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Royal Philharmonic Playing PlayStation Concert in London
PlayStation games' soundtracks are getting the royal treatment.
By Shabana Arif
Updated: 24 May 2018 10:46 am
Posted: 24 May 2018 10:46 am
The Royal Philharmonic is partnering with Classic FM and PlayStation for a concert in London, featuring music from games from the original PlayStation up to PlayStation 4.
PlayStation in Concert will feature the Royal Philharmonic Orchestra and City of London Choir, as well as a flashy light show, and will feature music from a slew of PS1, PS2, PS3, and PS4 games, including The Last of Us, The Last Guardian, God of War, Everybody's Gone to the Rapture, and Horizon Zero Dawn.
"PlayStation has been a force in the world of arts and entertainment for some of its fans' entire lives," said James Williams, Managing Director of the Royal Philharmonic Orchestra. "Many people today have grown up with the music video games, and have come to love the sound of the orchestra through playing."
Concert goers will be introduced to Encue, a "real time programme note app" that displays media content related to the performance on attendees' smartphones.
"This truly 21st century concert, accompanied by the latest technology to enhance the concert experience, is a signpost for where orchestral music is expanding," added Williams. "It is fantastic that one of the world's great orchestras will collaborate with other leaders in their fields to craft something very exciting for a new generation of concert goers."
BAFTA award-winning composer of Everybody's Gone to the Rapture, Jessica Curry, will be presenting the show. Classic FM listeners may be familiar with her show, High Score, the first radio series dedicated entirely to music from video games.
"The success of our two series of High Score, and the fact that over half a million under-25-year-olds now listen to Classic FM every week, is proof of a growing younger audience for classical music in the UK," said Sam Jackson, Classic FM's Managing Editor. "We're really pleased to be partnering with the Royal Philharmonic Orchestra and PlayStation on this, the first event of its kind, which promises to be an exciting night of world-class music."
PlayStation in Concert will take place at London's Royal Albert Hall on Wednesday, May 30, and tickets are available now.
Shabana is a freelance writer who enjoys JRPGs, wine, and not finishing games. Follow her on Twitter and Instagram.
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\section{INTRODUCTION}
\label{sec:introduction}
Open quantum theories, i.e., theoretical descriptions for quantum states/dynamics of a system exposed to fluctuation and dissipation of surrounding environments, are important subjects in a wide range of physics and chemistry because the environmental effects are ubiquitous.
Recent advances in experimental technologies have made it possible to observe electronic/vibrational dynamics in very short time and atomic spatial scales in which quantum properties of materials (e.g., quantum superposition and quantum tunneling) plays essential roles \cite{scholes2017nature, scholes2018jpcl, mancal2020cp}, and quantum theories becomes more significant as theoretical inputs to analyze such experiments \cite{ishizaki2007jpca, kreisbeck2012jpcl, dean2016chem, prokhorenko2016jpcl, miyata2017mc, rafiq2019jacs}.
In terms of dynamics, various quantum/semi--classical theories have been constructed to capture the quantum nature, including equations of motion for wave functions, density matrices, phase space distributions \cite{redfield1965inbook, tanimura1989jpsj, tanimura1994jcp, zhang1998jcp, kapral1999jcp, thoss2000jcp, kuhl2002jcp, xie2014jcp, ikeda2017jcp, ikeda2019jctc, chen2019jpcl}, and Gaussian quantum wavepackets \cite{ben1998jcp, ben2000jcpa, makhov2014jcp}, and approaches utilizing mixed quantum--classical trajectories \cite{tully1990jcp, hammes1994jcp, coker1995jcp, stock2005acp, subotnik2016arpc}.

Open quantum theories are needed to predict and analyze spectra of advanced non--linear spectroscopy experiments.
These spectra occasionally show complicated features caused by transitions and beatings among many quantum states and non--Markovian and non--perturbative effects of environments appear as, e.g., time--dependent Stokes shift and spectral diffusion, which requires more rigorous theories for prediction and interpretation \cite{tanimura2009acr, ishizaki2008cp, fujihashi2015jcp}.
Open quantum theories offering rigorous non--Markovian/non--perturbative numerical results include the quasiadiabatic propagator path integral (QUAPI) \cite{makri1992cpl, ilk1994jcp}, mapping Hamiltonian approach with the density matrix renormalization group (DMRG) \cite{chin2010jmp, prior2010prl}, multi--configuration time--dependent Hartree (MCTDH) theory \cite{meyer1990cpl, manthe1992jcp, wang2000jcp, wang2007jpca}, and hierarchical equations of motion (HEOM) theory \cite{tanimura1989jpsj, ishizaki2009jcp}.
In this article, we discuss a new generalization of the HEOM theory.

The HEOM theory was originally developed by Tanimura and Kubo to describe the dynamics of an open quantum system coupled to a high--temperature Drudian environment which could be characterized by a single--exponential environment correlation function \cite{tanimura1989jpsj}.
This theory was later extended into the cases for multi--exponential correlation functions \cite{tanimura1990pra, tanimura1994jpsj, ishizaki2005jpsj, tanaka2009jpsj, tanimura2012jcp}.
Hereafter, we refer to an element of the correlation function decomposition as a basis.
Although this original HEOM theory enables us to describe the effects of non--perturbative and non--Markovian system--environment coupling for environment correlation given in the form of a multi--exponential basis set, it may still fail under circumstance when an exponential function basis set is not the optimal expression of the correlation function, or when it is impossible to decompose the correlation functions into the basis set.

A generalization of the HEOM theory for a non--exponential basis set is given in an earlier work by Yan and co--workers \cite{xu2005jcp}.
Another generalization is made by Wu and co--workers in a systematic way as the extended HEOM \cite{tang2015jcp, duan2017prb}.
However, these generalizations have redundancies and are numerically challenging:
These generalizations lead up to a $2K$ dimensional hierarchy for the case of a $K$ general function basis, whereas the original HEOM theory forms a $K$ dimensional hierarchy in its time--dependent differential equations in the case of a $K$ exponential function basis set.
Such an increase in the hierarchy dimension results in huge demands of computational memory and long computational time.
Moreover, in the above generalization, the hierarchy structures for an exponential function basis are different from those for a non--exponential function basis, which make it difficult for discussing them in a unified framework.
Although a new approach including treatments using a non--exponential basis set with $K$ dimensional hierarchy has been recently proposed, the hierarchical Schr\"{o}dinger equations of motion (HSEOM) approach \cite{nakamura2018pra}, this theory has a limitation in the possible forms of basis functions, and is unstable in its long--time behavior.

In this article, we present a new generalization of the HEOM theory including treatments of non--exponential correlation functions, which could be more efficient and stable than the original theory and other generalizations.
Our generalization has a similar hierarchy structure to the original HEOM theory, and therefore it is easy to implement the theory by using a similar code to the original theory.
We demonstrate our new generalization by using three examples that demonstrate non--exponential behavior of the environment correlation functions.

The organization of this paper is as follows.
In section \ref{sec:theory}, we show our generalization and its relation to other theories.
In section \ref{sec:results}, we present numerical results for cases of super--Ohmic spectral density, critically--damped Brownian spectral density, and zero--temperature environment as demonstrations.
Section \ref{sec:conclusion} is devoted to concluding remarks.
The library and its Python binding we developed to perform simulations based on our approach, named LibHEOM and PyHEOM respectively, are provided as supplementary material, of which up--to--date versions may be found on GitHub.

\section{THEORY}
\label{sec:theory}

In this section, we show our generalization of the HEOM theory.
To construct the original HEOM theory, the path integral formulation with the Feynman--Vernon influence functional is typically used \cite{tanimura1989jpsj, tanimura2006jpsj}.
Here, we employ a cumulant expansion technique, which is used in Ishizaki and Fleming's work \cite{ishizaki2009jcp}.

\subsection{Hamiltonian}

We consider a system linearly coupled to a harmonic environment (bath).
Here, we assume that the system--bath interaction is characterized by a single system subspace operator $\hat{V}$ for simplicity.
An extension for multiple operators with multiple independent environments is trivial, and has been implemented in our codes (See supplementary material).
The total Hamiltonian of the system is expressed as
\begin{align}
  \hat{H}^{\mathrm{tot}}&\equiv \hat{H}+\hat{H}^{\mathrm{bath}}+\hat{H}^{\mathrm{int}}
  \label{eq:total-hamiltonian},
\end{align}
where $\hat{H}$, $\hat{H}^{\mathrm{bath}}$, and $\hat{H}^{\mathrm{int}}$ are the Hamiltonians of the system, bath, and interaction, respectively.
\begin{subequations}
  The bath Hamiltonian reads
  \begin{align}
    \hat{H}^{\mathrm{bath}}\equiv \sum _{\xi }\frac{\hbar \omega _{\xi }}{2}\left(\hat{p}_{\xi }^{2}+\hat{x}_{\xi }^{2}\right),
  \end{align}
  where $\hat{x}_{\xi }$, $\hat{p}_{\xi }$, and $\omega _{\xi }$ are the dimensionless coordinate, conjugate momentum, and characteristic frequency of the $\xi $th bath mode, and the interaction Hamiltonian is expressed as
  \begin{align}
    \hat{H}^{\mathrm{int}}\equiv -\sum _{\xi }g_{\xi }\hat{x}_{\xi }\hat{V},
  \end{align}
  where $g_{\xi }$ is the coupling strength between the system and $\xi $th bath model.
\end{subequations}
When our system is a spin system, the Hamiltonian $\hat{H}^{\mathrm{tot}}$ corresponds to the well--known spin--boson model \cite{leggett1987rmp}.

\subsection{Cumulant expansion of time evolution equation}
To evaluate the effects of the interaction Hamiltonian $\hat{H}^{\mathrm{int}}$ to the dynamics, we introduce the interaction picture based on the non--interacting Hamiltonian, $\hat{H}^{0}\equiv \hat{H}+\hat{H}^{\mathrm{bath}}$, as
\begin{align}
  \tilde{O}(t)\equiv e^{+i\hat{H}^{0}(t-t_{0})/\hbar }\hat{O}(t)e^{-i\hat{H}^{0}(t-t_{0})/\hbar }.
\end{align}
Here, a tilde on an operator indicates the operator is in the interaction picture.
In this picture, the time evolution equation of the total density operator, i.e., the Liouville--von Neumann equation, can be written as $\partial _{t}\tilde{\rho }^{\mathrm{tot}}(t)=-\tilde{\mathcal{L}}^{\mathrm{int}}(t)\tilde{\rho }^{\mathrm{tot}}(t)$, and the solution is
\begin{align}
  \tilde{\rho }^{\mathrm{tot}}(t)=\mathcal{T}_{+}\exp \left(-\int _{t_{0}}^{t}\!ds\,\tilde{\mathcal{L}}^{\mathrm{int}}(s)\right)\tilde{\rho }^{\mathrm{tot}}(t_{0}).
\end{align}
Here, we have defined a Liouvillian of a Hamiltonian $\hat{H}$ by $\mathcal{L}\equiv i[\hat{H},\dots ]/\hbar $, and have introduced the chronological time ordering operator $\mathcal{T}_{+}$.
We assume that the total density operator at $t=t_{0}$ can be written as
\begin{align}
  \hat{\rho }^{\mathrm{tot}}(t_{0})=\hat{\rho }(t_{0})\otimes \hat{\rho }_{\mathrm{eq}}^{\mathrm{bath}},
  \label{eq:fact-init}
\end{align}
where $\hat{\rho }(t_{0})=\tilde{\rho }(t_{0})$ is reduced density operator of the system subspace and $\hat{\rho }_{\mathrm{eq}}^{\mathrm{bath}}$ is the bath equilibrium density operator at temperature $T$, i.e.~, $\hat{\rho }_{\mathrm{eq}}^{\mathrm{bath}}=e^{-\beta \hat{H}^{\mathrm{bath}}}/\mathcal{Z}$.
Here, $\beta \equiv 1/k_{\mathrm{B}}T$ is the inverse temperature divided by the Boltzmann constant $k_{\mathrm{B}}$ and $\mathcal{Z}$ is the partition function of the bath.
Note that this factorized initial condition Eq.~\eqref{eq:fact-init} is merely temporarily introduced to evaluate time evolution of the reduced system and is not a restriction of numerical calculations.
If we want to start simulations with a correlated thermal equilibrium state, we simulate time evolution of the system from temporal initial state Eq.~\eqref{eq:fact-init} to a sufficiently long time $t_{\mathrm{i}}$, and then we regard the state at $t_{\mathrm{i}}$ as a initial state of the following calculations we want.
This technique is frequently used to calculate optical response functions \cite{tanimura2006jpsj, tanimura2012jcp}.
It is also possible to obtain the correlated thermal equilibrium state as a steady--state solution of the HEOM \cite{zhang2017jcp} or as an inverse temperature integration of the imaginary-time HEOM \cite{tanimura2014jcp, tanimura2015jcp}.

