text:
\section{Introduction}
\label{sec:introduction}

The concept of particles is known to depend on observers.
Even in the Poincar\'e-invariant Minkowski vacuum,
an observer with constant acceleration sees a thermal particle spectrum
\cite{Unruh:1976db}.
The Unruh-DeWitt detector \cite{Unruh:1976db,dewitt} was introduced
as a tool of thought experiment
to give an intuitive understanding of such thermal character of spacetime
(see also \cite{Birrell:1982ix,Takagi:1986kn} and references therein).
This is a detector weakly interacting with a matter quantum field
in a certain vacuum state,
and one can study the thermal character of spacetime
through the density distribution of the detector.

Various spacetimes have been examined with the Unruh-DeWitt detector,
including de Sitter space.
Although free scalar field in de Sitter space has a large family of de Sitter-invariant vacua
(called the $\alpha$-vacua) \cite{Mottola:1984ar,Allen:1985ux},
the Bunch-Davies vacuum (or the Euclidean vacuum) \cite{Bunch:1978yq}
is regarded as the most natural vacuum
because this satisfies the Hadamard condition.
It is actually only the Bunch-Davies vacuum which exhibits a thermal property
\cite{Birrell:1982ix,Spradlin:2001pw,Bousso:2001mw,Figari:1975km,Gibbons:1977mu};
If one places a detector at $\bx={\mathbf0}$
in the Poincar\'e patch,
\begin{align}
 \rmd s^2 = \ell^2\,\frac{{}-\rmd \eta^2+\rmd \bx^2}{\eta^2}
  \quad\bigl(-\infty<\eta<0\bigr)\,,
\end{align}
where scalar field is initially in the Bunch-Davies vacuum,
then the density distribution of the detector evolves
through the interaction with the scalar field
and eventually reaches the Gibbs distribution of temperature $T=1/2\pi\ell$
($\ell$ being the curvature radius of de Sitter space),
irrespectively of the initial form of the density distribution
[see Fig.~\ref{fig:medium} (a)].
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=6cm]{fig1a.eps}
\hspace{1cm}
\includegraphics[width=6cm]{fig1b.eps}
\begin{quote}
\caption{The Unruh-DeWitt detector in the Poincar\'e patch.
The scalar field is initially
in (a) the Bunch-Davies vacuum
or (b) a class of states deviated slightly from the Bunch-Davies vacuum.
\label{fig:medium}
}
\end{quote}
\end{center}
\vspace{-3ex}
\end{figure}
This implies that the detector behaves
as if it is in a thermal bath
of temperature $T=1/2\pi\ell$
when it is placed in the Bunch-Davies vacuum.

The specialty of the Bunch-Davies vacuum may be understood as follows.
We first notice that
there is no global timelike Killing vector in the Poincar\'e patch
and thus the Hamiltonian of scalar field has an explicit time dependence.
This implies that one cannot define a time-independent ground state
and can only introduce the instantaneous ground state $\ket{0_\eta}$
at each instant $\eta$\,.
As is investigated in detail in \cite{Fukuma:2013mx},
the Bunch-Davies vacuum $\ket{\rm BD}$
can be characterized as the ground state at the infinite past,
$\ket{\rm BD}=\lim_{\eta\to-\infty}\ket{0_\eta}$.
Thus, one may regard the Bunch-Davies vacuum
as a medium (surrounding the detector)
which already undergoes a sufficiently long time evolution
to reach a thermodynamic equilibrium state.

If instead the scalar field is initially in a certain class of states,
the Unruh-DeWitt detector may behave
as if it is surrounded by a medium
that is not in thermodynamic equilibrium
and that undergoes a relaxation to the equilibrium
corresponding to the Bunch-Davies vacuum [see Fig.~\ref{fig:medium} (b)].
It should be possible to investigate the relaxation process of the surrounding medium
by observing the time evolution of the density distribution of the detector.
This analysis may give useful information
on the nonequilibrium thermodynamic character
intrinsic to de Sitter space.

The main purpose of this paper is to develop a machinery
for describing such nonequilibrium dynamics
and to calculate the relaxation times of the surrounding media.
For this, we first develop a general framework
to treat an Unruh-DeWitt detector in arbitrary background geometry
and derive the master equation
which completely determines a finite time evolution
of the density matrix of the detector.
We then apply this framework to a detector in de Sitter space.
We show that if the initial state of the scalar field is chosen
such that its Wightman function has the same short distance behavior
as that of the Bunch-Davies vacuum,
then the density distribution of any detector placed there
exhibits a relaxation to the Gibbs distribution,
with a relaxation time proportional to $\ell$
(measured in the proper time of the detector).
In particular, if we take the initial state
as the instantaneous ground state
at a {\em finite} past (say, at $\eta_0$),
the relaxation time is always given by a universal value $\ell/2$.

In order to avoid possible confusions,
we here stress that there can be two kinds of relaxation times.
The first is the relaxation time that may exist
even when the detector is placed in a thermal bath
[see Fig.~\ref{fig:medium} (a)].
This is the period of time
it takes for the detector to reach the Gibbs distribution
from a given initial density distribution.
This kind of relaxation time can be neglected if one considers an ideal detector
which can get adjusted to its environment instantaneously.
Another kind of relaxation time, in which we are interested,
is the period of time it takes for the nonequilibrium environment
to reach a thermodynamic equilibrium state [see Fig.~\ref{fig:medium} (b)].
This relaxation time should not depend on details of the detector
or on the form of interaction between the detector and the scalar field,
and is related to the nonequilibrium dynamics
intrinsic to de Sitter space.

This paper is organized as follows.
In section \ref{sec:master},
adopting the method of projection operator \cite{kubo},
we first derive the master equation
which describes the time evolution of the density matrix of the detector.
Then, after justifying a Markovian approximation,
we derive a simplified form of the master equation
which enables us to study
the relaxation behavior of the density distribution analytically.
In section \ref{sec:dS},
we apply the framework to an Unruh-DeWitt detector
in the Poincar\'e patch of de Sitter space.
We consider a situation
where the initial state of scalar field is chosen
such that its Wightman function has the same short distance behavior
as that of the Bunch-Davies vacuum.
We compute the transition rate matrix of the density distribution
of the detector,
and show that the density distribution exhibits the expected relaxation
to the equilibrium corresponding to the Bunch-Davies vacuum
with the relaxation time of the form $\ell/\alpha$\,,
where the constant $\alpha$ is determined
by the asymptotic form of the change of the Wightman function
from that of the Bunch-Davies vacuum.
In section \ref{sec:fintie_past},
we consider a particular case
where the initial state of scalar field
is the instantaneous ground state $\ket{0_{\eta_0}}$
at a finite past $\eta=\eta_0$\,,
and show that the relaxation time is always given by a universal value $\ell/2$\,,
irrespectively of the value of $\eta_0$ or the form of interaction
between the detector and the scalar field.
Section \ref{sec:conclusion} is devoted to discussions and conclusion.
We collect miscellaneous formulas in appendices.

\section{Master equation for the density matrix of an Unruh-DeWitt detector}
\label{sec:master}

\subsection{Setup}
\label{subsec:master_setup}

We consider an Unruh-DeWitt detector
in $d$-dimensional spacetime with background metric
$\rmd s^2=g_{\mu\nu}(x)\,\rmd x^\mu\,\rmd x^\nu$
$(\mu,\nu=0,1,\ldots,d-1)$,
the detector being interacting with a scalar field
$\phi(x)$ of mass $m$\,.
We assume that the detector has a sufficiently large mass
so that it can be treated as moving along a classical trajector
\footnote{
We also write the trajectory as $x^\mu(\tau)=\bigl(t,\bx(t)\bigr)$
using the functional relation $t=t(\tau)$ or $\tau=\tau(t)$
with $\rmd t/\rmd \tau >0$\,.
\label{fn:trajectory}}
$x^\mu(\tau)=\bigl(t(\tau),\bx(\tau)\bigr)$,
where $\tau$ is a proper time of the trajectory
(see Fig.~\ref{fig:detector}).
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=4.5cm]{fig2.eps}
\begin{quote}
\caption{The trajectory of an Unruh-DeWitt detector,
which starts interacting with a scalar field $\phi(x)$ at time $t_1$\,.
The time coordinate $t$ will be denoted by $\eta$
when it is the conformal time in the Poincar\'e patch.
\label{fig:detector}
}
\end{quote}
\end{center}
\vspace{-3ex}
\end{figure}

