are compared with analytically exact results for the same quantities. This provides a critical test of the formalism for a non-factorized thermal state and elucidates the roles of fluctuation, dissipation, non-Markovian effects, and systembath coherence. Employing numerical solutions of the imaginary-time QHFP, we demonstrate the capability of this method to obtain thermodynamic quantities for any potential surface. It is shown that both types of QHFP equations can produce numerical results of any desired accuracy. The FORTRAN source codes that we developed, which allow for the treatment of Wigner space dynamics with any potential form (TanimuranFP15 and ImTanimuranFP15), are provided as the supplementary material. C 2015 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License. [http://dx

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Section: Reduced Hierarchal Equations Of Motion From Correlated Imentioning

confidence: 99%

Section: Real-time Quantum Hierarchal Fokker-planck Equationsmentioning

confidence: 99%

Section: Real-time Quantum Hierarchal Fokker-planck Equationsmentioning

confidence: 99%

Section: Imaginary-time Quantum Hierarchal Fokker-planck Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Real-time and imaginary-time quantum hierarchal Fokker-Planck equations

Tanimura

2015

The Journal of Chemical Physics

Self Cite

134

237

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Efficient calculation of open quantum system dynamics and time‐resolved spectroscopy with distributed memory HEOM (DM‐HEOM)

et al. 2018

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Time- and frequency-resolved optical signals provide insights into the properties of light-harvesting molecular complexes, including excitation energies, dipole strengths and orientations, as well as in the exciton energy flow through the complex. The hierarchical equations of motion (HEOM) provide a unifying theory, which allows one to study the combined effects of system-environment dissipation and non-Markovian memory without making restrictive assumptions about weak or strong couplings or separability of vibrational and electronic degrees of freedom. With increasing system size the exact solution of the open quantum system dynamics requires memory and compute resources beyond a single compute node. To overcome this barrier, we developed a scalable variant of HEOM. Our distributed memory HEOM, DM-HEOM, is a universal tool for open quantum system dynamics. It is used to accurately compute all experimentally accessible time- and frequency-resolved processes in light-harvesting molecular complexes with arbitrary system-environment couplings for a wide range of temperatures and complex sizes. © 2018 Wiley Periodicals, Inc.

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Influences of Quantum Mechanically Mixed Electronic and Vibrational Pigment States in 2D Electronic Spectra of Photosynthetic Systems: Strong Electronic Coupling Cases

Fujihashi

Fleming

Ishizaki

2015

J Chinese Chemical Soc

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In 2D electronic spectroscopy studies, long-lived quantum beats have recently been observed in photosynthetic systems, and several theoretical studies have suggested that the beats are produced by quantum mechanically mixed electronic and vibrational states. Concerning the electronic-vibrational quantum mixtures, the impact of protein-induced fluctuations was examined by calculating the 2D electronic spectra of a weakly coupled dimer with the Franck-Condon active vibrational modes in the resonant condition [Fujihashi et al., J. Chem. Phys. 142, 212403 (2015), arXiv:1505. This analysis demonstrated that quantum mixtures of the vibronic resonance are rather robust under the influence of the fluctuations at cryogenic temperatures, whereas the mixtures are eradicated by the fluctuations at physiological temperatures. However, this conclusion cannot be generalized because the magnitude of the coupling inducing the quantum mixtures is proportional to the interpigment electronic coupling. In this study, we explore the impact of the fluctuations on electronic-vibrational quantum mixtures in a strongly coupled dimer with an offresonant vibrational mode. Toward this end, we calculate energy transfer dynamics and 2D electronic spectra of a model dimer that corresponds to the most strongly coupled bacteriochlorophyll molecules in the Fenna-Matthews-Olson complex in a numerically accurate manner. The quantum mixtures are found to be robust under the exposure of protein-induced fluctuations at cryogenic temperatures, irrespective of the resonance. At 300 K, however, the quantum mixing is disturbed more strongly by the fluctuations, and therefore, the beats in the 2D spectra become obscure even in a strongly coupled dimer with a resonant vibrational mode. Further, the overall behaviors of the energy transfer dynamics are demonstrated to be dominated by the environment and coupling between the 0 − 0 vibronic transitions as long as the Huang-Rhys factor of the vibrational mode is small. The electronic-vibrational quantum mixtures do not necessarily play a significant role in electronic energy transfer dynamics despite contributing to the enhancement of long-lived quantum beating in the 2D spectra.

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Reduced hierarchical equations of motion in real and imaginary time: Correlated initial states and thermodynamic quantities

Cited by 195 publications

References 97 publications

Real-time and imaginary-time quantum hierarchal Fokker-Planck equations

Real-time and imaginary-time quantum hierarchal Fokker-Planck equations

Efficient calculation of open quantum system dynamics and time‐resolved spectroscopy with distributed memory HEOM (DM‐HEOM)

Influences of Quantum Mechanically Mixed Electronic and Vibrational Pigment States in 2D Electronic Spectra of Photosynthetic Systems: Strong Electronic Coupling Cases

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