The reduced system density operator at $t$,
\begin{align}
  \hat{\rho }(t)\equiv \mathrm{Tr}_{\mathrm{bath}}\{\hat{\rho }^{\mathrm{tot}}(t)\},
\end{align}
is expressed in the interaction picture as
\begin{align}
  \tilde{\rho }(t)=\mathcal{U}(t,t_{0})\tilde{\rho }(t_{0})\equiv \left\langle \mathcal{T}_{+}\exp \left(-\int _{t_{0}}^{t}\!ds\,\tilde{\mathcal{L}}^{\mathrm{int}}(s)\right)\right\rangle _{\mathrm{bath}}\tilde{\rho }(t_{0}),
  \label{eq:reduced-int}
\end{align}
where $\langle \dots \rangle _{\mathrm{bath}}\equiv \mathrm{Tr}_{\mathrm{bath}}\{\dots \rho _{\mathrm{eq}}^{\mathrm{bath}}\}$.
Because of the Gaussian property of the coordinate operator $\tilde{x}_{\xi }(s)$ in $\tilde{\mathcal{L}}^{\mathrm{int}}(s)$ via Wick's theorem, the above propagator $\mathcal{U}(t,t_{0})$ can be rewritten in the form of the second--order cumulant expansion as \cite{weiss2011book, ishizaki2009jcp}
\begin{align}
  \begin{split}
    \mathcal{U}(t,t_{0})&=\mathcal{T}_{+}\exp \Biggl[\int _{t_{0}}^{t}\!ds\,\frac{i}{\hbar }\tilde{V}(s)^{\times }\\
    &\quad \times \int _{t_{0}}^{s}\!du\,\frac{i}{\hbar }\left(\mathcal{C}(s-u)\tilde{V}(u)^{\rightarrow }-\mathcal{C}^{\ast }(s-u)\tilde{V}(u)^{\leftarrow }\right)\Biggr],
  \end{split}
\end{align}
where
\begin{align}
  \mathcal{C}(t)\equiv \langle \tilde{X}(t)\tilde{X}(0)\rangle _{\mathrm{bath}}
\end{align}
is the quantum correlation function of the collective environment coordinate $\hat{X}\equiv \sum _{\xi }g_{\xi }\hat{x}_{\xi }$, and we have introduced superoperators $\hat{A}^{\rightarrow }\hat{B}\equiv \hat{A}\hat{B}$, $\hat{A}^{\leftarrow }\hat{B}\equiv \hat{B}\hat{A}$, $\hat{A}^{\times }\hat{B}\equiv \hat{A}\hat{B}-\hat{B}\hat{A}$, and $\hat{A}^{\circ }\hat{B}\equiv \hat{A}\hat{B}+\hat{B}\hat{A}$.
Thus, the effects of the system--bath interaction is characterized by the second--order cumulant, i.e.,~$\mathcal{C}(t)$.
This correlation function is connected to the spectral density, $\mathcal{J}(\omega )\equiv \pi \sum _{\xi }g_{\xi }^{2}\delta (\omega -\omega _{\xi })/2$, as
\begin{align}
  \mathcal{C}(t)&=\frac{1}{\pi }\int _{-\infty }^{\infty }\!d\omega \,\mathcal{J}(\omega )\left(n_{\mathrm{BE}}(\omega ,T)+1\right)e^{-i\omega t}.
  \label{eq:C}
\end{align}
Here, $n_{\mathrm{BE}}(\omega ,T)=(e^{\beta \hbar \omega }-1)^{-1}$ is the Bose--Einstein distribution function.
The real and imaginary parts of the quantum correlation function, i.e., the symmetrized correlation function $\mathcal{S}(t)\equiv (\mathcal{C}(t)+\mathcal{C}^{\ast }(t))/2$ and anti--symmetrized correlation function $\mathcal{A}(t)\equiv (\mathcal{C}(t)-\mathcal{C}^{\ast }(t))/2i$, represent fluctuation and dissipation of the bath, respectively.
Note that
\begin{subequations}
  \begin{align}
    \mathcal{S}(t)&=\frac{2}{\pi }\int _{0}^{\infty }\!d\omega \,\mathcal{J}(\omega )\left(n_{\mathrm{BE}}(\omega ,T)+\frac{1}{2}\right)\cos \omega t
    \label{eq:S}\\
    \intertext{and}
    \mathcal{A}(t)&=-\frac{1}{\pi }\int _{0}^{\infty }\!d\omega \,\mathcal{J}(\omega )\sin \omega t
    \label{eq:A}.
  \end{align}
\end{subequations}

By using $\mathcal{S}(t)$ and $\mathcal{A}(t)$, the propagator can be rewritten as
\begin{align}
  \mathcal{U}(t,t_{0})&=\mathcal{T}_{+}\mathcal{F}(t,t_{0})\\
  \intertext{and}
  \mathcal{F}(t,t_{0})&\equiv \exp \Biggl[\int _{t_{0}}^{t}\!ds\,\tilde{\Phi }(s)\notag\\
    &\quad \quad \times \int _{t_{0}}^{s}\!du\,\left(\mathcal{S}(s-u)\tilde{\Phi }(u)-\mathcal{A}(s-u)\tilde{\Psi }(u)\right)\Biggr]
  \label{eq:cumulant}.
\end{align}
Here, $\hat{\Phi }\equiv i\tilde{V}^{\times }/\hbar $ and $\hat{\Psi }\equiv \tilde{V}^{\circ }/\hbar $.

\subsection{Hierarchical equations of motion}
We prepare a set of $K$ basis functions of time $t$, ${}^{t}\bm{\phi }(t)=(\dots ,\phi _{k}(t),\dots )$, which satisfy a set of time evolution equations
\begin{align}
  \partial _{t}\bm{\phi }(t)=-\bm{\gamma }\bm{\phi }(t).
  \label{eq:basis-evolution}
\end{align}
Here, the superscripts ${}^{t}$ on matrix/vector represent the transposes of the matrix/vector, and $(\bm{\gamma })_{jk}=\gamma _{jk}$ is a $K\times K$ complex matrix which can be non--diagonalizable.
Note that, to obtain stable time evolution of the basis $\bm{\phi }(t)$, the real parts of non--degenerate and degenerate eigenvalues of the coefficient matrix $\bm{\gamma }$ should be non-negative and positive, respectively.
To construct the ``hierarchy'' later, we need to express the symmetrized and anti--symmetrized correlation functions as linear combinations of $\bm{\phi }(t)$ and the Dirac delta function $\delta (t)$\footnote{ In this paper, we treat $\delta (t)$ as an even function, and therefore $\int _{0}^{\infty }\!dx\,f(x)\delta (x)=f(0)/2$.
}, i.e., $\mathcal{S}(t)=\sum _{k}S_{k}\cdot \phi _{k}(t)+S_{\delta }\cdot 2\delta (t)={}^{t}\bm{S}\bm{\phi }(t)+S_{\delta }\cdot 2\delta (t)$ and $\mathcal{A}(t)=\sum _{k}A_{k}\cdot \phi _{k}(t)={}^{t}\bm{A}\bm{\phi }(t)$.
In this paper, we assume that the symmetrized and anti--symmetrized correlation functions can be rewritten as
\begin{subequations}
  \begin{align}
    &\begin{aligned}
      \mathcal{S}(t)&=\sum _{jk}\sigma _{j}s_{jk}\phi _{k}(t)+S_{\delta }\cdot 2\delta (t)\\
      &={}^{t}\bm{\sigma }\bm{s}\bm{\phi }(t)+S_{\delta }\cdot 2\delta (t)
      \label{eq:S-expansion}
    \end{aligned}
    \intertext{and}
    &\begin{aligned}
      \mathcal{A}(t)&=\sum _{jk}\sigma _{j}a_{jk}\phi _{k}(t)\\
      &={}^{t}\bm{\sigma }\bm{a}\bm{\phi }(t),
      \label{eq:A-expansion}
    \end{aligned}
  \end{align}
\end{subequations}
i.e.,
\begin{align}
  {}^{t}\bm{S}={}^{t}\bm{\sigma }\bm{s}~\text{and}~{}^{t}\bm{A}={}^{t}\bm{\sigma }\bm{a}.
  \label{eq:S-s-relation}
\end{align}
Here, ${}^{t}\bm{\sigma }=(\dots ,\sigma _{k},\dots )$ is a constant vector which is common in $\mathcal{S}(t)$ and $\mathcal{A}(t)$, and $\bm{s}$ and $\bm{a}$ are $K\times K$ complex matrices, which commute with $\bm{\gamma }$.
Note that the parametrization of $\bm{\sigma }$, $\bm{s}$, and $\bm{a}$ is not unique.
In Appendix \ref{sec:commuting-s-a}, we give examples of possible constructions of $\bm{\sigma }$, $\bm{s}$, and $\bm{a}$ form $\bm{S}$ and $\bm{A}$.
By substituting Eqs.~\eqref{eq:S-expansion} and \eqref{eq:A-expansion} into cumulant expansion Eq.~\eqref{eq:cumulant},  we get
\begin{align}
  \mathcal{F}(t,t_{0})&=\exp \Biggl[\int _{t_{0}}^{t}\!ds\,\biggl(-\tilde{\Xi }(s)+\sum _{j}\tilde{\Phi }_{j}(s)\int _{t_{0}}^{s}\!du\,\tilde{\Theta }_{j}(s,u)\biggr)\Biggr],
\end{align}
where
\begin{subequations}
  \begin{align}
    \tilde{\Xi }(s)&\equiv -S_{\delta }\tilde{\Phi }(s)^{2}
  \end{align}
  is describing the effect of Markovian part of the bath correlation function, and
  \begin{align}
    \tilde{\Phi }_{j}(s)&\equiv \sigma _{j}\tilde{\Phi }(s)\\
    \intertext{and}
    \tilde{\Theta }_{j}(t,s)&\equiv \sum _{k}\left(s_{jk}\phi _{k}(t-s)\tilde{\Phi }(s)-a_{jk}\phi _{k}(t-s)\tilde{\Psi }(s)\right)
  \end{align}
  are describing non--Markovian effects by delay functions $\{\phi _{k}(t)\}$.
\end{subequations}
Because $\bm{s}$ and $\bm{a}$ commute with $\bm{\gamma }$, $\Theta _{k}(t,s)$ satisfies a time evolution equation similar to Eq.~\eqref{eq:basis-evolution} as follows:
\begin{widetext}
  \begin{align}
    \begin{split}
      \partial _{t}\tilde{\Theta }_{j}(t,s)&=-\sum _{k}\sum _{l}\left(s_{jk}\gamma _{kl}\phi _{l}(t-s)\tilde{\Phi }(s)-a_{jk}\gamma _{kl}\phi _{l}(t-s)\tilde{\Psi }(s)\right)\\
      &=-\sum _{k}\gamma _{jk}\sum _{l}\left(s_{kl}\phi _{l}(t-s)\tilde{\Phi }(s)-a_{kl}\phi _{l}(t-s)\tilde{\Psi }(s)\right)\\
      &=-\sum _{k}\gamma _{jk}\tilde{\Theta }_{k}(t,s).
    \end{split}
  \end{align}
\end{widetext}
This result is the key point of our new treatment.