For a quantum mechanical description of the system,
we introduce the Hilbert space $\cH^{\rm tot}$
which is the tensor product of those of the detector and the field,
\begin{align}
 \cH^{\rm tot}=\cH^{\rm d}\otimes\cH^{\phi}\,.
\end{align}
The total Hamiltonian then takes the following form in the Schr\"odinger picture:
\begin{align}
 H^{\rm tot}(t) = H^{\rmd}\,
 \frac{\rmd \tau(t)}{\rmd t} \otimes 1
 + 1\otimes H^{\phi}(t)+V(t)\,.
\end{align}
Here, $H^{\rm d}$ is the Hamiltonian of the detector
associated with its proper time,
and we assume that $H^{\rm d}$ does not depend on $\tau$ (or on $t$),
denoting its eigenstates and eigenvalues by $\ket{m}$ and $E_m$\,,
respectively:
$H^{\rmd}\ket{m}=E_m \ket{m}$\,.
$H^\phi(t)$ is the Hamiltonian of free scalar field
and may depend on time explicitly
through an explicit time dependence of the metric.
$V(t)$ stands for the interaction between the detector and the field,
and we assume that it suddenly starts at time $t_1$
in the form of monopole interaction
\footnote{
Recall that this is in the Schr\"odinger picture.
We later shall take an average over the start-up time $t_1$
since one usually needs a certain period of time
to specify the initial density distribution of the detector.}
\begin{align}
 V(t)=\lambda\,\mu\, \frac{\rmd \tau(t)}{\rmd t}
  \otimes\phi\bigl(\bx(t)\bigr)\,
  \theta(t-t_1)\,.
\end{align}
Here, $\lambda$ is a dimensionless coupling constant,
$\bx(t)$ represents the position of the detector at time $t$
(see footnote \ref{fn:trajectory}),
and $\mu$ is an operator acting on $\cH^{\rm d}$\,,
which we again assume to be time independent.
The time evolution of the total density matrix $\rho^{\rm tot}(t)$
is given by
\begin{align}
 \rho^{\rm tot}(t)
  = U^{\rm tot}(t,t')\,\rho^{\rm tot}(t')\,\bigl[U^{\rm tot}(t,t')\bigr]^{-1}\,,
\end{align}
where $U^{\rm tot}(t,t')$ is the time evolution operator in the Schr\"odinger picture,
\begin{align}
 U^{\rm tot}(t,t')\equiv
  {\rm T}\exp\Bigl(-\ii\int_{t'}^t\!\!\rmd t'\,H^{\rm tot}(t')\Bigr)\,.
\end{align}

We consider a situation where one can only measure
observables associated with the detector,
such as the matrix elements $\mu_{mn}=\bra{m} \mu \ket{n}$.
Then the maximum information one can get from the system
is the reduced density matrix $\rho(t)$
which is defined as the partial trace of the total density matrix $\rho^{\rm tot}(t)$
over $\cH^\phi$\,,
\begin{align}
 \rho(t) \equiv \Tr_{\phi}\rho^{\rm tot}(t)\,.
\end{align}
The time evolution of $\rho(t)$ should be uniquely determined
once one specifies the form of interaction
and the initial condition for $\rho^{\rm tot}(t)$\,.
Since there had been no interaction
between the detector and the field before time $t_1$\,,
we may let the total density matrix take the following factorized form at $t_1$\,:
\begin{align}
 \rho^{\rm tot}(t_1) \equiv \rho^{\rm d}(t_1) \otimes \rho^{\rm \phi}(t_1)
  =\rho(t_1) \otimes \rho^{\rm \phi}(t_1)\,.
\label{init_rho}
\end{align}
The latter equality can be easily seen by noting that
$
 \rho(t_1)=\Tr_{\phi}\rho^{\rm tot}(t_1)=\rho^{\rm d}(t_1)
$\,.

\subsection{Master equation}
\label{subsec:master_eqn}

The time evolution of $\rho(t)$ can be best analyzed
if we go over to the interaction picture
by decomposing the total Hamiltonian to
\begin{align}
 H^{\rm tot}(t) &=H^{\rm tot}_0(t)+V(t)\,,
\\
 H^{\rm tot}_0(t) &\equiv H^{\rmd}\,\frac{\rmd\tau(t)}{\rmd t}\otimes 1
 + 1\otimes H^{\phi}(t)
\end{align}
and treat $V(t)$ as a perturbation.
The time evolution operator $U^{\rm tot}(t,t_1)$
is then decomposed to the unperturbed and perturbed parts as
\begin{align}
 U^{\rm tot}(t,t_1) = U^{\rm tot}_0(t,t_1)\,\,U^{\rm tot}_I(t,t_1)
\end{align}
with
\begin{align}
 U^{\rm tot}_0(t,t_1) &\equiv  {\rm T}\exp\Bigl(
  -\ii\int^{t}_{t_1}\!\!\rmd t^{\prime}\, H^{\rm tot}_0(t^{\prime})\Bigr)
  \nn\\
  &= e^{{}-\ii H^{\rm d}\cdot\, (\tau-\tau_1)} \otimes
  {\rm T} e^{-\ii\int_{t_1}^t\rmd t' H^\phi(t')} \,,
\\
 U^{\rm tot}_I(t,t_1) &\equiv {\rm T}\exp\Bigl(-\ii\int^{t}_{t_1}\!\!\rmd t'\,
  V_I( t')\Bigr)\,.
\end{align}
Here, $V_I(t)$ is defined by
\begin{align}
 V_I(t)&\equiv \bigl[U^{\rm tot}_0(t,t_1)\bigr]^{-1}
  V(t)\, U^{\rm tot}_0(t,t_1)
\nn\\
 &= \lambda\,\frac{\rmd \tau}{\rmd t}\,\mu_I(\tau) \otimes
  \phi_I\bigl(x(\tau)\bigr)\,\theta(t-t_1)\,,
\label{vint}
\end{align}
where $\mu_I(\tau)$ is given by
\begin{align}
 \mu_I(\tau)
  &\equiv e^{\ii H^{\rm d}\cdot\,(\tau-\tau_1)}\,\mu\,e^{{}-\ii H^{\rm d}\cdot\,(\tau-\tau_1)}
  \nn\\
  &=\sum_{m,n}\,e^{\ii (E_m-E_n)(\tau-\tau_1)}\,\mu_{mn}\,
  \ket{m}\bra{n}\,,
\end{align}
and the operator
\begin{align}
 \phi_I\bigl(x(\tau)\bigr)&\equiv \phi_I\bigl(t,\bx(t)\bigr)
 \nn\\
  &= \bigl[{\rm T} e^{-\ii\int_{t_1}^t\rmd t' H^\phi(t')} \bigr]^{-1}\,
  \phi\bigl(\bx(t)\bigr)\,{\rm T} e^{-\ii\int_{t_1}^t\rmd t' H^\phi(t')}
\end{align}
satisfies the free Klein-Gordon equation associated with the metric
$\rmd s^2=g_{\mu\nu}(x)\,\rmd x^\mu\,\rmd x^\nu$\,.
Accordingly, the density matrix in the interaction picture is given by
\begin{align}
 \rho^{\rm tot}_I(t)\equiv \bigl[U^{\rm tot}_0(t,t_1)\bigr]^{-1}
  \rho^{\rm tot}(t)\, U^{\rm tot}_0(t,t_1)
  = U^{\rm tot}_I(t,t_1)\,\rho^{\rm tot}(t_1)\,\bigl[U^{\rm tot}_I(t,t_1)\bigr]^{-1}\,,
\end{align}
and satisfies the von Neumann equation of the form
\begin{align}
 \frac{\rmd}{\rmd t}\rho^{\rm tot}_I(t)
  ={}-\ii\,[V_I(t),\rho^{\rm tot}_I(t)]
  \equiv -\ii\ad_{V_I(t)} \rho^{\rm tot}_I(t) \,.
\label{vn}
\end{align}

This von Neumann equation can be rewritten
to an equation involving only $\rho_I(t)\equiv\Tr_\phi\rho^{\rm tot}_I(t)$
by adopting the \emph{projection operator method} (see, e.g., \cite{kubo}).
We first introduce a linear operator
$\PP\! : {\rm End}\,\cH^{\rm tot} \to {\rm End}\,\cH^{\rm tot}$
which acts linearly on elements belonging to ${\rm End}\,\cH^{\rm tot}$
(the set of linear operators acting on $\cH^{\rm tot}$)
and has the form
\begin{align}
 \PP : \, O \, \mapsto \, \PP O \equiv (\Tr_{\phi}O) \otimes X^\phi
  \quad \bigl( O \in {\rm End}\,\cH^{\rm tot}\bigr)\,.
\end{align}
Here, $X^\phi \in {\rm End}\,\cH^{\phi}$ can be any operator acting on $\cH^\phi$
as long as it satisfies
\begin{align}
 \Tr_\phi X^\phi =1
\label{xphi}
\end{align}
and does not depend on time.
From \eqref{xphi} one can easily see that $\PP$ is a projection operator,
$\PP^2=\PP$.
For $O=\rho^{\rm tot}_I(t)$,
we obtain
\begin{align}
 \PP \rho^{\rm tot}_I(t)
  = (\Tr_\phi \rho^{\rm tot}_I(t)) \otimes X^\phi
  =\rho_I(t) \otimes X^\phi \,.
\end{align}
We further introduce $\QQ\equiv1-\PP$\,,
which is also a projection operator, $\QQ^2=\QQ$\,,
and satisfies $\PP\QQ = 0 = \QQ\PP$.

