Applications of Real and Imaginary time Hierarchical Equations of Motion

概要

学位論⽂の要約
題⽬

Applications of Real and Imaginary time Hierarchical Equations of Motion
（実時間と虚時間の階層⽅程式の実⽤）

氏名

張 嘉驥

序論
The hierarchical equations of motion (HEOM) approach plays an important role in the study
of open quantum dynamics. Its current formulation has been applied to various chemical and
physical processes, for instance proton transfer reaction. Original studies are limited to the real
time dynamics. Recently extension has been made to the thermodynamic problems. The socalled imaginary time HEOM serves as a powerful tool on the evaluation of various
thermodynamic quantities.

Proton tunneling in a two-dimensional potential energy surface with a non-linear
system–bath interaction: Thermal suppression of reaction rate
We consider a proton transfer (PT) system described by a proton transfer reaction (PTR)
coordinate and a rate promoting vibrational (RPV) coordinate interacting with a non-Markovian
heat bath. While dynamics of PT processes has been widely discussed using two-dimensional
potential energy surfaces, the role of the heat bath, in particular, in a realistic form of the
system–bath interaction has not been well explored. Previous studies are largely based on a
one-dimensional model and linear-linear system–bath interaction.
In the present study, we introduce an exponential-linear (EL) system–bath interaction,
which is derived from the analysis of a PTR–RPV system in a realistic situation. This interaction
mainly causes vibrational dephasing in the PTR mode and population relaxation in the RPV
mode. Numerical simulations were carried out using the hierarchical equations of motion
approach. We analyze the role of the heat bath interaction in the chemical reaction rate as a
function of the system–bath coupling strength at different temperatures and for different values
of the bath correlation time.
A prominent feature of the present result is that while the reaction rate predicted from
classical and quantum Kramers theory increases as the temperature increases, the present EL
interaction model exhibits opposite temperature dependence. The Kramers turn-over profile of
the reaction rate as a function of the system–bath coupling is also suppressed in the present EL
model, turning into a plateau-like curve for larger system bath interaction strength. Such
features arise from the interplay of the vibrational dephasing process in the PTR mode and the
population relaxation process in the RPV mode.

1

Probing photoinduced proton-coupled electron transfer process by means of twodimensional electronic-vibrational spectroscopy
We develop a detailed theoretical model of photo-induced proton-coupled electron transfer
(PPCET) processes, which are at the basis of solar energy harvesting in biological systems and
photovoltaic materials. Our model enables us to analyze the dynamics and the efficiency of a
PPCET reaction under the influence of a thermal environment by disentangling the contribution
of the fundamental electron transfer and proton transfer steps.
In order to study quantum dynamics of the PPCET process under an interaction with the
non-Markovian environment, we employ the hierarchical equations of motion. We calculate
transient absorption spectroscopy (TAS) and a newly defined two-dimensional resonant
electronic–vibrational spectroscopy (2DREVS) signals in order to study the non-equilibrium
reaction dynamics. Our results show that different transition pathways can be separated by TAS
and 2DREVS.
Since we use the eigenstate representation of the system, it is also possible to improve the
description of the reacting system by increasing the dimension of its configuration space and
by introducing a more complex and structured system–bath interaction, for example, with the
help of machine learning approaches. This provides a powerful tool to analyze the nonequilibrium reaction dynamics for rather complex PPCET reactions.

Imaginary-time hierarchical equations of motion for thermodynamic variables
The partition function (PF) plays a key role in the calculation of quantum thermodynamic
properties of a system that interacts with a heat bath. The imaginary-time hierarchical equations
of motion (imHEOM) approach was developed to evaluate in a rigorous manner the PF of a
system strongly coupled to a non-Markovian bath.
In this thesis, we present a more detailed discussion of imaginary-time HEOM (imHEOM).
We also introduce an additional β-differentiated imHEOM (BD-imHEOM) that are obtained
from the derivative of the elements of imHEOM with respect to inverse temperature β. We use
both imHEOM and BD-imHEOM to evaluate the thermodynamic properties of not only the
reduced system but also the system–bath interaction and the bath, in a numerically rigorous
manner. In addition, we introduce the polyharmonic decomposition (PHD) method to construct
both the imHEOM and the BD-imHEOM in a concise hierarchical structure that makes it
possible to improve the numerical performance with high precision.
The capability of this formalism has been verified through numerical demonstrations. We
have employed a spin-boson model and a 2 × 2 transverse Ising model to clearly reveal the
behavior of various quantities in a straightforward manner. Although this work has been
restricted to an Ohmic form of the SDF, we note here that the PHD method imposes no
restrictions on the SDF and could have broader applications. To make the present approach
more useful, further computational efforts will have to be made to treat larger systems. For
example, we can extend our imHEOM to the wave function based case by utilizing some other
techniques. The scalability of all the above-mentioned approaches is the same as that of the
typical Schrödinger equation, which incurs less computational cost than the density-matrixbased approach. ...