To evaluate the time evolution of the reduced density operator $\hat{\rho }(t)$, we introduce the auxiliary density operators (ADOs) defined by
\begin{align}
  \begin{split}
    \tilde{\rho }_{\bm{n}}(t)&\equiv \mathcal{T}_{+}\prod _{k}\left(-\int _{t_{0}}^{t}\!ds\,\tilde{\Theta }_{k}(t,s)\right)^{n_{k}}\mathcal{F}(t,t_{0})\tilde{\rho }(t_{0}),
    \label{eq:ado}
  \end{split}
\end{align}
where $\bm{n}=(\dots ,n_{k},\dots )$ is a $K$--dimensional multi--index whose components are non--negative integers.
Cleary, $\tilde{\rho }_{\bm{0}}(t)$ equals to the reduced density operator $\tilde{\rho }(t)$, i.e., Eq.~\eqref{eq:reduced-int}.
By calculating first--order time derivative of Eq.~\eqref{eq:ado} with respect to time $t$, we obtain a set of time evolution equations in the interaction picture as
\begin{align}
  \begin{split}
    \partial _{t}\tilde{\rho }_{\bm{n}}(t)&=-\tilde{\Xi }(t)\tilde{\rho }_{\bm{n}}(t)-\sum _{j,k}n_{j}\gamma _{jk}\tilde{\rho }_{\bm{n}-\bm{1}_{j}+\bm{1}_{k}}(t)\\
    &\quad -\sum _{k}\tilde{\Phi }_{k}(t)\tilde{\rho }_{\bm{n}+\bm{1}_{k}}(t)-\sum _{k}n_{k}\tilde{\Theta }_{k}(t,t)\tilde{\rho }_{\bm{n}-\bm{1}_{k}}(t).
  \end{split}
  \label{eq:heom-int}
\end{align}
Here, $\bm{1}_{k}=(0,\dots ,1,0,\dots )$ is the $k$th unit vector.
These can be rewritten in the Schr\"odinger picture as
\begin{align}
  \begin{split}
    \partial _{t}\hat{\rho }_{\bm{n}}(t)&=-(\mathcal{L}+\hat{\Xi })\hat{\rho }_{\bm{n}}(t)-\sum _{j,k}n_{j}\gamma _{jk}\hat{\rho }_{\bm{n}-\bm{1}_{j}+\bm{1}_{k}}(t)\\
    &\quad -\sum _{k}\hat{\Phi }_{k}\hat{\rho }_{\bm{n}+\bm{1}_{k}}(t)-\sum _{k}n_{k}\hat{\Theta }_{k}\hat{\rho }_{\bm{n}-\bm{1}_{k}}(t).
    \label{eq:heom}
  \end{split}
\end{align}
Here,
\begin{subequations}
  \begin{align}
    \mathcal{L}&\equiv -\frac{i}{\hbar }\hat{H}^{\times },\quad \hat{\Xi }\equiv -S_{\delta }\hat{\Phi }^{2},\quad \hat{\Phi }_{k}\equiv \sigma _{k}\hat{\Phi },\\
    \intertext{and}
    \hat{\Theta }_{k}&\equiv \sum _{l}s_{kl}\phi _{l}(0)\hat{\Phi }-\sum _{l}a_{kl}\phi _{l}(0)\hat{\Psi }.
  \end{align}
\end{subequations}
This is our generalization of the HEOM.

\begin{figure}
  \centering
  \includegraphics[scale=\SingleColFigScale]{fig1_hierarchy.pdf}
  \caption{ (a) The hierarchical structure of Eq.~\eqref{eq:heom} in the case of $K=2$ and $\mathcal{N}_{\mathrm{max}}=2$.
    Here, the first term in Eq.~\eqref{eq:heom} is omitted.
    Starting and end points of arrows indicate the left--hand side and right--hand side terms of Eq.~\eqref{eq:heom}.
    Orange, black, and blue arrows represent the second, third, and fourth terms.
    The dashed orange arrows refer to connections caused by off--diagonal elements of $\bm{\gamma }$, which do not exist in the original formulation.
    (b) One of the conventional generalizations, the extended HEOM Eq.~\eqref{eq:eheom}, in the case of $K=2$ and $\mathcal{N}_{\mathrm{max}}=2$.
    Orange, red, black, blue, and green arrows represent the second, third, fourth, fifth, and sixth terms of Eq.~\eqref{eq:eheom}, respectively.
  }
  \label{fig:hierarchy}
\end{figure}
In Fig.~\ref{fig:hierarchy}(a), a schematic structure of connections of ADOs in Eq.~\eqref{eq:heom} is depicted.
The set of equations forms a hierarchical structure, of which top element is the reduced density operator $\hat{\rho }_{\bm{0}}(t)=\hat{\rho }(t)$.
The first term of Eq.~\eqref{eq:heom} represents Markovian dynamics caused by the free propagation of the system and the Markovian part of the bath.
The second term means time evolution of the basis set $\bm{\phi }(t)$, which expresses non--Markovian free propagation of the bath.
The third and fourth terms represent system--environment interactions, which include the effects of fluctuation and dissipation of the bath via $\mathcal{S}(t)$ and $\mathcal{A}(t)$.

Because Eq.~\eqref{eq:heom} consists of an infinite number of differential equations, we need to truncate $\bm{n}$ to carry out numerical calculations.
The method of truncation affects the efficiency of calculations, and many advanced methods are proposed within the original HEOM framework \cite{tanimura1994jpsj, ishizaki2005jpsj, shi2009jcp3, hartle2013prb, hartle2015prb}.
In this article, in order to make the validation of theories simple, we truncate them in accordance with the condition that $\bm{n}$ satisfies the relation $\mathcal{N}\equiv \sum _{k}n_{k}>\mathcal{N}_{\mathrm{max}}$, and we regard the ADOs which satisfy $\mathcal{N}>\mathcal{N}_{\mathrm{max}}$ as zero.
Here, $\mathcal{N}$ is referred to as a tier of an ADO.
In this manner, we need to judge convergence of the numerical results by changing $\mathcal{N}_{\mathrm{max}}$.

In numerical implementations, it is more convenient to define ADOs by $\hat{\rho }'_{\bm{n}}=\hat{\rho }_{\bm{n}}/\prod _{k}\sqrt {\mathstrut n_{k}!}$ to suppress divergence of ADOs in deep tiers $\mathcal{N}\gg 1$ in the case of strong system--bath coupling \cite{shi2009jcp3}.
Then the equations of motion are rewritten as
\begin{align}
  \begin{split}
    \partial _{t}\hat{\rho }'_{\bm{n}}(t)&=-(\mathcal{L}+\hat{\Xi })\hat{\rho }_{\bm{n}}'(t)-\sum _{k}n_{j}\gamma _{kk}\hat{\rho }_{\bm{n}}'(t)\\
    &\quad -\sum _{j\neq k}\sqrt {n_{j}}\sqrt {\mathstrut n_{k}+1}\gamma _{jk}\hat{\rho }_{\bm{n}-\bm{1}_{j}+\bm{1}_{k}}'(t)\\
    &\quad -\sum _{k}\sqrt {\mathstrut n_{k}+1}\hat{\Phi }_{k}\hat{\rho }_{\bm{n}+\bm{1}_{k}}'(t)-\sum _{k}\sqrt {\mathstrut n_{k}}\hat{\Theta }_{k}\hat{\rho }_{\bm{n}-\bm{1}_{k}}'(t).
    \label{eq:heom-prime}
  \end{split}
\end{align}

\subsection{Relation to conventional theories}
In this section, we show the relation among our new generalization, the original theory, and extended HEOM.
The relation among our approach and some other generalizations of HEOM is given in Appendix~\ref{eq:rel-yheom}.

\subsubsection{Reduction to the original HEOM}
\label{sec:rel-oheom}
When the coefficient matrix $\bm{\gamma }$ has only diagonal elements (i.e., $\gamma _{kk}\equiv \gamma _{k}$), Eq.~\eqref{eq:basis-evolution} is solved as
\begin{align}
  \phi _{k}(t)=e^{-\gamma _{k}t} &&(t\geq 0).
\end{align}
Here, we have set $\bm{\phi }(0)$ as $\phi _{k}(0)=1$.
In this case, $\bm{s}$ and $\bm{a}$, which commute with $\bm{\gamma }$, should also have only diagonal elements (i.e., $s_{kk}\equiv s_{k}$ and $a_{kk}\equiv a_{k}$), and $\mathcal{S}(t)$ and $\mathcal{A}(t)$ can be rewritten as $\mathcal{S}(t)=\sum _{k}s_{k}\cdot e^{-\gamma _{k}\left|t\right|}+S_{\delta }\cdot 2\delta (t)$ and $\mathcal{A}(t)=\sum _{k}a_{k}\cdot e^{-\gamma _{k}\left|t\right|}$ (We have fixed the parameter $\bm{\sigma }$ as $\sigma _{k}=1$).
As a result, Eq.~\eqref{eq:heom} reduces to
\begin{align}~
  \begin{split}
    \partial _{t}\hat{\rho }_{\bm{n}}(t)&=-\left(\mathcal{L}+\hat{\Xi }+\sum _{k}n_{k}\gamma _{k}\right)\hat{\rho }_{\bm{n}}(t)\\
    &\quad -\sum _{k}\hat{\Phi }\hat{\rho }_{\bm{n}+\bm{1}_{k}}(t)-\sum _{k}n_{k}\hat{\Theta }_{k}\hat{\rho }_{\bm{n}-\bm{1}_{k}}(t).
    \label{eq:oheom}
  \end{split}
\end{align}
Here,
\begin{align}
  \hat{\Theta }_{k}&\equiv s_{k}\hat{\Phi }-a_{k}\hat{\Psi }.
\end{align}
This is the original HEOM.
Thus, the original HEOM can be regarded as a special case of our generalization.

\subsubsection{Relation to the extended HEOM}
\label{sec:rel-eheom}
In the extended HEOM approach, we need to duplicate the basis $\bm{\phi }(t)$ as ${}^{t}\bm{\phi }'(t)=({}^{t}\bm{\phi }(t), {}^{t}\bm{\phi }(t))$.
This satisfies time evolution
\begin{align}
  \partial _{t}
  \bm{\phi }'(t)
  &=
  -\bm{\gamma }'\bm{\phi }'(t)
  \equiv
  -\begin{pmatrix}
  \bm{\gamma }&\bm{0}\\
  \bm{0}&\bm{\gamma }\\
  \end{pmatrix}
  \begin{pmatrix}
    \bm{\phi }(t)\\
    \bm{\phi }(t)\\
  \end{pmatrix}.
  \label{eq:gamma_prime}
\end{align}
Expansion of correlation functions $\mathcal{S}(t)$ and $\mathcal{A}(t)$ for the extended HEOM can be written in the form of Eqs.~\eqref{eq:S-expansion} and \eqref{eq:A-expansion} as
\begin{subequations}
  \begin{align}
    &\begin{aligned}
       \mathcal{S}(t)&=
       {}^{t}\bm{\sigma }'\bm{s}'\bm{\phi }'(t)\equiv
       \begin{pmatrix}
         {}^{t}\bm{\sigma }_{\mathcal{S}}&{}^{t}\bm{\sigma }_{\mathcal{A}}\\
       \end{pmatrix}
       \begin{pmatrix}
         \bm{1}&\bm{0}\\
         \bm{0}&\bm{0}\\
       \end{pmatrix}
       \begin{pmatrix}
         \bm{\phi }(t)\\
         \bm{\phi }(t)\\
       \end{pmatrix}\\
       &=\sum _{k}\sigma _{\mathcal{S},k}\phi _{k}(t)
     \end{aligned}
    \intertext{and}
    &\begin{aligned}
       \mathcal{A}(t)&=
       {}^{t}\bm{\sigma }'\bm{a}'\bm{\phi }'(t)\equiv
      \begin{pmatrix}
        {}^{t}\bm{\sigma }_{\mathcal{S}}&{}^{t}\bm{\sigma }_{\mathcal{A}}\\
      \end{pmatrix}
      \begin{pmatrix}
        \bm{0}&\bm{0}\\
        \bm{0}&\bm{1}\\
      \end{pmatrix}
      \begin{pmatrix}
        \bm{\phi }(t)\\
        \bm{\phi }(t)\\
      \end{pmatrix}\\
      &=\sum _{k}\sigma _{\mathcal{A},k}\phi _{k}(t)
    \end{aligned}
  \end{align}
\end{subequations}
Here, $\bm{0}$ and $\bm{1}$ are the $K\times K$ zero and identity matrices, respectively, and we have assumed that $S_{\delta }=0$.
Obviously, $\sigma _{\mathcal{S},k}=S_{k}$ and $\sigma _{\mathcal{A},k}=A_{k}$, and $\bm{a}'$ and $\bm{s}'$ commute with $\bm{\gamma }'$.
Then the number of the indices of the ADOs are also doubled as $\bm{n}=(\bm{l},\bm{m})$, and we get the corresponding HEOM from Eq.~\eqref{eq:heom} as
\begin{widetext}
  \begin{align}
    \begin{split}
      \partial _{t}\hat{\rho }_{\bm{l},\bm{m}}(t)&=-\mathcal{L}\hat{\rho }_{\bm{l},\bm{m}}(t)\\
      &\quad -\sum _{j,k}\gamma _{jk}l_{j}\hat{\rho }_{\bm{l}-\bm{1}_{j}+\bm{1}_{k},\bm{m}}(t)-\sum _{j,k}\gamma _{jk}m_{j}\hat{\rho }_{\bm{l},\bm{m}-\bm{1}_{j}+\bm{1}_{k}}(t)\\
      &\quad -\sum _{k}\hat{\Phi }\left(\sigma _{\mathcal{S},k}\hat{\rho }_{\bm{l}+\bm{1}_{k},\bm{m}}(t)+\sigma _{\mathcal{A},k}\hat{\rho }_{\bm{l},\bm{m}+\bm{1}_{k}}(t)\right)\\
      &\quad -\sum _{k}\phi _{k}(0)l_{k}\hat{\Phi }\hat{\rho }_{\bm{l}-\bm{1}_{k},\bm{m}}(t)+\sum _{k}\phi _{k}(0)m_{k}\hat{\Psi }\hat{\rho }_{\bm{l},\bm{m}-\bm{1}_{k}}(t).
      \label{eq:eheom}
    \end{split}
  \end{align}
\end{widetext}
This is equivalent to the extended HEOM.
As shown here, the extended HEOM can be regarded as a subset of our new treatment.