Using the fact that operator $X^\phi$ can be chosen arbitrarily
without changing the time evolution of $\rho_I(t)$
(as far as it satisfies the aforementioned conditions),
we here set $X^\phi \equiv \rho^\phi(t_1) =\rho^\phi_I(t_1)$\,.
This certainly satisfies the condition \eqref{xphi},
and due to the initial condition \eqref{init_rho}
the following equations hold:
\begin{align}
 \PP\rho^{\rm tot}_I(t_1)=\rho^{\rm tot}_I(t_1)\,,\quad
 \QQ\rho^{\rm tot}_I(t_1)=0 \,.
 \label{pq_simple}
\end{align}
Then, if the one-point function of scalar field vanishes
(as we assume hereafter),
\begin{align}
\Tr_\phi\bigl( \phi_I\bigl(x(\tau)\bigr) \rho_I^\phi(t_1) \bigr) =0\,,
\label{1pt0}
\end{align}
we obtain the equation
\begin{align}
 \frac{\rmd \rho_I(t)}{\rmd t} \otimes \rho^\phi_I(t_1) &=
 -\PP\ad_{V_I(t)}\int^{t}_{t_1}\rmd t'\,
 {\rm T} e^{-\ii\int^{t}_{t'}\rmd t^{\prime\prime}\QQ\ad_{V_I(t^{\prime\prime})}}
 \ad_{V_I(t')}\,\bigl( \rho_I(t') \otimes \rho^\phi_I(t_1) \bigr)\,.
\label{master_full}
\end{align}
We give a proof of Eq.~\eqref{master_full} in appendix \ref{app:master}.
This is the master equation
which with the initial condition \eqref{init_rho}
and the assumption \eqref{1pt0}
completely determines the time evolution of the reduced density matrix $\rho_I(t)$\,.

\subsection{Approximation of the master equation}
\label{subsec:master_approx}

We expand the right-hand side of \eqref{master_full}
to the second order in perturbation to obtain
\begin{align}
 \frac{\rmd \rho_I(t)}{\rmd t}
 =-\Tr_\phi \Big(\ad_{V_I(t)}\,\int^{t}_{t_1}\rmd t'
 \ad_{V_I(t')}\,\bigl( \rho_I(t') \otimes \rho^\phi_I(t_1) \bigr) \Bigr)
 +\mathcal{O}(\lambda^3)\,.
\end{align}
This can be further rewritten in terms of the proper time $\tau$
to the following form
[denoting $\rho\bigl(t(\tau)\bigr)$ by $\rho(\tau)$
and derivatives with respect to $\tau$ by dots]:
\begin{align}
 &\dot{\rho}_I(\tau)
\nn\\
  &= {}-\lambda^2 \int^{\tau}_{\tau_1}\!\!\rmd \tau'\, \Tr_\phi
  \bigl[\mu_I(\tau) \otimes \phi_I\bigl(x(\tau)\bigr),\,
  \bigl[\mu_I(\tau')\otimes \phi_I\bigl(x(\tau')\bigr),\,
  \rho_I(\tau') \otimes \rho^\phi_I(\tau_1) \bigr] \bigr]
  +\mathcal{O}(\lambda^3)
\nn\\
 &= \lambda^2\int^{\tau}_{\tau_1}\rmd \tau' \Bigl(
 \bigl[\mu_I(\tau),\,\rho_I(\tau')\,\mu_I(\tau')\bigr]\,G_X^+\bigl(x(\tau'),x(\tau)\bigr)
\nn\\
 &\qquad -\bigl[\mu_I(\tau),\,\mu_I(\tau')\,\rho_I(\tau')\bigr]\,
 G_X^+\bigl(x(\tau),x(\tau')\bigr)
\Bigr) + \mathcal{O}(\lambda^3)\,,
\label{dense-int}
\end{align}
where
\begin{align}
 G_X^+(x,x')\equiv \Tr_\phi
  \bigl( \phi_I(x)\,\phi_I(x')\,\rho^\phi_I(\tau_1) \bigr)
  \label{Wightman_X}
\end{align}
is the Wightman function of free scalar field
with respect to the density matrix $X^\phi= \rho^\phi_I(\tau_1)$\,.

Since the reduced density matrix in the interaction picture, $\rho_I(\tau)$,
is related to that in the Schr\"odinger picture, $\rho(\tau)$,
as
\begin{align}
 \rho_I(\tau)= e^{\ii H^{\rm d}\cdot\,(\tau-\tau_1)}\,
 \rho(\tau)\,e^{{}-\ii H^{\rm d}\cdot\,(\tau-\tau_1)}\,,
\end{align}
we can rewrite \eqref{dense-int} to the following form in the Schr\"odinger picture:
\begin{align}
 \dot{\rho}(\tau) +\ii\,[H^{\rm d},\rho(\tau)]
 &=\lambda^2\int^{\tau}_{\tau_1}\rmd \tau' \bigl[
 \mu\, e^{{}-\ii H^{\rm d}\cdot\,(\tau-\tau')}\,\rho(\tau')\,\mu\,
 e^{\ii H^{\rm d}\cdot\,(\tau-\tau')}\,
 G_X^+\bigl(x(\tau'),x(\tau)\bigr)
\nn\\
 &\qquad\qquad
 + e^{{}-\ii H^{\rm d}\cdot\,(\tau-\tau')}\,\mu\,\rho(\tau')\,\mu\,
 e^{\ii H^{\rm d}\cdot\,(\tau-\tau')}\,
 G_X^+\bigl(x(\tau),x(\tau')\bigr)
\nn\\
 &\qquad\qquad
 - e^{{}-\ii H^{\rm d}\cdot\,(\tau-\tau')}\,\rho(\tau')\,\mu\,
 e^{\ii H^{\rm d}\cdot\,(\tau-\tau')}\,\mu\,
 G_X^+\bigl(x(\tau'),x(\tau)\bigr)
\nn\\
 &\qquad\qquad
 -\mu\, e^{{}-\ii H^{\rm d}\cdot\,(\tau-\tau')}\,\mu\,\rho(\tau')\,
 e^{\ii H^{\rm d}\cdot\,(\tau-\tau')}\,
 G_X^+\bigl(x(\tau),x(\tau')\bigr)
\bigr]
\nn\\
&\quad + \mathcal{O}(\lambda^3)\,.
\label{dense'}
\end{align}

The integro-differential equation \eqref{dense'} can be further simplified
as follows.
Since the Wightman function $G_X^+\bigl(x(\tau),x(\tau')\bigr)$ in the integral
is singular at $\tau'=\tau$
and decreases exponentially for large separations of $\tau$ and $\tau'$\,
\footnote{
We see in appendix \ref{app:Wightman-function}
that for a scalar field in de Sitter space
the Wightman function certainly exhibits this property
if the mass is large enough.
In general, there can be a case
where the Wightman function has a long tail
and one needs to take account of memory effects carefully.
We do not deal with such cases in the present paper.
}
the main contributions to the integral should come
only from the region $\tau'\sim\tau$\,.
This implies that the memory effect in the equation is highly suppressed,
and thus we may replace $\rho_{kl}(\tau')$ in the integral
by its boundary value $\rho_{kl}(\tau)$ to a good accuracy,
assuming that $\rho(\tau')$ slowly changes.
Equation \eqref{dense'} can thus be rewritten (in terms of matrix elements) as
\begin{align}
 &\dot{\rho}_{mn}(\tau) +\ii(E_m-E_n)\rho_{mn}(\tau)
\nn\\
 &\simeq \lambda^2\sum_{k,l} \int^{\tau}_{\tau_1}\rmd\tau' \bigl[\,
 e^{-\ii(E_n-E_k)(\tau'-\tau)}\mu_{mk}\,\mu_{ln}\,
 \rho_{kl}(\tau)\,G_X^+\bigl(x(\tau'),x(\tau)\bigr)
\nn\\
 &\qquad\qquad\quad\quad
 +e^{-\ii(E_m-E_l)(\tau-\tau')}\mu_{mk}\,\mu_{ln}\,
 \rho_{kl}(\tau)\,G_X^+\bigl(x(\tau),x(\tau')\bigr)
\nn\\
 &\qquad\qquad\quad\quad
 -e^{-\ii(E_l-E_m)(\tau'-\tau)}\mu_{kl}\,\mu_{ln}\,
 \rho_{mk}(\tau)\,G_X^+\bigl(x(\tau'),x(\tau)\bigr)
\nn\\
 &\qquad\qquad\quad\quad
 -e^{-\ii(E_k-E_n)(\tau-\tau')}\mu_{mk}\,\mu_{kl}\,
 \rho_{ln}(\tau)\,G_X^+\bigl(x(\tau),x(\tau')\bigr)
 \,\bigr]\,.
\label{dense}
\end{align}
If the off-diagonal elements of $\rho(\tau)$
can be further neglected
\footnote{
We will see in section \ref{sec:thermal}
that we can set the off-diagonal elements of $\rho(\tau)$
can be set to zero without losing generality
if the detector is a two-level system
and $\mu$ has an off-diagonal form.}
then \eqref{dense} becomes
\begin{align}
 \dot{\rho}_{mm}(\tau)
&= \sum_{k\neq m} \bigl[w^X_{mk}(\tau,\tau_1)\rho_{kk}(\tau)
 -w^X_{km}(\tau,\tau_1)\rho_{mm}(\tau)\bigr]\,.
\label{master}
\end{align}
Here, we have introduced the transition rate matrix
\begin{align}
 w^X_{mk}(\tau,\tau_1) \equiv \lambda^2\,\abs{\mu_{mk}}^2 \,
 \dot{\cF}_X(E_m-E_k;\tau,\tau_1)\,,
\label{wmk}
\end{align}
where
\begin{align}
 \dot{\cF}_X(\Delta E;\tau,\tau_1)&\equiv\int^{\tau}_{\tau_1}\rmd\tau' \bigl[\,
 e^{-\ii\Delta E(\tau'-\tau)} \,G_X^+\bigl(x(\tau'),x(\tau)\bigr)
 + e^{-\ii\Delta E(\tau-\tau')} \,G_X^+\bigl(x(\tau),x(\tau')\bigr)
 \bigr]\,.
\label{fdot}
\end{align}
Equation \eqref{master} now has the standard form of the master equation.