In Fig.~\ref{fig:hierarchy}(b), a schematic structure of connections of ADOs in Eq.~\eqref{eq:eheom} is depicted within the same condition of Fig.~\ref{fig:hierarchy}(a).
In comparison with the extended HEOM, our new approach displays a simple hierarchical structure without doubling the number of indices of the hierarchy, and the simple structure has huge theoretical and numerical advantages, especially when $\mathcal{K}$ and $\mathcal{N}_{\mathrm{max}}$ are large.

\section{NUMERICAL RESULTS}
\label{sec:results}
Our generalization Eq.~\eqref{eq:heom} works for arbitrary spectral densities at arbitrary temperature as far as accurate expansions bath correlation functions Eqs.~\eqref{eq:S-expansion} and \eqref{eq:A-expansion} are given and the required ADOs converge in a finite number.
In this section, we demonstrate our new treatment for non--exponential correlation functions by using three examples.
Hereafter, we employ the dimensionless units $\hbar =1$ and $k_{\mathrm{B}}=1$ for simplicity.
Numerical calculations were carried out to integrate time differential equations by using the fourth--order low--storage Runge--Kutta (LSRK4) method \cite{williamson1980jcp}.
The time step for the LSRK4 method was set to $\delta t=10^{-2}$.

\subsection{Super--Ohmic spectral density with Bessel function basis}
\label{sec:example1}
\begin{figure}
  \centering
  \includegraphics[scale=\SingleColFigScale]{fig2_semicircle.pdf}
  \caption{ (a) Donor--acceptor model system of an exciton/electron transfer problem in this article.
    Gray, orange, and blue curves represent free energy surfaces of $|g\rangle $, $|D\rangle $, and $|A\rangle $, respectively.
    Here, $E_{\mathrm{D}}-\mathrm{E}_{A}=1$, and in the cases of $\lambda =0.01$, $0.1$, $1$ are depicted.
    (b) Super--Ohmic spectral density model with a semicircle cutoff function, Eq.~\eqref{eq:semicircle}.
    Here, $\lambda =1$ and $\gamma _{\mathrm{c}}=1.5$.
    (c) The symmetrized correlation function of the spectral density Eq.~\eqref{eq:semicircle}.
    Blue triangles represent results of the expansion Eq.~\eqref{eq:S-bessel} with the truncated basis set $\phi (t)$.
    Here, the parameters of the spectral density are the same as (b), and the other parameters are $T=1$ and $K=16$.
  }
  \label{fig:semicircle}
\end{figure}
In this section, we consider exciton/electron transfer problems in donor--acceptor system with a super--Ohmic spectral density.
We consider a three--level system which has ground state $|g\rangle $, reactant state of the transfer $|D\rangle $ (donor state), and product state of the transfer $|A\rangle $ (acceptor state).
In the case of an electron transfer problem, the states are, e.g., $\mathrm{D}\mathrm{A}$, $\mathrm{D}^{\ast }\mathrm{A}$, and $\mathrm{D}^{+}\mathrm{A}^{-}$, respectively.
Here, $\mathrm{D}$ is a donor molecule and $\mathrm{A}$ is an acceptor molecule.
Typically, degrees of freedom of reorganization after the transitions $|g\rangle \rightarrow |D\rangle $ and $|D\rangle \rightarrow |A\rangle $ are different, and therefore we need to consider two sets of phonon degrees of freedom to describe them.
In this article, to make our demonstration simple, we assume that the transition $|g\rangle \rightarrow |D\rangle $ does not cause reorganization of the molecule (i.e., the stable points on $|g\rangle $ and $|D\rangle $ are the same) as depicted in Fig.~\ref{fig:semicircle}(a).
Then, we need only a single set of phonon degrees of freedom, and the total Hamiltonian can be expressed as \cite{yang2002cp}
\begin{align}
  \begin{split}
    \hat{H}^{\mathrm{tot}}&=E_{g}|g\rangle \langle g|+E_{D}|D\rangle \langle D|+E_{A}|A\rangle \langle A|\\
    &\quad +J(|D\rangle \langle A|+|A\rangle \langle D|)\\
    &\quad +\left(|g\rangle \langle g|+|D\rangle \langle D|\right)\sum _{\xi }\frac{\hbar \omega _{\xi }}{2}\left(\hat{p}_{\xi }^{2}+\hat{x}_{\xi }^{2}\right)\\
    &\quad +|A\rangle \langle A|\sum _{\xi }\frac{\hbar \omega _{\xi }}{2}\left[\hat{p}_{\xi }^{2}+\left(\hat{x}_{\xi }-\frac{g_{\xi }}{\hbar \omega _{\xi }}\right)^{2}\right].
    \label{eq:hamiltonian-et}
  \end{split}
\end{align}
Here, $E_{g}$, $E_{D}$, and $E_{A}$ are stable equilibrium energies of each state, and $J$ is the electronic coupling between donor and acceptor states.
Spontaneous transitions between $|g\rangle $ and $\{|D\rangle , |A\rangle \}$ are not considered.
The displacement $g_{\xi }/\hbar \omega _{\xi }$ represents the difference of the stable points between $|D\rangle $ and $|A\rangle $ in the $\xi $th phonon degree of freedom, and total reorganization energy after $|D\rangle \rightarrow |A\rangle $ transition can be written as $\lambda =g_{\xi }^{2}/2\hbar \omega _{\xi }$.
As far as we consider transfer problem after excitation $|g\rangle \rightarrow |D\rangle $, we can ristrict the space of the problem in $\{|D\rangle , |A\rangle \}$, and then Eq.~\eqref{eq:hamiltonian-et} can be rewritten in the form of Eq.~\eqref{eq:total-hamiltonian} with $\hat{H}=E_{\mathrm{D}}|D\rangle \langle D|+(E_{A}+\lambda )|A\rangle \langle A|+J(|D\rangle \langle A|+|A\rangle \langle D|)$ and $\hat{V}=|A\rangle \langle A|$.
Here, we adopt a super--Ohmic spectral density model with a semicircle cutoff function,
\begin{align}
  \mathcal{J}(\omega )=
  \begin{cases}
    \displaystyle \frac{16\lambda }{\gamma _{\mathrm{c}}^{3}}\omega ^{3}\sqrt {1-\omega ^{2}/\gamma _{\mathrm{c}}^{2}}&(\left|\omega \right|\leq \gamma _{\mathrm{c}})\\
    0,&(\left|\omega \right|>\gamma _{\mathrm{c}})
  \end{cases}
  \label{eq:semicircle}
\end{align}
which is proposed to describe electron transfer in bacterial photosynthesis \cite{ando1998jpcb}.
Here, $\gamma _{\mathrm{c}}$ is the cutoff frequency of the phonon.
In this spectral density model, the density has a rigid cutoff at $\omega =\gamma _{\mathrm{c}}$ as depicted in Fig.~\ref{fig:semicircle}(b), and this describes the upper--limit of the frequencies in a molecule--environment system, which should have a finite value.
In the case of this spectral density model, the correlation functions Eqs.~\eqref{eq:S} and \eqref{eq:A} are analytically evaluated in the forms of linear combinations the Bessel functions of the first kind $J_{k}(x)$ as
\begin{subequations}
  \begin{align}
    \mathcal{S}(t)&=\sum _{k=0}S_{k}\cdot J_{k}(\gamma _{\mathrm{c}}t)
    \label{eq:S-bessel}
    \intertext{and}
    \mathcal{A}(t)&=\sum _{k=0}A_{k}\cdot J_{k}(\gamma _{\mathrm{c}}t)
    \label{eq:A-bessel}.
  \end{align}
\end{subequations}
For details of the coefficients, see Appendix \ref{sec:decomposition-superohmic}.
Thus, we can employ the Bessel functions as the basis functions for HEOM, $\phi _{k}(t)=J_{k}(\gamma _{\mathrm{c}}t)$, which satisfy time evolution equations
\begin{subequations}
  \begin{align}
    \partial _{t}\phi _{0}(t)&=+\gamma _{\mathrm{c}}\phi _{1}(t)\\
    \intertext{and}
    \partial _{t}\phi _{k}(t)&=-\gamma _{\mathrm{c}}\phi _{k-1}(t)/2+\gamma _{\mathrm{c}}\phi _{k+1}(t)/2&(k\geq 1).
    \label{eq:bessel-basis}
  \end{align}
\end{subequations}
These are an infinite number of simultaneous equations, so we need to truncate them in a finite number $K$.
Although this truncation introduces an approximation in HEOM calculations, we can test the accuracy of the results by changing $K$  and we can obtain solution as accurate as we need.
In the calculations in this section, we simply ignore $\phi _{K}(t)$.
Note that, this truncation also makes long time behavior of the basis functions unstable.
Therefore we need to increase $K$ when we want to increase the simulation times of our calculations.

We set the parameters as $E_{\mathrm{D}}-E_{\mathrm{A}}=1$, $J=0.5$, $\gamma _{\mathrm{c}}=1.5$, and $T=1$.
The truncation number of Eq.~\eqref{eq:bessel-basis} was chosen to be $K=15$, which sufficiently reproduces analytical correlation functions in our simulation time $0\leq t\leq 15$ as depicted in Fig.~\ref{fig:semicircle}(c).
Because $\bm{\gamma }$ from Eq.~\eqref{eq:bessel-basis} is diagonalizable, we constructed $\bm{\sigma }$, $\bm{s}$, and $\bm{a}$ by using the method shown in Appendix \ref{sec:commuting-s-a-1}

\begin{figure}
  \centering
  \includegraphics[scale=\SingleColFigScale]{fig3_superohmic.pdf}
  \caption{ Donor population dynamics after excitation at $t=0$ in the case of (a) weak coupling (reorganization energy) $\lambda =0.01$, (b) moderate coupling $\lambda =0.1$, and (c) strong coupling $\lambda =1$.
    Orange curves, blue triangles, and black dots represent results of new treatment (Eq.~\eqref{eq:heom}), extended HEOM (Eq.~\eqref{eq:eheom}), and original HEOM (Eq.~\eqref{eq:oheom}) with diagonalized $\bm{\gamma }$.
    Gray curves represent results of the Redfield equation without the secular approximation. }
  \label{fig:superohmic}
\end{figure}
In Fig.~\ref{fig:superohmic}, donor populations as a function of time $t$ are depicted for (a) weak, moderate, and strong coupling cases ($\lambda =0.01$, $0.1$, and $1$, respectively).
We performed simulations by using our new treatment (Eq.~\eqref{eq:heom}), the extended HEOM, and Redfield equations \cite{redfield1965inbook, yang2002cp, ishizaki2009jcp2} for comparison.
The truncation tiers of HEOM were chosen as $\mathcal{N}_{\mathrm{max}}=3$ for weak and moderate couplings and $\mathcal{N}_{\mathrm{max}}=8$ for strong coupling.
Under this truncation, the numbers of ADOs for Eq.~\eqref{eq:heom} are $968$ and $735,470$, and those of the extended HEOM \eqref{eq:eheom} are $6,544$ and $76,904,684$.
Because the extended HEOM calculation with $\mathcal{N}_{\mathrm{max}}=8$ was numerically high--cost, we did not perform it.
As shown, our new treatment completely reproduced the results of the extended HEOM, while computational costs are much smaller.
In Fig.~\ref{fig:superohmic}(a), the results of HEOM theories and Redfield theory are with almost coincident, because the Born and Markov approximations in the Redfield theory is valid when the coupling sufficiently small.
On the other hand, in Figs.~\ref{fig:semicircle}(b) and \ref{fig:superohmic}(c), the results of Redfield theory are different from the other calculations.
This indicates that the results of our new treatment are equivalent to that of the extended HEOM beyond a perturbative Markovian regime.

As shown here, our new treatment can describe non--exponential behavior of correlation functions.
However, the coefficient matrix $\bm{\gamma }$ in this section is diagonalizable as mentioned above, and it is possible to construct an equivalent original HEOM by diagonalization (black dots in Fig.~\ref{fig:superohmic}), which is more efficient than the calculations with non--diagonalized $\bm{\gamma }$.
Generally, as far as $\bm{\gamma }$ is diagonalizable, the original framework of the HEOM with diagonalized $\bm{\gamma }$ works well, and there is no need to use the other generalizations.
The generalizations for non--exponential correlation functions make a difference when $\bm{\gamma }$ is non--diagonalizable.
In the following two sections, we show examples with non--diagonalizable $\bm{\gamma }$.

\subsection{Critically damped Brownian oscillator}
\label{sec:example2}
In this section, we show exciton/electron transfer models coupled to damped vibrational degrees of freedom, expressed by using a Brownian spectral density \cite{tanimura1994jpsj, tanaka2009jpsj}.
The model Hamiltonian and parameters are the same as those in Sec.~\ref{sec:example1}, while we employ a Brownian spectral density
\begin{align}
  \mathcal{J}(\omega )&=2\lambda \frac{\zeta \omega _{0}^{2}\omega }{(\omega ^{2}-\omega _{0}^{2})^{2}+\zeta ^{2}\omega ^{2}}
  \label{eq:brownian}
\end{align}
instead of Eq.~\eqref{eq:semicircle}.
This spectral density represents a Brownian motion of a harmonic oscillator with frequency $\omega _{0}$ and friction constant $\zeta $.
The conditions $\zeta <2\omega _{0}$, $\zeta =2\omega _{0}$, and $\zeta >2\omega _{0}$ correspond to the underdamped, critically damped, and overdamped cases of the oscillator, respectively.
When the system $\hat{H}$ is coupled to a primary harmonic mode and the mode is coupled to another Ohmic bath, the effective spectral density the mode engenders is analytically reduced to Eq.~\eqref{eq:brownian} \cite{garg1985jcp}, and this can be regarded as a simplest model of vibronic phenomena in a dissipative environment.

The anti--symmetrized correlation function Eq.~\eqref{eq:A} is evaluated as
\begin{align}
  \mathcal{A}(t)&=\frac{\lambda \omega _{0}^{2}i}{2\omega _{1}}\left(e^{-\gamma _{+}\left|t\right|}-e^{-\gamma _{-}\left|t\right|}\right),
\end{align}
where $\omega _{1}\equiv \sqrt {\mathstrut \omega _{0}^{2}-\zeta ^{2}/4}$ and $\gamma _{\pm }\equiv \zeta /2\mp i\omega _{1}$, and typicall we employ $\phi _{+}(t)\equiv e^{-\gamma _{+}\left|t\right|}$ and $\phi _{-}(t)\equiv e^{-\gamma _{-}\left|t\right|}$ as basis functions $\bm{\phi }(t)$ (Regarding $\mathcal{S}(t)$, see Appendix \ref{sec:decomposition-brownian1}).
However, this basis set is degenerated in the critical--damped case, $\zeta =2\omega _{0}$, and this cannot express the non--exponential behavior of a critical--damped oscillator,
\begin{align}
  \mathcal{A}(t)&=-\lambda \omega _{0}^{2}\cdot te^{-(\zeta /2)\left|t\right|}.
\end{align}
Instead, we employ the basis
\begin{subequations}
  \begin{align}
    \phi _{p}(t)&=-\frac{\omega _{0}}{\omega _{1}}\sin (\omega _{1}\left|t\right|)e^{-\zeta \left|t\right|/2}\\
    \intertext{and}
    \phi _{q}(t)&=\left(\frac{\zeta }{2\omega _{1}}\sin (\omega _{1}\left|t\right|)+\cos (\omega _{1}\left|t\right|)\right)e^{-\zeta \left|t\right|/2},
  \end{align}
\end{subequations}
which satisfy $\phi _{p}(0)=0$, $\phi _{q}(0)=1$, and
\begin{align}
  \partial _{t}
  \begin{pmatrix}
    \phi _{p}(t)\\
    \phi _{q}(t)\\
  \end{pmatrix}
  &=-
  \begin{pmatrix}
    \zeta &\omega _{0}\\
    -\omega _{0}&0\\
  \end{pmatrix}
  \begin{pmatrix}
    \phi _{p}(t)\\
    \phi _{q}(t)\\
  \end{pmatrix}.
  \label{eq:brownian-basis}
\end{align}
Then $\mathcal{A}(t)$ is expressed as $\mathcal{A}(t)=\lambda \omega _{0}\phi _{p}(t)$.
When $\zeta =2\omega _{0}$, the coefficient matrix of Eq.~\eqref{eq:brownian-basis} is non--diagonalizable, while it is diagonalizable otherwise.
Note that, Eq.~\eqref{eq:brownian-basis} is equivalent to the phase space equations of motion of a damped harmonic oscillator with coordinate $q(t)$ and momentum $p(t)$,
\begin{subequations}
  \begin{align}
    p(t)&=-\zeta p(t)-\omega _{0}q(t)\\
    \intertext{and}
    q(t)&=+\omega _{0}p(t).
  \end{align}
\end{subequations}
This representation is related to phase--space Fokker--Planck equations \cite{tanimura2015jcp, ikeda2019jctc}.

\begin{figure}
  \centering
  \includegraphics[scale=\SingleColFigScale]{fig4_critically_damped.pdf}
  \caption{
    Donor population dynamics after excitation at $t=0$ in the case of (a) underdamped ($\zeta =0.1$), overdamped ($\zeta =2$), and (b) near critically damped cases $\zeta =0.999, 1, 1.001$.
    Orange, blue, and gray curves represent results of new treatment (Eq.~\eqref{eq:heom}), while orange and blue triangles represent results of the original HEOM (Eq.~\eqref{eq:oheom}).
    Here, $\lambda =2$, $\omega _{0}=0.5$, and $T=1$.
  }
  \label{fig:critically_damped}
\end{figure}
In Fig.~\ref{fig:critically_damped}, we show the numerical results in the cases of underdamped, critically damped, overdamped regions with our generalization Eq.~\eqref{eq:heom} and original HEOM Eq.~\eqref{eq:oheom}.
Here, we employed the Pad\`{e} spectral decomposition (PSD) $[N-1/N]$ scheme to express the Bose--Einstein distribution in $\mathcal{S}(t)$ \cite{hu2011jcp}, and adopted a single pole ($N=1$).
This causes an additional exponential function basis in $\mathcal{S}(t)$, and therefore $K=3$ including $\phi _{p}(t)$ and $\phi _{q}(t)$ (or $\phi _{+}(t)$ and $\phi _{-}(t)$).
The truncation tiers of the HEOM were chosen as $\mathcal{N}_{\mathrm{max}}=20$, $15$, and $25$ for $\lambda =0.01$, $0.1$ and $1$, respectively, which cause $1,770$, $815$, and $3,275$ ADOs.
For details, see Appendix \ref{sec:decomposition-brownian1}.

In the overdamped and underdamped regions, our generalization and original HEOM showed consistent results.
However, near critical--damped region, the original HEOM became very unstable, and was unsolvable with a finite $\mathcal{N}$.
This is because, the basis set used in the original HEOM, $\phi _{+}(t)$ and $\phi _{-}(t)$, are almost denegerate near the critically damped condition.
$\mathcal{A}(t)$ is described as a difference between $\phi _{+}(t)$ and $\phi _{-}(t)$, it causes numerical errors from loss of significance.
On the other hand, our generalization worked well in all regions because our method does not depend on diagonalization of the basis functions and therefore the stability is not related to degeneration.

As shown here, our new treatment can describe the dynamics caused by non--exponential correlation functions, which has a non--diagonalizable coefficient matrix $\bm{\gamma }$.
Moreover, even if the matrix $\bm{\gamma }$ is diagonalizable, the original framework can be unstable like the case of nearly critically damping regime.
Thus, our new treatment enables us to make more stable analysis for non--perturbative and non--Markovian regime.
Note that, in our treatment, the choice of basis set is not unique while the choice relates the stability of the equations.

\subsection{Spontaneous de--excitation under zero--temperature environment}
\label{sec:example3}
Finally, we discuss a spontaneous de--excitation process of a two--level system $\{|g\rangle , |e\rangle \}$ caused by zero--temperature bosonic environment.
Here, we assume that the Hamiltonian of the system and the system--bath interaction operator are expressed as $\hat{H}=\hbar \Omega _{e}\hat{a}^{+}\hat{a}^{-}$ and $\hat{V}=\hat{a}^{+}+\hat{a}^{-}$, respectively, where $\hat{a}^{+}=|e\rangle \langle g|$ and $\hat{a}^{-}\equiv |g\rangle \langle e|$ are creation/annihilation operators of the system.
As the environment model, we employ a Brownian spectral density
\begin{align}
  \mathcal{J}(\omega )&=2\alpha \frac{\zeta \omega _{0}^{2}\omega }{(\omega _{0}^{2}-\omega ^{2})^{2}+\zeta ^{2}\omega ^{2}},
  \label{eq:brownian-alpha}
\end{align}
which has a sharp peak at $\omega =\omega _{1}=\sqrt {\mathstrut \omega _{0}^{2}-\zeta ^{2}/4}$ in the underdamped condition $\zeta <2\omega _{0}$.
To make the system and bath characteristic frequencies resonant, we set $\Omega _{e}=\omega _{1}$.
Note that, although the coupling strength $\alpha $ has the same form as the reorganization energy $\lambda $ in Sec.~\ref{sec:example2}, physical meaning of the constants are quite different because of the difference of $\hat{V}$:
The interaction $\hat{V}$ here has no diagonal elements, and does not cause reorganization of the quantum states $|g\rangle $ and $|e\rangle $.

We set the initial state of the system as the pure excited state $\hat{\rho }(t_{0})=|e\rangle \langle e|$, and calculate population dynamics connected to zero--temperature bath $T=0$.
In this model, transitions among the two states are purely caused by fluctuation of the bath coordinates.
Because there is no thermal fluctuation in the zero--temperature limit, the spontaneous transition is caused by quantum fluctuations of the environment.

To apply the HEOM, we need to express $\mathcal{C}(t)$ in a basis set $\bm{\phi }(t)$.
It is difficult to treat $\mathcal{C}(t)$ in the case of low temperatures because we need to incorporate many bosonic Matsubara frequencies $2\pi k/\beta \hbar $ near a system frequency, which come from the Bose--Einstein distribution and provide quantum low--temperature corrections.
In the case of the zero--temperature, the bosonic Matsubara frequencies are almost degenerate and form a continuum.
There are many investigations to propose improved treatment for low--temperature situations and to eliminate unphysical artifacts caused by insufficient low--temperature corrections \cite{ishizaki2005jpsj, hu2011jcp, duan2017prb, dijkstra2017jcp, erpenbeck2018jcp, lambert2019nc, ishizaki2019jpsj}.
Here, we adopt the Fano spectral decomposition (FSD) technique recently made by Yan and co--workers \cite{cui2019jcp, zhang2020jcp}, which enables us to describe the low--temperature effects accurately with a few number of basis functions $\bm{\phi }(t)$.
This method divides the Bose--Einstein distribution into a high--temperature part with a reference temperature $T_{0}$ and a low--temperature part as $n_{\mathrm{BE}}(\omega ,T)=n_{\mathrm{BE}}(\omega ,T_{0})+\Delta n_{\mathrm{BE}}(\omega ,T,T_{0})$.
The high--temperature part is evaluated by using the conventional PSD framework, and the low--temperature part is accurately expressed by using summation of modified Fano functions.
The modified Fano functions result in basis functions in the forms of $t^{l}\cdot e^{-\gamma t}$ ($l\geq 0$), and $\bm{\gamma }$ caused by these functions are non--diagonalizable.
Hence, we cannot employ the original HEOM Eq.~\eqref{eq:oheom} and need generalizations for non--exponential functions.
In this article, we employ our new treatment Eq.~\eqref{eq:heom} to describe the dynamics under zero--temperature environment.
For details, see Appendix \ref{sec:decomposition-brownian2}.
Note that, it is impossible to perform the strict zero--temperature parametrization $T=0$ within the FSD framework.
However, if the temperature parameter we use is sufficiently small, $\mathcal{C}(t)$ asymptotically arrives the analytical solution of $T=0$ as discussed below.