If, in particular,
$\displaystyle \dot\cF^{\rm eq}(\Delta E)\equiv
\lim_{\tau-\tau_1\to\infty} \dot{\cF}_X(\Delta E;\tau,\tau_1)$
satisfies the relatio
\footnote{
The relation \eqref{thermaltransit} indeed holds
with $\beta=2\pi\ell$
when $\rho^\phi(t_1)$ corresponds to the Bunch-Davies vacuum
in de Sitter space.
See Eq.~\eqref{equib_relation}.}
\begin{align}
 \frac{\dot{\cF}^{\rm eq}(\Delta E)}{\dot{\cF}^{\rm eq}(-\Delta E)}
  =e^{-\beta \Delta E}\,,
\label{thermaltransit}
\end{align}
the transition rate matrix satisfies the relation
\begin{align}
 \frac{w_{mk}(\tau;\tau_1)}{w_{km}(\tau;\tau_1)}
  \,\overset{\tau-\tau_1\to\infty}{\longrightarrow} \,e^{{}-\beta\, (E_m-E_k)}\,.
\end{align}
Then, the distribution of the detector in equilibrium, $\rho^{\rm eq}$\,,
may be determined by the detailed balance condition
$\displaystyle\lim_{\tau-\tau_1\to\infty}w_{m k}(\tau;\tau_1)\,\rho^{\rm eq}_{k k}
=\displaystyle\lim_{\tau-\tau_1\to\infty}w_{k m}(\tau;\tau_1)\,\rho^{\rm eq}_{m m}$\,,
and we obtain
\begin{align}
 \rho^{\rm eq}_{mm}=\frac{e^{-\beta E_m}}{Z}\quad\bigl(Z=\sum_n e^{-\beta E_n}\bigr)\,,
\label{eq:Gibbs}
\end{align}
which is nothing but the Gibbs distribution at temperature $1/\beta$.

We close this section
by making a comment on the relationship between our formalism and the literature.
One can easily show that the transition rate matrix $w^X_{mk}$\,,
\eqref{wmk}, is the $\tau$-derivative of
\begin{align}
 \lambda^2\, \abs{\mu_{mk}}^2\, \cF_X(E_m-E_k;\tau,\tau_1)\,,
\label{transition_prob}
\end{align}
where
\begin{align}
 \cF_X(\Delta E;\tau,\tau_1)=
  \int^{\tau}_{\tau_1}\rmd \tau'\int^{\tau}_{\tau_1}\rmd \tau^{\prime\prime}
  e^{-\ii\Delta E(\tau'-\tau^{\prime\prime})}\,
  G_X^+\bigl(x(\tau'),x(\tau^{\prime\prime})\bigr)\,.
\label{response}
\end{align}
In the literature (e.g.\ \cite{Birrell:1982ix}),
one often considers a process
from an initial state $\ket{k}\otimes\ket{\alpha}$ at time $t_1$
(usually taken to be the infinite past)
to a final state $\ket{m} \otimes \ket{\beta}$ at time $t$\,,
and sums over the final states $\ket{\beta}$ of the scalar field.
The transition probability has the same form
as \eqref{transition_prob}
if we set $X^\phi=\rho^\phi(t_1)=\rho_I^\phi(t_1)=\ket{\alpha}\bra{\alpha}$
for which the Wightman function becomes
$G_X^+(x,x')=\bra{\alpha}\,\phi_I(x)\,\phi_I(x')\,\ket{\alpha}$\,.
We thus again see that \eqref{wmk} represents the transition probability
per unit proper time of the detector.
$\cF_X$ in \eqref{response} is often called
the response function (see, e.g., \cite{Birrell:1982ix}).
As we have seen, $\cF_X$ or its derivative $\dot{\cF}_X$
does not depend on details of the detector
and can be thoroughly determined by the Wightman function of free scalar field.

\section{Unruh-DeWitt detector in de Sitter space}
\label{sec:dS}

In this section we consider an Unruh-DeWitt detector
in $d$-dimensional de Sitter space,
which is weakly interacting with a massive scalar field $\phi(x)$
of mass $m$\,.
We exclusively consider the Poincar\'e patch
denoting the time variable by $\eta$\,,
\begin{align}
 \rmd s^2 = \ell^2\,\frac{{}-\rmd \eta^2 + \rmd \bx^2}{\eta^2}
  \quad \bigl(-\infty < \eta < 0\bigr)\,,
\end{align}
and set the classical trajectory of the detector to be the geodesic
\begin{align}
 x^{\mu}(\tau) = \bigl({}-\ell\, e^{-\tau/\ell},\,{\mathbf0}\bigr)\,.
\label{eq:geodesic}
\end{align}
We denote by $\eta_1$
the time when the detector
starts the interaction with the field $\phi(x)=\phi(\eta,\bx)$,
which has the following form in the interaction picture
[see \eqref{vint}]:
\begin{align}
 V_I(\eta)=\lambda\,\mu_I(\tau)\, \frac{\rmd \tau(\eta)}{\rmd \eta}
  \otimes\phi_I\bigl(x(\tau)\bigr)\,
  \theta(\eta-\eta_1)\,.
\end{align}

We will show that the density distribution of the detector
exhibits a relaxation to the Gibbs distribution with $\beta=2\pi\ell$
when the initial condition $X^\phi=\rho^\phi_I(\eta_1)$ satisfies the condition \eqref{1pt0}
and the corresponding Wightman function $G_X^+(x,x')$
has the same short distance behavior ($\abs{\bx-\bx'}\to 0$ with $\eta=\eta'$)
as that of the Bunch-Davies vacuum, $G_{\rm BD}^+(x,x')$.
In the following, we set the curvature radius $\ell=1$.

Let the initial density matrix $X^\phi=\rho_I^\phi(\eta_1)$ of scalar field have the form
\begin{align}
 X^\phi=\ket{\rm BD}\bra{\rm BD} + \Delta X^\phi\,,
\end{align}
for which the Wightman function takes the form
\begin{align}
 G_X^+(x,x')=\Tr_\phi\bigl(\phi_I(x) \phi_I(x') X^\phi\bigr)
 =G_{\rm BD}^+(x,x') + \Delta G^+(x,x')\,.
\label{Wightman-decomposition}
\end{align}
We assume that the deformed Wightman function is invariant
under spatial translations and rotations
for fixed $\eta$ and $\eta'$\,,
and write its Fourier transform as ($k\equiv |\bk|$)
\begin{align}
 G_X^+(\eta,\bx,\eta',\bx')
 &\equiv \int\!\frac{\rmd^{d-1}\bk}{(2\pi)^{d-1}}\,e^{\ii\,\bk\cdot(\bx-\bx')}\,
 G_{X,k}^+(\eta,\eta')
\nn\\
 &\equiv \int\!\frac{\rmd^{d-1}\bk}{(2\pi)^{d-1}}\,e^{\ii\,\bk\cdot(\bx-\bx')}\,
 \bigl[G^+_{{\rm BD},k}(\eta,\eta') + \Delta G_{k}^+(\eta,\eta')\bigr]\,.
\end{align}
This takes the following form for the geodesic \eqref{eq:geodesic}
[with $\eta=-e^{-\tau}$ and $\eta'=-e^{-\tau'}$]:
\begin{align}
 G_X^+\bigl(x(\tau),x(\tau')\bigr)
 &=\int\!\frac{\rmd^{d-1}\bk}{(2\pi)^{d-1}}\,G_{X,k}^+(\eta,\eta')
\nn\\
 &=\frac{2}{(4\pi)^{\frac{d-1}{2}}\,\Gamma\bigl(\frac{d-1}{2}\bigr)}\,\int_0^\infty\!\rmd k\,k^{d-2}\,
 G_{X,k}^+(\eta,\eta')
\nn\\
 &\equiv G_{\rm BD}^+\bigl(x(\tau),x(\tau')\bigr)
 +\Delta G^+\bigl(x(\tau),x(\tau')\bigr)\,.
\end{align}