To discuss the effects of the rigorous description based on our treatment, we introduce two approximation for comparison:

\paragraph{Rotating wave approximation}
In the zero--temperature limit, all of the bath modes are in their ground state when it is not connected to the system, and the factorized initial condition Eq.~\eqref{eq:fact-init} can be rewritten in the form of a wavefunction of the total system as
\begin{align}
  |\Psi ^{\mathrm{tot}}(t_{0})\rangle &=|e\rangle \otimes |\vec{0}\rangle .
  \label{eq:init-wf}
\end{align}
Here, $|\vec{0}\rangle $ represents the vaccum state of the bath.
By using creation and annihilation operators of the $\xi $th bath mode, $\hat{b}_{\xi }^{+}$ and $\hat{b}_{\xi }^{-}$, the interaction Hamiltonian of the total system can be rewritten as
\begin{align}
  \hat{H}^{\mathrm{int}}&=-\sum _{\xi }\frac{g_{\xi }}{\sqrt {\mathstrut 2}}\left(\hat{a}^{+}+\hat{a}^{-}\right)\left(\hat{b}_{\xi }^{+}+\hat{b}_{\xi }^{-}\right),
\end{align}
which includes resonant transitions $\hat{a}^{-}\hat{b}_{\xi }^{+}$ and $\hat{a}^{+}\hat{b}_{\xi }^{-}$ and non--resonant transitions $\hat{a}^{+}\hat{b}_{\xi }^{+}$ and $\hat{a}^{-}\hat{b}_{\xi }^{-}$.
By neglecting non--resonant terms, we obtain
\begin{align}
  \hat{H}^{\mathrm{int}}_{\mathrm{RWA}}&=-\sum _{\xi }\frac{g_{\xi }}{\sqrt {\mathstrut 2}}\left(\hat{a}^{-}\hat{b}_{\xi }^{+}+\hat{a}^{+}\hat{b}_{\xi }^{-}\right),
\end{align}
which is the well--known rotating wave approximation (RWA) form.

In this approximation, the number of excitations of the total wavefunction $|\Psi ^{\mathrm{tot}}(t)\rangle $ is preserved in the time evolution, and in the case of Eq.~\eqref{eq:init-wf}, the wavefunction is restricted in the single Fock state space as
\begin{align}
  |\Psi ^{\mathrm{tot}}(t)\rangle =c_{e,0}(t)|e\rangle \otimes |\vec{0}\rangle +\sum _{\xi }c_{g,\xi }(t)|g\rangle \otimes |\vec{1}_{\xi }\rangle .
  \label{eq:total-wavefunction}
\end{align}
Here, $|\vec{1}_{\xi }\rangle $ represents a state in which the $\xi $th mode is singly excited.
The dynamics of the coefficient $c_{e,0}(t)$ in the interaction picture,
\begin{align}
  \tilde{c}_{e,0}(t)&\equiv e^{+i\left(\hbar \omega _{0}+E_{\mathrm{vac}}\right)(t-t_{0})/\hbar }c_{e,0}(t),
\end{align}
is determined by a Volterra--type integro--differential equation (see Appendix \ref{sec:volterra})
\begin{align}
  \frac{\partial }{\partial t}\tilde{c}_{e,0}(t)&=-\frac{1}{\hbar ^{2}}\int _{t_{0}}^{t}\!ds\,\mathcal{C}(t-s)e^{+i\Omega _{e}(t-s)}\tilde{c}_{e,0}(s),
  \label{eq:volterra}
\end{align}
in which $\mathcal{C}(t)$ acts as a memory kernel.
Here, $E_{\mathrm{vac}}=\sum _{\xi }\hbar \omega _{\xi }/2$ is the summation of the zero--point energies of the bath.
This is equivalent to theories used in quantum electrodynamics \cite{dung2000pra, wang2019jcp}.
The population dynamics of the excited state can be calculated as $P_{e}(t)=\left|\tilde{c}_{0}(t)\right|^{2}$.

While it is possible to solve Eq.~\eqref{eq:volterra} directly, we can construct simpler time differential equations to solve the same problem as follows:
By substituting Eqs.~\eqref{eq:S-expansion} and \eqref{eq:A-expansion} into the Eq.~\eqref{eq:volterra},
\begin{widetext}
  \begin{align}
    \partial _{t}\tilde{c}_{e,0}(t)&=-\frac{1}{\hbar ^{2}}S_{\delta }\tilde{c}_{e,0}(s)-\frac{1}{\hbar ^{2}}\int _{t_{0}}^{t}\!ds\,(\vec{\sigma }^{t}(\bm{s}+i\bm{a})\vec{\phi }(t-s))e^{+i\omega _{0}(t-s)}\tilde{c}_{e,0}(s)\notag\\
    &=-\frac{1}{\hbar ^{2}}S_{\delta }\tilde{c}_{e,0}(t)-\frac{i}{\hbar }\sum _{k}\sigma _{k}\tilde{d}_{k}(t).
    \label{eq:diff-coeff-c}
  \end{align}
  Here, we have introduced auxiliary coefficients defined as
  \begin{align}
    \tilde{d}_{k}(t)&\equiv -\frac{i}{\hbar }\int _{t_{0}}^{t}\!ds\,\sum _{l}(s_{kl}+ia_{kl})\phi _{l}(t-s)e^{+i\Omega _{e}(t-s)}\tilde{c}_{e,0}(s).
  \end{align}
  The time evolution equation of $\tilde{d}_{j}(t)$ is evaluated as
  \begin{align}
    \partial _{t}\tilde{d}_{k}(t)&=+i\Omega _{e}\tilde{d}_{k}(t)-\sum _{l}\gamma _{kl}\tilde{d}_{k}(t)-\left[\sum _{l}\left(\frac{i}{\hbar }s_{kl}-\frac{1}{\hbar }a_{kl}\right)\phi _{l}(0)\right]\tilde{c}_{e,0}(t).
    \label{eq:diff-coeff-d}
  \end{align}
\end{widetext}
Hence, the set of equations Eqs.~\eqref{eq:diff-coeff-c} and \eqref{eq:diff-coeff-d} are equivalent to Eq.~\eqref{eq:volterra}.
This can be regarded a generalization of a method to solve a Volterra--type integro--differential equation by using a second--order differential equation in Ref \onlinecite{wang2019jcp}, and a similar method is found in Ref.~\onlinecite{elattari2000pra} in the case of a multi--exponential function basis set.
Remarkably, this set of equations has the same structure to our HEOM Eq.~\eqref{eq:heom-int} truncated by the first--tier $\mathcal{N}\leq 1$.
Because of the RWA, the hierarchical elements which are higher--order tiers are not occupied.

\paragraph{Single Lorentzian component approximation}
In the case of underdamped condition $\omega _{0}>\zeta /2$, the Brownian spectral density can be decomposed into two Lorentzian components as
\begin{align}
  \mathcal{J}(\omega )&=\frac{\alpha }{2\omega _{1}}\left(\frac{\gamma _{\mathrm{c}}\omega _{0}^{2}}{(\omega -\omega _{1})^{2}+(\gamma _{\mathrm{c}}/2)^{2}}-\frac{\gamma _{\mathrm{c}}\omega _{0}^{2}}{(\omega +\omega _{1})^{2}+(\gamma _{\mathrm{c}}/2)^{2}}\right)\notag\\
  &\equiv \mathcal{J}_{+}(\omega )+\mathcal{J}_{-}(\omega ),
\end{align}
where $\omega _{1}\equiv \sqrt {\mathstrut \omega _{0}^{2}-\gamma _{\mathrm{c}}^{2}/2}$ and $\mathcal{J}_{\pm }(\omega )$ are resonant to $\omega =\pm \omega _{1}$.
When the Lorenzian components are sufficiently narrow, i.e.~$\gamma _{\mathrm{c}}$ is sufficiently small in comparison with $\omega _{1}$, we can neglect $\mathcal{J}_{-}(\omega )$ in the region $\omega >0$ and we can regard $\mathcal{J}_{+}(\omega )\simeq 0$ in the region $\omega <0$.
Under this approximation, the quantum correlation function $\mathcal{C}(t)$ in the zero--temperature is evaluated as
\begin{align}
  \mathcal{C}(t)&\simeq \mathcal{C}_{+}(t)\equiv \frac{1}{\pi }\int _{-\infty }^{\infty }\!d\omega \,\mathcal{J}_{+}(\omega )e^{-i\omega t}.
  \label{eq:single-lorentz}
\end{align}
Here, we have used $n_{\mathrm{BE}}(\omega ,+0)+1=\theta _{\mathrm{H}}(\omega )$, where $\theta _{\mathrm{H}}(x)$ is the Heaviside step function.
Note that while this approximate form of spectral density, $\mathcal{J}_{+}(\omega )$, is sometimes adopted in open quantum theories, the spectral density does not satisfy $\mathcal{J}_{+}(-\omega )=-\mathcal{J}_{+}(\omega )$.

Hereafter, we discuss the effects of rigorous description (New approach, Eq.~\eqref{eq:heom}), wavefunction description with the rotating wave approximation (WF--RWA, Eqs.~\eqref{eq:diff-coeff-c} and \eqref{eq:diff-coeff-d}), and Born--Markov approximation (Redfield equation).

\begin{figure}
  \centering
  \includegraphics[scale=\SingleColFigScale]{fig5_zerotemp_correlation.pdf}
  \caption{ (a) Symmetrized correlation function $\mathcal{S}(t)$ in the case of zero--temperature limit with a Brownian spectral density Eq.~\eqref{eq:brownian-alpha}.
    Orange curves and blue triangles represent the results of the PSD$[0/1]$ +FSD$[9]$ scheme and PSD$[9/10]$ scheme for $T=10^{-3}$, respectively.
    These two cause $12$ basis functions $\bm{\phi }(t)$.
    Blue curves mean the results of single Lorentzian approximation Eq.~\eqref{eq:single-lorentz}, and gray and black curves represent analytical solutions with strict zero--temperature $T=0$ and relatively high--temperature $T=10^{-2}$ cases, respectively.
    (b) Long--time tail behavior of (a).
  }
  \label{fig:zerotemp_correlation}
\end{figure}
To check the accuracy of correlation function expansions, we first plotted the symmetrized correlation function $\mathcal{S}(t)$ in Fig.~\ref{fig:zerotemp_correlation}.
Here, we employ the FSD scheme with temperature $T=10^{-3}$ and a single PSD$[N-1/N]$ pole on the reference temperature $T_{0}=1$ (PSD$[0/1]$+FSD$[9]$), which yields $12$ basis functions $\bm{\phi }(t)$.
As seen here, the results of this expansion (orange curves) sufficiently reproduces the analytical solution of the strict zero--temperature case (gray curves).
The analytical solution and expansion has an algebraic long--time tail with power $t^{-2}$, which is a characteristic feature of a Brownian motion near zero--temperature under an Ohmic friction \cite{jung1985pra, weiss2011book}.
Although the single Lorentzian approximation Eq.~\eqref{eq:single-lorentz} also well reproduces the analytical solution, the amplitude near $t=0$ is different from the correct solution, and this cannot reproduce the long--time tail behavior.

\begin{figure}
  \centering
  \includegraphics[scale=\SingleColFigScale]{fig6_zerotemp_population.pdf}
  \caption{ Spontaneous de--excitation processes of the excited population $P_{e}(t)\equiv \rho _{ee}(t)$ under a zero--temperature environment for (a) weak coupling $\alpha =0.004$, (b) moderate coupling $\alpha =0.1$, and (c) strong coupling $\alpha =0.4$ cases.
    Orange curves represent our generalization of the HEOM Eq.~\eqref{eq:heom} with the PSD$[0/1]$+FSD$[9]$ parametrization, and blue curves mean the results of the WF--RWA calculations, Eqs.~\eqref{eq:diff-coeff-c} and \eqref{eq:diff-coeff-d}, with the same basis set $\bm{\phi }(t)$.
    Black solid and dashed curves represent the results of the HEOM and WF--RWA calculations with single Lorentzian approximation Eq.~\eqref{eq:single-lorentz}.
    Gray curves mean the results of the Redfield theory.
  }
  \label{fig:zerotemp_population}
\end{figure}
In Fig.~\eqref{fig:zerotemp_population}, we depict the numerical results of the spontaneous de--excitation of the excited state population in the case of weak, moderate, and strong coupling cases.
The parameters were set to $\omega _{0}=1$ and $\zeta =0.5$, and this causes $\omega _{1}=\Omega _{e}=0.968$.
The truncation tiers of the HEOM were chosen as $\mathcal{N}_{\mathrm{max}}=3$, $5$, and $8$ for $\alpha =0.004$, $0.1$ and $0.4$, respectively, which generates $454$, $6,187$, and $125,969$ ADOs.

In Fig.~\ref{fig:zerotemp_population}(a), the results of the Redfield theory and other calculations are already different even though it is in a weak coupling regime.
This is because, the non--Markovian feature of the environment correlation is important in this problem:
In the cases of Sections~\ref{sec:example1} and \ref{sec:example2}, the population transfer processes are caused by the electronic coupling of the system states, $J$, and the beating behavior in the weak coupling regime is dominantly determined by the coupling.
On the other hand, in the problem of this section, the transitions are caused by the bath fluctuations and their behavior reflect details of the bath correlation functions in comparison with the previous problems.
Because both HEOM and WF--RWA calculations capture the non--Markovian behavior, the results are coincident.

In the moderate and strong coupling cases, Figs.~\ref{fig:zerotemp_population}(b) and (c), the WF--RWA results differ from the HEOM results.
This is because, when the system--bath coupling is strong, the non--resonant terms $\hat{a}^{+}\hat{b}_{\xi }^{+}$ and $\hat{a}^{-}\hat{b}_{\xi }^{-}$ which are neglected in the WF--RWA calculations contribute de--excitation processes.
In this regime, the excited state has a finite equilibrium population caused by quantum fluctuations of the bath even though it is zero--temperature (In the strong--coupling limit, the population should be $0.5$ because the enegy difference between ground and excited states becomes negligible).
The WF--RWA calculations underestimate the population owing to the lack of non--resonant terms.
The results of single Lorentzian approximation become different from the correct calculations when the coupling is strong, because details of the correlation functions affect in the regime.

\section{CONCLUDING REMARKS}
\label{sec:conclusion}
In this paper, we developed a new generalization of the HEOM theory including treatments of non--exponential basis sets for environment correlation functions, Eq.~\eqref{eq:heom}.
We showed that our generalization was unified with the original HEOM theory and other generalizations, and our generalization could be more efficient and stable than the conventional theories.
We demonstrated our new generalization by using three examples, in which non--exponential behavior of environment correlation functions plays essential roles, and we further examined the validity of our approach.

Our generalization is based on a new, simple design of expansion forms of environment correlation functions, Eqs.~\eqref{eq:S-expansion} and \eqref{eq:A-expansion}, and while other elements of the theory are almost the same as the original HEOM theory.
Therefore, other techniques developed within the framework of the original theory, e.g.,~the HEOM theory for fermion environments \cite{jin2007jcp, jin2008jcp, hartle2013prb, hartle2015prb, schinabeck2018prb}, should be easily incorporated with our generalization.

In our demonstration calculations, we employed a simple truncation of tiers of the HEOM, which could be inefficient.
An extension of more advanced truncation scheme for our generalization, e.g., those in Refs.~\onlinecite{hartle2013prb, hartle2015prb}, should be worthful for more efficient, stable, and general calculations.
In addition, in this paper, we discussed only the instability of HEOM in the case of near degenerate basis functions.
Besides this, instabilities of HEOM theories in the case of strongly non--Markovian bath correlation function are known.
Although a numerical approach to remove the instabilities has been proposed\cite{dunn2019jcp}, the physics underlying the instabilities is an unsolved problem.
Improving the stability of HEOM will be worthful for many practical applications.
These problems are left for future investigations.

\section*{SUPPLEMENTARY MATERIAL}
See supplementary material for the LibHEOM library and its Python binding (PyHEOM) which we developed to perform simulations based on our approach.
Their documents and up--to--date codes may be found on GitHub.

\begin{acknowledgments}
  T.~Ikeda is thankful to B.~Fu for manuscript reading.
  T.~Ikeda was supported by JSPS Overseas Challenge Program for Young Researchers.
  G.~D.~Scholes is a CIFAR Fellow in the Bio--Inspired Energy Program.
  This research is funded by the Gordon and Betty Moore Foundation through Grant GBMF7114.
\end{acknowledgments}

\section*{DATA AVAILABILITY}
Data available on request from the authors.

redpajama_set_name: RedPajamaArXiv


Rabbitohs and Indigenous All Stars skipper, Greg Inglis, joined more than 30 past and present Indigenous rugby league players, including members of this year's men's Indigenous All Stars team at a two-day leadership camp, heralding the start of a week-long celebration leading up to the 2016 Harvey Norman Rugby League All Stars match on Saturday 13 February.
The group gathered on Stradbroke Island on Sunday, with past and present Indigenous All Stars players as part of what has become a tradition in the lead up to the All Stars match.
Players spent the day bonding and learning from one another, with the camp designed to strengthen player's cultural identity and wellbeing, along with developing off-field leadership and future skills.
Indigenous All Stars captain, Greg Inglis said he was honoured to not only lead his team this year, but also to be amongst the camp and develop stronger ties across the game.
"The All Stars week is a really special week and occasion for me and it is important for me as a proud Aboriginal man to represent my family, community and people," Inglis said.
Former NRL Indigenous player and current Indigenous All Stars women's coach, Dean Widders said he felt an enormous sense of pride whenever the All Stars week came around.
"This is the sixth time that we have had an All Stars match and every year, the spirit and support from everyone grows stronger," Widders said.
o Career Choices: with players and participants identifying career paths outside of rugby league and developing plans aimed at gaining qualifications and experience for the future.
o Mental Wellbeing: with a number of workshops held whilst in camp, led by wellbeing experts and NRL wellbeing partners.
o Development of a cultural dance: with players to form their own dance to be used in the future across the game and representing Indigenous people.
The Indigenous All Stars and World All Stars men's teams will officially head into camp in Brisbane on Monday 8 February, with the women's teams, touch football teams and under 16s teams heading into their camps later in the week.


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Stevens, Robert D. and Sutter, Raoul. (2013) Prognosis in severe brain injury. Critical Care Medicine, 41 (4). pp. 1104-1123.
Sutter, Raoul and Bruder, Elisabeth and Weissenburg, Mandy and Balestra, Gianmarco M.. (2013) Thyroid hemorrhage causing airway obstruction after intravenous thrombolysis for acute ischemic stroke. Neurocritical care, Vol. 19, Nr. 3. pp. 381-384.
Sutter, Raoul and Grize, Leticia and Fuhr, Peter and Rüegg, Stephan and Marsch, Stephan. (2013) Acute-phase proteins and mortality in status epilepticus: a 5-year observational cohort study. Critical care medicine, Vol. 41, H. 6. pp. 1526-1533.
Sutter, Raoul and Kaplan, Peter W.. (2013) The neurophysiological types of nonconvulsive status epilepticus - EEG patterns of different phenotypes. In: Current advances and future directions in status epilepticus, 54, Special issue. Malden, Mass., pp. 23-27.
Sutter, Raoul and Kaplan, Peter W. and Rüegg, Stephan. (2013) Independent external validation of the status epilepticus severity score. Critical care medicine, Vol. 41, H. 12 , e475-479.
Sutter, Raoul and Marsch, Stephan and Fuhr, Peter and Rüegg, Stephan. (2013) Mortality and recovery from refractory status epilepticus in the intensive care unit: a 7-year observational study. Epilepsia, Vol. 54, H. 3. pp. 502-511.
Sutter, Raoul and Marsch, Stephan and Rüegg, Stephan. (2013) Safety and efficacy of intravenous lacosamide for adjunctive treatment of refractory status epilepticus: a comparative cohort study. CNS drugs, Vol. 27, H. 4. pp. 321-329.
Sutter, Raoul and Rüegg, Stephan. (2013) Predicting outcome of adults of status epilepticus. Zeitschrift für Epileptologie, Vol. 26, H. 4. pp. 79-84.
Sutter, Raoul and Sharshar, Tarek and Stevens, Robert D.. (2013) Clinical neurological assessment of the critically ill patient. In: Brain Disorders in Critical Illness: Mechanisms, Diagnosis, and Treatment. Cambridge, pp. 219-228.
Sutter, Raoul and Stevens, Robert D.. (2013) The epidemiology of critical illness brain dysfunction. In: Brain Disorders in Critical Illness: Mechanisms, Diagnosis, and Treatment. Cambridge, pp. 1-14.
Sutter, Raoul and Stevens, Robert D. and Kaplan, Peter W.. (2013) Clinical and imaging correlates of EEG patterns in hospitalized patients with encephalopathy. Journal of Neurology, 260 (4). pp. 1087-1098.
Sutter, Raoul and Stevens, Robert D. and Kaplan, Peter W.. (2013) Continuous electroencephalographic monitoring in critically ill patients: indications, limitations, and strategies. Critical Care Medicine, 41 (4). pp. 1124-1132.
Sutter, Raoul and Stevens, Robert D. and Kaplan, Peter W.. (2013) Significance of triphasic waves in patients with acute encephalopathy: a nine-year cohort study. Clinical Neurophysiology, 124 (10). pp. 1952-1958.
Ferguson, Matthew and Bianchi, Matt T. and Sutter, Raoul and Rosenthal, Eric S. and Cash, Sydney S. and Kaplan, Peter W. and Westover, M. Brandon. (2012) Calculating the risk benefit equation for aggressive treatment of non-convulsive status epilepticus. Neurocritical Care, 18 (2). pp. 216-227.
Hottinger, Annalena and Sutter, Raoul and Marsch, Stephan and Rüegg, Stephan. (2012) Topiramate as an adjunctive treatment in patients with refractory status epilepticus: an observational cohort study. CNS drugs, Vol. 26, H. 9. pp. 761-772.
Stevens, Robert D. and Sutter, Raoul. (2012) Coma emergence and recovery: the new and the old. Critical Care Medicine, 40 (4). pp. 1352-1353.
Sutter, Raoul and Kaplan, Peter W.. (2012) Electroencephalographic criteria for nonconvulsive status epilepticus: synopsis and comprehensive survey. Epilepsia, 53 Suppl 3. pp. 1-51.
Sutter, Raoul and Kaplan, Peter W.. (2012) Electroencephalographic patterns in coma: when things slow down. Epileptologie, 29. pp. 201-209.
Sutter, Raoul and Rüegg, Stephan. (2012) Refractory status epilepticus: epidemiology, clinical aspects and management of a persistent epileptic storm. Epileptologie, Vol. 29. pp. 186-193.
Sutter, Raoul and Rüegg, Stephan and Kaplan, Peter W.. (2012) Epidemiology, diagnosis, and management of nonconvulsive status epilepticus: opening Pandora's box. Neurology Clinical Practice, 2 (3). pp. 275-286.
Sutter, Raoul and Tschudin-Sutter, Sarah and Grize, Leticia and Fuhr, Peter and Bonten, Marc J. M. and Widmer, Andreas F. and Marsch, Stephan and Rüegg, Stephan. (2012) Associations between infections and clinical outcome parameters in status epilepticus : a retrospective 5-year cohort study. Epilepsia : journal of the International League against Epilepsy, Vol. 53, H. 9. pp. 1489-1497.