Note that both $G_{X,k}^+(\eta,\eta')$ and $G^+_{{\rm BD},k}(\eta,\eta')$
[and thus $\Delta G_{k}^+(\eta,\eta') $ also]
satisfy the homogeneous Klein-Gordon equation,
$\eta^2\,\ddot{f}(\eta)+(d-2)\,\eta\,\dot{f}(\eta)+(k^2\,\eta^2+m^2)\,f(\eta)=0$\,,
with respect to each of the arguments $\eta$ and $\eta'$\,.
The solutions to this equation are given by linear combinations of
$ (-\eta)^{(d-1)/2}\,H_\nu^{(a)}(-k\eta)$ $(a=1,2)$\,,
where $H_\nu^{(1,2)}(z)$ are the Hankel functions
and
\begin{align}
  \nu &\equiv \left\{
  \begin{array}{ll}
 \displaystyle
   \sqrt{\Bigl(\frac{d-1}{2}\Bigr)^2- m^2}
     &\displaystyle \Bigl(0< m < \frac{d-1}{2}\Bigr)
\\[3mm]
 \displaystyle
   \ii\,\sqrt{m^2 - \Bigl(\frac{d-1}{2}\Bigr)^2} \quad
     &\displaystyle \Bigl(m \geq \frac{d-1}{2}\Bigr)
  \end{array}
  \right..
\end{align}
Thus the Wightman function is generically given by a linear combination of
\begin{align}
 [(-\eta)(-\eta')]^{\frac{d-1}{2}}\,
 H^{(a)}_{\nu}(-k\,\eta) \,H^{(b)}_{\nu}(-k\,\eta')
 \quad (a,b=1,2)\,.
\end{align}
For example, $G^+_{{\rm BD},k}(\eta,\eta')$ is given by
(see \cite{Bunch:1978yq,Birrell:1982ix,Fukuma:2013mx})
\begin{align}
\label{BD_Wightman}
 G^+_{{\rm BD},k}(\eta,\eta')
 &=
 \frac{\pi}{4}\,\bigl[(-\eta)\,(-\eta')\bigr]^{\frac{d-1}{2}} \,
 H^{(1)}_\nu(-k\,\eta)\,H^{(2)}_\nu(-k\,\eta')  \,.
\end{align}
Due to the condition that $G_{X}^+(x,x')$ and $G_{{\rm BD}}^+(x,x')$
have the same short distance behavior,
$\Delta G^+_k(\eta,\eta')$ takes the following form
($c_0$ is a constant with the dimension of time):
\begin{align}
 \Delta G_{k}^+(\eta,\eta')
 = [(-\eta)(-\eta')]^{\frac{d-1}{2}}\,
  \sum_{a,b=1}^2 f_{ab}(c_0\,k)\, H^{(a)}_{\nu}(-k\,\eta) \,H^{(b)}_{\nu}(-k\,\eta')
\label{eq:DeltaG+}
\end{align}
wit
\footnote{
$\alpha_{ab}$ will need to be integers
when imposing analyticity on the Wightman functions. }
\begin{align}
 f_{ab}(z) = {\rm const.} \, z^{-\alpha_{ab}}\,\bigl[1+\mathcal{O}(z^{-1})\bigr] \qquad
 \bigl(\alpha_{ab}>0 \bigr) \,.
\label{f_ab-alpha}
\end{align}
Note that we only require that the leading singularities
be the same for $G_{X}^+(x,x')$ and $G_{\rm BD}^+(x,x')$\,,
and the functions $f_{ab}(z)$ control the subleading singularities.

The derivative of the response function, $\dot{\cF}_X$
[defined in Eq.~\eqref{fdot}], can now be written as
\begin{align}
 \dot{\cF}_X(\Delta E;\tau,\tau_1) =
 \dot{\cF}_{\rm BD}(\Delta E;\tau,\tau_1)+\Delta\dot{\cF}(\Delta E;\tau,\tau_1)\,,
\end{align}
where
\begin{align}
 \dot{\cF}_{\rm BD}(\Delta E;\tau,\tau_1)
 &\equiv \int_{-(\tau-\tau_1)}^0\rmd s\,
  e^{-\ii\Delta E\,s}\,G^{+}_{\rm BD}\bigl(x(\tau+s),x(\tau)\bigr)
\nn\\
 &\quad +\int_0^{(\tau-\tau_1)}\rmd s\,
  e^{-\ii\Delta E\,s}\,G^{+}_{\rm BD}\bigl(x(\tau),x(\tau-s)\bigr)\,,
  \label{fdot_BD_int}
\\
\Delta\dot{\cF}(\Delta E;\tau,\tau_1)
 &\equiv \int_{-(\tau-\tau_1)}^0\rmd s\,
  e^{-\ii\Delta E\,s}\,\Delta G^{+}\bigl(x(\tau+s),x(\tau)\bigr)
\nn\\
 &\quad +\int_0^{(\tau-\tau_1)}\rmd s\,
  e^{-\ii\Delta E\,s}\,\Delta G^{+}\bigl(x(\tau),x(\tau-s)\bigr)\,.
  \label{delta_fdot_int}
\end{align}
The integrals can be evaluated analytically
as shown in appendix \ref{app:integration},
and we find that they take the following asymptotic forms
in the limit $\tau \to \infty$ [see Eqs.~\eqref{eq:BD-asympt} and \eqref{eq:DF-asympt}]:
\begin{align}
 \dot{\cF}_{\rm BD}(\Delta E;\tau,\tau_1)
 &\sim \dot{\cF}^{\eq} (\Delta E)
 +{\rm const.} \, e^{-(\frac{d-1}{2}\pm\nu\pm\ii\Delta E)\,(\tau-\tau_1)}\,,
\label{BD_fdot_asympt}
\\
 \Delta\dot{\cF}(\Delta E;\tau,\tau_1)
 &\sim {\rm const.}\,e^{-\alpha\,\tau}
  + {\rm const.}\,e^{-(\frac{d-1}{2}\pm\nu\pm\ii\Delta E)\,(\tau-\tau_1)}
\label{delta_fdot_asympt}
\end{align}
with $\alpha \equiv \underset{a,b}{\rm min}(\alpha_{ab})$\,.

The $\tau$-independent term $\dot{\cF}^{\eq} (\Delta E)$
is given by [see Eq.~\eqref{F_eq}]
\begin{align}
 \dot{\cF}^{\rm eq}(\Delta E)
 &= \frac{e^{-\pi\ell\,\Delta E}\,
 \Gamma\Bigl(\frac{\frac{d-1}{2}+\nu+\ii \ell\Delta E}{2}\Bigr)\,
 \Gamma\Bigl(\frac{\frac{d-1}{2}-\nu+\ii \ell\Delta E}{2}\Bigr)\,
 \Gamma\Bigl(\frac{\frac{d-1}{2}+\nu-\ii \ell\Delta E}{2}\Bigr)\,
 \Gamma\Bigl(\frac{\frac{d-1}{2}-\nu-\ii \ell\Delta E}{2}\Bigr)}
         {8\,\pi^{\frac{d+1}{2}}\,\Gamma\bigl(\frac{d-1}{2}\bigr)}\,,
 \label{eq:Fdot-eq}
\end{align}
which agrees with the known result
obtained in \cite{Garbrecht:2004du,Higuchi:1986ww}
(we have restored the curvature radius $\ell$).
One easily finds that this satisfies the relation
\begin{align}
 \frac{\dot{\cF}^{\rm eq}(\Delta E)}{\dot{\cF}^{\rm eq}(-\Delta E)}
  =e^{-2\pi\ell \Delta E}\,.
\label{equib_relation}
\end{align}
Thus, from the argument following \eqref{thermaltransit},
we confirm that, as $\tau$ becomes large,
the density distribution $\rho_{mm}(\tau)$ approaches the Gibbs distribution
at temperature $1/2\pi\ell$\,,
\begin{align}
 \rho^{\eq}_{mm}
 = \frac{e^{-2\pi\ell E_m}}{Z} \quad
 \bigl(Z=\sum_n e^{-2\pi\ell E_n}\bigr)\,.
\end{align}

The $\tau$-dependent terms, $e^{-\alpha\,\tau/\ell}$
and $e^{-[(\frac{d-1}{2}\pm\nu)/\ell\pm\ii\Delta E]\,\tau}$\,,
in the asymptotic forms \eqref{BD_fdot_asympt} and \eqref{delta_fdot_asympt}
represent relaxation modes of the detector
with relaxation times $\ell/\alpha$ and $\abs{(d-1)/2\pm {\rm Re}\,\nu}^{-1}\ell$,
respectively.
Note that the modes with the latter relaxation time
exist even when the initial state is the Bunch-Davies vacuum,
in which we are not interested here.
On the other hand, the mode with relaxation time $\ell/\alpha$ arises
only when the initial state of scalar field
is deviated from the Bunch-Davies vacuum.
We thus identify with $\ell/\alpha$
the relaxation time
for the surrounding medium [see Fig.~\ref{fig:medium} (b)]
which relaxes from a nonequilibrium state
to the equilibrium of temperature $1/2\pi\ell$\,.

We close this section with writing down the full expression of \eqref{BD_fdot_asympt}
for comparison with the known results in the literature [see Eq.~\eqref{eq:BD-full}]:
\begin{align}
 &\dot{\cF}_{\rm BD}(\Delta E,\tau,\tau_1)
\nn\\
 &= \dot\cF^{\rm eq}(\Delta E)
\nn\\
  &\ +\frac{e^{\ii\pi\,\frac{d-1}{2}}\,e^{\ii\pi\,\nu}\,
                   e^{-(\frac{d-1}{2}+\nu-\ii\Delta E)\,(\tau-\tau_1)}}
                  {8\,\pi^{\frac{d-1}{2}}\,\Gamma\bigl(\frac{d-1}{2}\bigr)\,\sin(\pi\,\nu)} \,
  \, _3\widehat{F}_2\Biggl(
  \afrac{\frac{d-1}{2},\,
                      \frac{\frac{d-1}{2}+\nu-\ii\Delta E}{2} ,\,
                      \frac{d-1}{2}+\nu}
                     {1+\nu,\,
                      \frac{\frac{d+3}{2}+\nu-\ii\Delta E}{2}};\,
                      e^{-2 (\tau-\tau_1)} \Biggr)
\nn\\
 & \ +\frac{e^{\ii\pi\,\frac{d-1}{2}}\,e^{-\ii\pi\,\nu}\,
                 e^{-(\frac{d-1}{2}-\nu-\ii\Delta E)\,(\tau-\tau_1)}}
                {8\,\pi^{\frac{d-1}{2}}\,\Gamma\bigl(\frac{d-1}{2}\bigr)\,\sin(-\pi\,\nu)} \,
  \, _3\widehat{F}_2\Biggl(
  \afrac{\frac{d-1}{2},\,
                      \frac{\frac{d-1}{2}-\nu-\ii\Delta E}{2},\,
                      \frac{d-1}{2}-\nu}
                     {1-\nu,\,
                      \frac{\frac{d+3}{2}-\nu-\ii\Delta E}{2}};\,
                     e^{-2 (\tau-\tau_1)}\Biggr)
\nn\\
 & \ +\frac{e^{-\ii\pi\,\frac{d-1}{2}}\,e^{-\ii\pi\,\nu}\,
                   e^{-(\frac{d-1}{2}+\nu+\ii\Delta E)\,(\tau-\tau_1)}}
                  {8\,\pi^{\frac{d-1}{2}}\,\Gamma\bigl(\frac{d-1}{2}\bigr)\,\sin(\pi\,\nu)} \,
  \, _3\widehat{F}_2\Biggl(
  \afrac{\frac{d-1}{2},\,
                      \frac{\frac{d-1}{2}+\nu+\ii\Delta E}{2} ,\,
                      \frac{d-1}{2}+\nu}
                     {1+\nu,\,
                      \frac{\frac{d+3}{2}+\nu+\ii\Delta E}{2}};\,
                     e^{-2 (\tau-\tau_1)} \Biggr)
\nn\\
 & \ +\frac{e^{-\ii\pi\,\frac{d-1}{2}}\,e^{\ii\pi\,\nu}\,
                 e^{-(\frac{d-1}{2}-\nu+\ii\Delta E)\,(\tau-\tau_1)}}
                {8\,\pi^{\frac{d-1}{2}}\,\Gamma\bigl(\frac{d-1}{2}\bigr)\,\sin(-\pi\,\nu)} \,
  \, _3\widehat{F}_2\Biggl(
  \afrac{\frac{d-1}{2},\,
                      \frac{\frac{d-1}{2}-\nu+\ii\Delta E}{2} ,\,
                      \frac{d-1}{2} -\nu}
                     {1-\nu,\,
                      \frac{\frac{d+3}{2}-\nu+\ii\Delta E}{2}};\,
                     e^{-2 (\tau-\tau_1)}\Biggr) \Biggr]\,.
\label{eq:Fdot-n=0}
\end{align}
If we set $d=4$ and $\nu=1/2$,
Eq.~\eqref{eq:Fdot-n=0} correctly reproduces
the result obtained in section~IV-D of \cite{Garbrecht:2004du},
where the case $\tau_0=-\infty$ is considered
so that the higher-order corrections
$\Delta\dot{\cF}(\Delta E,\tau,\tau_1)$
are all zero.

\section{The case of the instantaneous ground state at a finite past}
\label{sec:fintie_past}

In this section,
we consider the case where the initial state $X^\phi=\rho^\phi_I(\eta_1)$
is the instantaneous  ground state at an earlier time $\eta_0$\,.
In this case, $\Delta\dot{\cF}(\Delta E;\tau,\tau_1)$
can be calculated explicitly,
and we will see that the relaxation time takes a universal value $\ell/2$
(i.e., $\alpha=2$),
irrespectively of the values of $\eta_0$ and $\eta_1$
or the form of interaction between the detector and the scalar.

\subsection{The instantaneous ground states}
\label{subsec:dS_instantaneous}

Let $\ket{0_{\eta_0}, \eta_0}$ denote the Schr\"odinger picture state
which is the instantaneous ground state at an early time $\eta_0$
(see Fig.~\ref{fig:Poincare-detector}).
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=5.5cm]{fig3.eps}
\begin{quote}
\caption{The trajectory of an Unruh-DeWitt detector in the Penrose diagram
of de Sitter space.
The shaded region corresponds to the Poincar\'e patch,
and the dashed line represents the future event horizon for the detector.
The scalar field is in the ground state at time $\eta_0$\,.
\label{fig:Poincare-detector}
}
\end{quote}
\end{center}
\vspace{-3ex}
\end{figure}
We then introduce the state $\ket{0_{\eta_0}}\equiv\ket{0_{\eta_0},\eta_1}$
as an interaction picture state
(or as a Heisenberg picture state of free field theory)
which is obtained by applying the free-field time evolution operator
from $\eta_0$ to $\eta_1$\,:
$\ket{0_{\eta_0}}=
{\rm T}\exp\bigl(-\ii\int_{\eta_0}^{\eta_1}\!\!\rmd \eta'\,H^\phi(\eta')\bigr)\,
\ket{0_{\eta_0},\eta_0}$\,.
In this section, we set $X^\phi=\rho^\phi_I(\eta_1)=\ket{0_{\eta_0}} \bra{0_{\eta_0}}$\,.
See \cite{Fukuma:2013mx}
for a detailed discussion on defining such time-dependent ground states
for free scalar field in curved spacetimes.

Note that the one-point function vanishes,
$\Tr_\phi \bigl(\phi_I(x)X^\phi\bigr)
=\bra{0_{\eta_0}}\,\phi_I(x)\,\ket{0_{\eta_0}}=0$\,.
Then, with the approximations made in subsection \ref{subsec:master_approx},
we have the master equation of the form [see \eqref{master}--\eqref{fdot}]
\begin{align}
 \dot{\rho}_{mm}(\tau)
 =\sum_{k\neq m} \bigl[w_{mk}(\tau,\tau_1;\eta_0)\,\rho_{kk}(\tau)
 -w_{km}(\tau,\tau_1;\eta_0)\,\rho_{mm}(\tau)\bigr]
\label{master_P}
\end{align}
with the transition rate matrix
\begin{align}
 w_{mk}(\tau,\tau_1;\eta_0) \equiv \lambda^2\,\abs{\mu_{mk}}^2 \,
 \dot{\cF}(E_m-E_k;\tau,\tau_1;\eta_0)\,.
\label{wmk_P}
\end{align}
Here, the derivative of the response function is given by
\begin{align}
 \dot{\cF}(\Delta E;\tau,\tau_1;\eta_0)
 &\equiv\int^{\tau}_{\tau_1}\rmd\tau' \bigl[\,
 e^{-\ii\Delta E(\tau'-\tau)} \,G^+\bigl(x(\tau'),x(\tau);\eta_0\bigr)
\nn\\
 &\qquad\qquad + e^{-\ii\Delta E(\tau-\tau')} \,G^+\bigl(x(\tau),x(\tau');\eta_0\bigr)
 \bigr]
\label{fdot_P}
\end{align}
with the Wightman function
\begin{align}
 G^+(x,x';\eta_0)\equiv \bra{0_{\eta_0}}\phi_I(x)\phi_I(x')\ket{0_{\eta_0}}\,.
\label{Wightman_P}
\end{align}
In the following, we compute $G^+(x,x';\eta_0)$ for finite $\eta_0$
using a technique developed in \cite{Fukuma:2013mx},
and will find that $G^+(x,x';\eta_0)$ indeed has the form \eqref{Wightman-decomposition}
and \eqref{eq:DeltaG+}
with
\begin{align}
 f_{12}(z)=f_{21}(z)=
 \frac{\pi}{4}\Bigl(\frac{d-2}{4}\Bigr)^2\,z^{-2}+{\cal O}(z^{-4})\,,
 \quad f_{11}(z)=f_{22}(z)={\cal O}(z^{-4})\,.
\label{eq:fab}
\end{align}
Thus, $\alpha=2$ in this model,
and the relaxation time of a medium surrounding the detector
is given by $\ell/2$\,.
Since an explicit functional form can be obtained for $\Delta G^+(x,x';\eta_0)$ in this model,
we can further determine the coefficients of the damping term $e^{-\alpha\,\tau}$
in Eq.~\eqref{delta_fdot_asympt}.