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With the Phillies not participating in the MLB post-season for the first time in six years, regional baseball fans are confused about what to do with their time, how much playoff action they can enjoy and who they might root for. Rest assured, fans, it is perfectly okay to cheer for and have been happy when former Phillie Raul Ibanez single-handedly downed the Orioles in game three of the American League Division Series, giving the hated New York Yankees a 2-1 series lead.
On Wednesday night, the 40-year-old outfielder/designated hitter pinch hit for superstar Alex Rodriguez in the 9th inning and launched a solo homerun that tied the game at two runs apiece and sent the contest into extra frames. In the bottom of the 12th inning, Ibanez smacked another blast that gave his team a 3-2 victory.
Philadelphia fan reaction on Twitter was mixed. Some Philly faithful were happy that the veteran, who has never won a World Series, could help his team in such a tremendous fashion. Others cursed and expressed displeasure with the Phillies for letting him walk away as a free agent last off-season.
Phillies hurler Michael Schwimer pointed out, with a tweet on Thursday, that he was rooting for players this year, not teams. My advice to Phillies fans would be for them to do the same. Feelings of hatred or jealousy for the team Ibanez plays for should not impact the affinity that fans feel toward the player they grew so fond of when he became an All-Star in Philadelphia.
Former two-time Phillie Jim Thome is another veteran without a championship ring that is battling in the playoffs. The future Hall of Famer, who ranks 7th all-time in home runs, with 612, began this season with the Phils before a June trade sent him to Baltimore.
Other former Phillies still active in the MLB post-season that are chasing their first World Series title include Wilson Valdez (Reds), Brandon Moss (Athletics) and Hunter Pence (Giants).
Although, if you're a stereotypical Philadelphia sports fan that simply can't bear to cheer on a guy in any other uniform and would prefer to boo something, there are always the likes of Scott Rolen as well as Jayson Werth and his beloved beard still going for their respective clubs, the Reds and the Nationals, to root against. Each of those guys already have a World Series ring, so do they really need another one?


The principle of a room-in-room scenario has been chosen in order to illustrate the idea of light. All the spatial volumes dictated by the radiation angle serve as daylight regulators. Sliding walls shape the spaces within the exhibition area through a systematic organisation of the guide rails on the floor and on the suspended ceiling.
Further flexible space-shaping elements emphasize the general concept while creating a museumlike experience. The shadows generated by the lighting design accompany the visitor almost like a personal guide through the gallery.
Construction in existing contexts – needs and demands of the home are always changing and have to be adjusted over time.
We help and are happy about our partners who are doing a great job!
We are super happy! Our work with and for Black Yak on the ISPO 2018 was a strong piece of work! The realization earned the Red Dot Award, Communication Design 2018! A special thanks to all who took part of it, especially Simon Vorhammer, Ligeo by GiP, Viscons und 507nanometer.
During the last years, we did not only design the fair stands for BLACK YAK, but also the collection books for the sales department.
To develop the brand´s design language as an overall image from season to season, we adapted the layout, typography, type and created special illustrations from the clothes.
We did the new law office for KSPP in Munich throughout all project phases.
The client wanted a bright and open office with scandinavian inspired furniture.
Snapshot from the brand new Ego Movement Shop in Hamburg, Eppendorfer Baum 42. The new shop is practically finished and we can´t wait for the grand opening very soon!!
Big thanks again to Mono Tee for the drinks and Sandra Forster from Blitz Restaurant for the perfect after ride barbecue and the cosy evening in your beergarden!
HAPPY ABOUT THE INVITATON FROM FRAME MAGAZINE TO BE A PART OF THE UPCOMING "NIGHT FEVER ISSUE 6"
We will be inside the upcoming Night Fever Issue 6 from Frame Magazine with our project Blitz Club. The book offers readers more than 100 fresh and exciting projects in the field of hospitality design and is scheduled to be released in February next year. Looking forward to it!
The single-family house FLOMO is finished. You will find the whole project documentation soon here on our website.
Architektouren 2018 were great! We are very happy that so many interested people came over to visit our project Blitz Club and listened to Simon Vorhammer (www.simonvorhammer.de) and Andreas Müller (Studio Knack) talking about the interior design concept. Thanks to Bayerische Architektenkammer for giving us the chance to represent one of our favorite projects!
We are very happy to announce, that Blitz Club is part of this year´s Architektouren. A big thanks to our clients who are opening the club´s doors for us during hours that are quite unconventional for a nocturnal venue. Our Computational Designer, Simon Vorhammer, will, together with Andreas Müller, present the architecture and the acoustic concept of the location. Come over, peeps!
Welcome! In our newsfeed, we will from now on keep you updated on what's happening at STUDIO KNACK.


We combine our business security systems with customizable automation solutions that ensure our services meet the needs of your growing business. You can be assured that your business will be truly customized to your specific needs.
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Our certified technicians can design and customize surveillance systems that fit your specific needs.
We provide a large amount of CCTV equipment with Pan-Tilt-Zoom cameras that allow specific zones to be targeted in a particular area. A DVR is used to store the footage detected from the cameras. We also offer Internet Protocol (IP) cameras that can be accessed using your smartphone or laptop.
Elite installs AirVac central vacuum system, a complete home cleaning system. The AirVac's sleek, new modular design houses a powerful, yet efficient motor that lowers operating costs and reduces allergy causing agents & noise. This protects against dirt, dirt mites, pet dander, and other pollutantsby using high-efficiency, two-stage filtration.
We design access control systems that fit to individual or commercial needs. An electronic access control system offers several advantages over traditional locks and keys. Electronic keys are difficult to duplicate, you never have to change your locks and you don't have to remember your keys. Access control systems can limit the entry of un-authorized intruders or personnel. Protecting your business against damage, theft and protecting your employees from harm.
Our auto GPS tracking devices are designed to keep you family members safe when they are away by providing details of where your vehicle is at all times.
Environmental sensors to alert you to extreme temperature changes. An average supermarket or grocery store may sell up to 42,000 unique items, ranging from ready-to-eat meals to basic produce, poultry, meats, and fish. Of those 42,000 items, the nationwide shipment value of frozen foods, fruits, vegetables and ice cream approaches nearly $50 billion, all of which require around-the-clock remote monitoring & alerting systems to maintain quality and prevent spoilage during transit, but also during storage in-store. Stores may use a wide variety of cooling equipment and techniques to store goods, such as display cases, walk-in refrigerators & freezers. As a result, there is a significant opportunity within supermarkets to improve older monitoring techniques (such as manual data logging) to increase operational efficiency, but also to maintain better quality control over perishable and temperature-sensitive products.
According to floodsmart.gov, "At least 25 percent of businesses that close after destructive events such as floods never reopen. From 2010 to 2014, the average commercial flood claim amounted to nearly $89,000." An automated flood and leak monitoring system can help catch leaks and flooding early before significant damages and/or losses can settle in. In most cases, server rooms, equipment rooms, and basement areas should all be outfitted with a remote flood monitoring system that sends notifications to individuals within the business that can remedy and/or solve the problem.
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I just sued the school system… start with your history teacher
Posted on October 17, 2016. Filed under: What History is For | Tags: American public schools, I just sued the school system, Prince Ea |
We've been hearing that argument about public schools in the U.S. being modeled on factories more and more lately. You've likely heard it; it's summed up in the Prince Ea video "I just sued the school system!!!" The idea is that grade schools were meant to run like factories, brutalizing students by making them sit in rows and raise their hands to talk and move when a bell rang, all so they would be good factory workers. And we, sadly, continue to run them this way today.
Where to start. First, factory workers in the early 20th century did not need to know how to read or write or do arithmetic, so why would future factory workers go to school at all?
Next, children worked in factories at ages as young as four years old, so there was no going to school first, working in a factory on graduation. (See our post Why was there child labor in America?)
Last, factories and child labor within them were established long before the 20th-century version of public school education was created.
So let's look at U.S. public schools at the turn of the 20th century. These are the sort of images we find of them:
These schools developed in our early 20th century cities. It's not how schools in America have always been (the cherished one-room country schoolhouse was the norm). When immigration to the U.S. increased exponentially in the 1910s, we suddenly had millions of children in cities, and most of them had parents who wanted at least some of them to go to school. Sending all of one's children to school, not just the eldest son or smartest boy, was possible for the first time in America because school was free. If the parents were both working and didn't need their children to work, too, all of the children could actually go to school.
So because we were—and still are—the only nation on Earth to promise a free public school education to all, we built big schools with lots of big classrooms and put lots of desks in them. How else could a teacher manage a class of 30-40 students? And since classes had to be big to educate everyone, there had to be rules like raising your hand to talk and sitting still and moving only when the bell rang or it would be chaos. It wasn't to mimic factories. It was the only way at the time to educate everyone. Some big-city tenement blocks in the 1920s had 500+ kids living on them—just one block! The hundreds of neighborhood schools that were built to educate them all had to operate a little like machines just to get all their students through.
To try to shame present-day American schools for still following this pattern, to a certain extent, is ridiculous. First, most grade schools have abandoned sitting in rows at desks all day to allow students to work in small groups, have "rug time", and other ways of moving around during the day. To a lesser extent, many junior high and high schools do this as well. It's been a long time since most American students sat down in the morning, got up at lunch, sat down after it, then got up to go home. (In fact, students today are the ones who don't get recess—a once-standard part of the American school day.)
And another reason it's unfair to shame the U.S. is that one reason we still have rows of desks is that we are still one of the few nations trying to educate all. To compare us, as is always done, with Finland or Singapore is crazy. Those are small, racially and ethnically homogenous states with no vast income inequalities. It's easy to teach students who all start at the same place and have the same background and language. And in most of the world, education stops for most children after grade school; in those nations where it continues, by the time students reach the equivalent of U.S. high school, they are divided into students who are going on to college and those who aren't, and they are educated separately and pretty unequally. The test scores that Finland and Singapore present to the world are just from their college-bound students.
But that's not how we are. We still try to educate everyone, no matter the differences in race, income, language, ethnicity, and learning ability. We fail. But we still try. It's still our goal. So any solution we come up with has to work for our situation, which is unique in the world.
Can we change our public schools to make them better? Are there better ideas out there than rows of desks? Yes. But to Prince Ea and all the others, we say reform all you want, but don't tell people that American schools were developed by evil heartless people to indoctrinate and crush children when it was completely the other way around.
(That said, we liked Prince Ea's video debunking the concept of race, which is indeed completely made up and not real.)
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SURPRISE!! IT'S A 15% OFF EVERYTHING DAY!!
We were so excited to see our new awning pop up this morning, we decided to pass on the Surprise with 15% off everything in the shop!! SO, get out in the beautiful weather and pay us a visit!
In the meantime, here is our new and improved store front!
Special thanks to Cain Awning!!
Monday was Marjorie's Birthday and there is no doubt she celebrated in style. Gallery 1930 was the venue and the attire was anything pink! Not knowing where she was going or to wear pink, she was greeted at the front entrance by floating pink and red balloons and some of her favorite girls.
Check out how Perfectly Pink her party was!
Marjorie with her jojo girls!
Customers are always coming in with pictures from magazines wanting to know where they can find this lamp, that pillow or those chairs. So we decided to start a new blog series called, "Get the Look". We will find beautifully designed rooms and then provide these items in our store or make them available for special order. How easy is this?!
If you have a room you would like us to feature, send an email to info@mhjinteriors.com.
Do you ever get in one of those moods when a certain design style just "speaks" to you? Well, we do and right now we are loving some Chinoiserie. Take and look at the wonderful works of Bungalow 5 and how mod and fabulous a little Chinese fretwork can be!
We carry Bungalow 5 at jojo Home, so make sure you take advantage of our special ordering!


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Mohamed Salah competes with Messi and Lewandowski in the final list of "The Best" award
The International Football Association "FIFA" announced the final list that will compete for the award for the best player in the world for 2021, and the Egyptian star Mohamed Salah's competition was limited to the first three advanced places.
He will compete with Argentine star Lionel Messi and Poland's Lewandowski, and this will be announced during a ceremony to be organized by FIFA on January 17th.
More than a month ago, FIFA revealed a list of 11 players competing to enter the final list.
The Egyptian Mohamed Salah, the Liverpool star, came at the top of the list, which also included strong names, the Argentine Lionel Messi, the Paris Saint-Germain star and the Portuguese Cristiano Ronaldo, the Manchester United striker.
And Lewandowski was the winner of last year's award 2020, after leading Bayern Munich to win the German League treble, the German Cup and the European Champions League.
International Mohamed Salah is preparing to lead the Egyptian national team in the 2021 African Cup of Nations, which will be hosted in Cameroon from January 9 to February 6.
Egypt is competing in Group D in the 2021 Africa Cup of Nations, with Nigeria, Guinea Bissau and Sudan, respectively.
Mohamed Salah dreams of achieving a championship with the Egyptian national team by winning the 2022 African Nations title in Cameroon, especially since he has not achieved this title before with the Pharaohs, as the cup has been absent from the team's coffers since 2010.
Egypt is scheduled to open its 2021 Africa Cup of Nations campaign in Cameroon, against Nigeria on January 11th.