We comment that the instantaneous ground states $\ket{0_{\eta_0}}$
differ from the adiabatic vacuum state
\cite{Parker:PhD,Parker:1974qw} (see also \cite{Birrell:1982ix,PT})
at second adiabatic order
for finite $\eta_0$\,.
This rapid departure may be expected from the fact that
the condition of the validity of adiabatic approximation
[$\omega_k \gg \dot{\omega_k}/\omega_k$ with $\omega_k$ given by \eqref{omega_k}]
does not hold for some modes $\bk$ if $\eta_0$ is finite.
Actually, while the adiabatic vacuum state has the same asymptotic expansion
as the Bunch-Davies vacuum,
the wave function of an instantaneous ground state
agrees with that of the Bunch-Davies vacuum
only at the leading asymptotic order [see \eqref{wave-function_asympto}].
This observation then means that we cannot use the adiabatic subtraction scheme
to regularize the expectation value of the energy-momentum tensor,
$\langle T_{\mu\nu}\rangle$\,,
for instantaneous ground states.
In fact,
it is shown in \cite{Anderson:2000wx, Anderson:2005hi}
that, in order for a state in $d$-dimensional de Sitter space
to admit such adiabatic regularization of $\langle T_{\mu\nu}\rangle$\,,
the state must be of $d$-th or higher adiabatic order;
in other words, the wave function $\varphi_k(\eta)$ of the state must have the form
\begin{align}
 \varphi_k(\eta)=(-\eta)^{\frac{d-1}{2}}\,\bigl[c_1(k)\,H^{(1)}_\nu(-k \eta)
 +c_2(k)\,H^{(2)}_\nu(-k \eta) \bigr]
\end{align}
with $c_1(k)={\rm const.}+o(k^{-d})$ and $c_2(k)=o(k^{-d})$\,
\footnote{
For such states,
the exponents $\alpha_{ab}$ in \eqref{f_ab-alpha} are greater than $d$\,,
and thus the relaxation time will take a value less than $\ell/d$\,.
}
which is clearly different from the form given in \eqref{wave-function_asympto}.
We expect that one can yet use the point-splitting regularization
to remove divergences of $\langle T_{\mu\nu}\rangle$ for instantaneous ground states
although one then needs to establish an algorithm
to determine the finite part of $\langle T_{\mu\nu}\rangle$\,.
This issue is beyond the scope of this paper and is left for the future work.

\subsection{Wightman function for the instantaneous ground state}
\label{subsec:dS_Wightman}

In the Poincar\'e patch, there is a translational invariance in the spatial directions,
and we assume that all spatial directions are compactified with radius $L/2\pi$
\footnote{
The introduction of $L$ is for dealing with the zero mode carefully.
$L$ will be taken to infinity in the end. }
Then the wave vector $\bk$ takes discrete values
$\bk =(2\pi/L)\,\bn$ with $\bn\in\bZZ^{d-1}$\,.
We denote $\bk>0$ (or $\bk<0$)
if the first nonvanishing element of the vector
$\bk=(k_1,\,k_2,\,\ldots\,,k_{d-1})$ is positive (or negative).
Now, we consider the mode expansion
\begin{align}
 \phi(x) = \phi(\eta,\bx) = \sum_{\bk\geq0}\,\sum_{a} \phi_{\bk,a}(\eta)\, Y_{\bk,\,a}(\bx) \,,
\label{eq:mode-expansion}
\end{align}
where the mode functions $\bigl\{Y_{\bk,\,a}(\bx)\bigr\}$ are given by
\begin{align}
 \bk=0:\quad&Y_{\bk=0,\,a=1}\equiv\frac{1}{\sqrt{V}}\quad
  \bigl(V\equiv L^{d-1}\bigr)\,, \\
 \bk>0:\quad&
  Y_{\bk,\, a=1}(\bx)=\sqrt{\dsfrac{2}{V}}\,\cos(\bk\cdot\bx)\,,\quad
  Y_{\bk,\, a=2}(\bx)=\sqrt{\dsfrac{2}{V}}\,\sin(\bk\cdot\bx)\,,
\end{align}
which form a complete set of (real-valued)
eigenfunctions of the spatial Laplacian
$\Delta_{d-1}=\sum_{i=1}^{d-1}\partial_i^2$\,.
Then, the Wightman function \eqref{Wightman_X} is given by
\begin{align}
 G_X^+(x,x')
 &= \sum_{\bk\geq 0}\,G_{X,k}^+(\eta,\eta')\,
    \sum_a\,Y_{\bk,a}(\bx)\,Y_{\bk,a}(\bx')
\nn\\
 &= \frac{1}{(2\pi)^{\frac{d-1}{2}}\,\abs{\bx-\bx'}^{\frac{d-3}{2}}}\,
    \int_0^\infty \!\!\!\rmd k\,k^{\frac{d-1}{2}}\,
             J_{\frac{d-3}{2}}\bigl(k\,\abs{\bx-\bx'}\bigr)\,
    G_{X,k}^+(\eta,\eta')
\label{G+_general}
\end{align}
with
\begin{align}
 G_{X,k}^+(\eta,\eta')
 = \bra{0_{\eta_0}} \phi_{\bk,a}(\eta)\,\phi_{\bk,a}(\eta') \ket{0_{\eta_0}}\,,
  \label{G+_mode}
\end{align}
where we have taken the limit $L\to\infty$
and integrated over angular variables
in the second equality of Eq.~\eqref{G+_general}.

With the mode expansion \eqref{eq:mode-expansion},
the Hamiltonian (in the Heisenberg picture of free field theory)
is expressed as a sum of the Hamiltonians
for independent harmonic oscillators:
\begin{align}
 H(\eta) &= \sum_{\bk,a}
   \omega_{k}(\eta)\,a_{\bk,a}^\dag(\eta)\,a_{\bk,a}(\eta)
   + \mbox{const.}\,,
\\
 a_{\bk,a}(\eta) &\equiv \frac{(-\eta)^{-\frac{d-2}{2}}}{\sqrt{2}}\,\Bigl[\,
         \omega_{k}^{1/2}(\eta) \,\phi_{\bk,a}(\eta)
       +\ii\, \omega_{k}^{-1/2}(\eta)\,\dot{\phi}_{\bk,a}(\eta)\,\Bigr]\,,
\\
 \omega_k(\eta)&\equiv \sqrt{m^2 \,(-\eta)^{-2}+k^2}
\label{omega_k}
\,.
\end{align}
The instantaneous ground state $\ket{0_{\eta_0}}$
is then defined by the condition
that $a_{\bk,a}(\eta_0)\,\ket{0_{\eta_0}} = 0$
$(\forall\, \bk,\,a)$.

By expanding the operator $\phi_{\bk,a}(\eta)$
using the annihilation and creation operators $a_{\bk,a}(\eta_0)$ and $a^\dag_{\bk,a}(\eta_0)$
as
\begin{align}
 \phi_{\bk,a}(\eta) = \varphi_k(\eta;\eta_0)\,a_{\bk,a}(\eta_0)
                   +\varphi_k^*(\eta;\eta_0)\,a_{\bk,a}^\dag(\eta_0) \,,
\end{align}
the ground state wave function $\varphi_k(\eta;\eta_0)$
is given by (see \cite{Fukuma:2013mx})
\begin{align}
 \varphi_k(\eta;\eta_0)
  ={} -\frac{\pi}{2\sqrt{2\omega_k(\eta_0)}}\,
    \frac{(-\eta)^{\frac{d-1}{2}}}{(-\eta_0)^{-\frac{d-2}{2}}}\,
    \bigl[ v_0\,J_{\nu}(-k\,\eta) - u_0\,N_{\nu}(-k\,\eta) \bigr] \,,
\label{eq:wave-function}
\end{align}
where
$J_\nu(x)$ and $N_\nu(x)$ are the Bessel and Neumann functions, respectively,
and
$u_0$ and $v_0$ are given by
\begin{align}
 u_0
  &= -(-\eta_0)^{-\frac{d-1}{2}}\,
      \Bigl[\Bigl(\frac{d-1}{2}+\nu +\ii\omega_k(\eta_0)\,\eta_0\Bigr)\,
            J_{\nu}(-k\,\eta_0)
            +k\,\eta_0\,J_{1+\nu}(-k\,\eta_0)\Bigr] \,,
\\
 v_0
  &= -(-\eta_0)^{-\frac{d-1}{2}}\,
      \Bigl[\Bigl(\frac{d-1}{2}+\nu +\ii\omega_k(\eta_0)\,\eta_0\Bigr)\,
            N_{\nu}(-k\,\eta_0)
            +k\,\eta_0\,N_{1+\nu}(-k\,\eta_0)\Bigr] \,.
\end{align}
If we expand the wave function $\varphi_k(\eta;\eta_0)$
around $\eta_0=-\infty$,
we find that the Wightman function for each mode,
\begin{align}
 G_k^+(\eta,\eta';\eta_0)
   = \bra{0_{\eta_0}}\,\phi_{\bk,\,a}(\eta)\,\phi_{\bk,\,a}(\eta')\,\ket{0_{\eta_0}}
   = \varphi_k(\eta;\eta_0)\,\varphi^*_k(\eta';\eta_0) \,,
\label{mode_sum}
\end{align}
takes the following form (see appendix \ref{app:asympto_Wightman-function}):
\begin{align}
 &G^{+}_{k}(\eta,\eta';\eta_0)
\nn\\
 &= \frac{\pi}{4}\,\bigl[(-\eta)\,(-\eta')\bigr]^{\frac{d-1}{2}} \,
 \Bigl\{H^{(1)}_\nu(-k\,\eta)\,H^{(2)}_\nu(-k\,\eta')
\nn\\
 &\quad +\Bigl(\frac{d-2}{4}\Bigr)^2\,\frac{1}{(-k\,\eta_0)^2} \,
 \bigl[H^{(1)}_\nu(-k\,\eta)\,H^{(2)}_\nu(-k\,\eta')
               + H^{(2)}_\nu(-k\,\eta)\,H^{(1)}_\nu(-k\,\eta')\bigr]
\nn\\
 &\quad +
   e^{2\ii[k\,\eta_0+\frac{\pi(1+2\nu)}{4}]}\,\Bigl[\ii\frac{d-2}{4(-k\,\eta_0)}
      +\frac{\bigl(\nu^2-\frac{1}{4}\bigr)\,(d-3)
   -\bigl(\frac{d-2}{2}\bigr)^2-m^2}{4(-k\,\eta_0)^2}\Bigr]\,H^{(1)}_\nu(-k\,\eta)\,H^{(1)}_\nu(-k\,\eta')
\nn\\
 &\quad +
   e^{-2\ii[k\,\eta_0+\frac{\pi(1+2\nu)}{4}]}\,\Bigl[-\ii\frac{d-2}{4(-k\,\eta_0)}
      +\frac{\bigl(\nu^2-\frac{1}{4}\bigr)\,(d-3)
  -\bigl(\frac{d-2}{2}\bigr)^2-m^2}{4(-k\,\eta_0)^2}\Bigr] \,H^{(2)}_\nu(-k\,\eta)\,H^{(2)}_\nu(-k\,\eta')
\nn\\
 &\quad +\mathcal{O}((-k\,\eta_0)^{-3})\Bigr\} \,.
\label{Wightman_k_asympt}
\end{align}
Here, the first term, which can be written as
$\displaystyle\lim_{\eta_0\to -\infty}G^{+}_{k}(\eta,\eta';\eta_0)$\,,
is the Wightman function for the Bunch-Davies vacuum,
$G^{+}_{{\rm BD},k}(\eta,\eta')$\,.
The last two terms include rapidly oscillating factors
$e^{\pm 2\ii k\,\eta_0}$
and can be dropped from the expression
because they do not contribute to the integral in \eqref{G+_general}
for a sufficiently large $\abs{\eta_0}$\,.
We thus find that the Wightman function has the form
\begin{align}
 G_{X,k}^+(\eta,\eta';\eta_0)
 = G^+_{{\rm BD},k}(\eta,\eta')
  + \Delta G_{k}^+(\eta,\eta';\eta_0)
\label{eq:G+expansion}
\end{align}
wit
\footnote{
Terms of order $(-k\eta_0)^{-3}$ also disappear
since they necessarily include the rapidly oscillating factors. }
\begin{align}
 \Delta G_{k}^+(\eta,\eta';\eta_0)
 &= [(-\eta)(-\eta')]^{\frac{d-1}{2}}\, f(-k\,\eta_0)\,
\nn\\
 &\quad\times
 \bigl[H^{(1)}_\nu(-e^{\ii\varepsilon}\,k\,\eta)\,
 H^{(2)}_\nu(-e^{-\ii\varepsilon}\,k\,\eta')
\nn\\
 &\qquad
 +H^{(2)}_\nu(-e^{-\ii\varepsilon}\,k\,\eta)\,
  H^{(1)}_\nu(-e^{\ii\varepsilon}\,k\,\eta')\bigr]
 +\mathcal{O}\bigl( (-k\,\eta_0)^{-4}\bigr)\,,
\label{DeltaG_k}
\\
 f(z)&\equiv
 \frac{\pi}{4}\Bigl(\frac{d-2}{4}\Bigr)^2\,z^{-2} \,.
\label{f-alpha}
\end{align}
Here, an infinitesimal positive constant $\varepsilon$ is introduced
in order to make the $k$ integration in Eq.~\eqref{G+_general} finite
(see footnote \ref{fn:regularization} in appendix \ref{app:Wightman-function}).
Then, comparing \eqref{DeltaG_k} and \eqref{f-alpha} with \eqref{f_ab-alpha},
we find that $\alpha = 2$
and thus the relaxation time is $\ell/\alpha = \ell/2$\,.
Note that the reason why the relaxation time is given by $\ell/2$ (not by $\ell$)
is the disappearance of the order $(-k\,\eta_0)^{-1}$ terms from \eqref{Wightman_k_asympt}
due to highly oscillatory integrals.

We close this subsection
with a comment that the state $\ket{0_{\eta_0}}$ does not have a de Sitter invariance
since it introduces an extra scale $\eta_0$ as a cutoff
[even though the corresponding Wightman function
has the same short distance behavior
as $G_{\rm BD}^+(x,x')$].
This can be seen from the fact that $G^+(x,x';\eta_0)$
does not have a de Sitter invariant form
(see appendix \ref{app:Wightman-function}).

\subsection{Higher-order corrections in $\dot{\cF}$}
\label{subsec:dS_cFdot}

With the Wightman function \eqref{eq:G+expansion},
$\dot{\cF}$ defined in \eqref{fdot_P} takes the form
\begin{align}
 \dot{\cF}(\Delta E,\tau,\tau_1;\,\eta_0)
 &=\dot{\cF}_{\rm BD}(\Delta E;\tau,\tau_1)
   +\Delta\dot{\cF}(\Delta E;\tau,\tau_1;\,\eta_0)\,,
\label{eq:cF-expansion}
\end{align}
where $\dot{\cF}_{\rm BD}(\Delta E;\tau,\tau_1)$
and $\Delta\dot{\cF}(\Delta E;\tau,\tau_1;\,\eta_0)$
are given by the integrals \eqref{fdot_BD_int} and \eqref{delta_fdot_int},
respectively.
The integrals
can again be performed analytically as is done in appendix \ref{app:integration},
and we obtain
[see \eqref{BD_fdot_asympt} and \eqref{delta_fdot_asympt}
and recall that $\alpha=2$]
\begin{align}
 \dot{\cF}_{\rm BD}(\Delta E,\tau,\tau_1)
 &\sim \dot{\cF}^{\eq} (\Delta E)
 + {\rm const.}\, e^{-(\frac{d-1}{2}\pm\nu\pm\ii\Delta E)\,(\tau-\tau_1)} \,,
\label{eq:cF-0-tau1}
\\
 \Delta\dot{\cF}(\Delta E,\tau,\tau_1;\,\eta_0)
 &\sim e^{-2(\tau-\tau_0)}\,\Delta\dot{\cF}^{(0)} (\Delta E)
 +{\rm const.}\, e^{-(\frac{d-1}{2}\pm\nu\pm\ii\Delta E)\,(\tau-\tau_1)} \,,
\label{eq:cF-2-tau1}
\end{align}
where $\tau_0\equiv -\log(-\eta_0)$
and the coefficient of the leading term
in $\Delta\dot{\cF}(\Delta E,\tau,\tau_1;\,\eta_0)$ is given b
\footnote{
$ _3\widehat{F}_2$ is defined from
the generalized hypergeometric function $_3F_2$
(see appendix \ref{app:G-function}) as
\begin{align}
 \, _3\widehat{F}_2\Bigl(
  \afrac{a_1,\, a_2,\, a_3}
                     {b_1,\, b_2};\, z\Bigr) \equiv
 \frac{\Gamma(a_1)\,\Gamma(a_2)\,\Gamma(a_3)}{\Gamma(b_1)\,\Gamma(b_2)}\,\, _3F_2\Bigl(
  \afrac{a_1,\, a_2,\, a_3}
                     {b_1,\, b_2};\, z\Bigr) \,.
\nn
\end{align}
}
\begin{align}
\Delta\dot{\cF}^{(0)} (\Delta E)&=\Bigl(\frac{d-2}{4}\Bigr)^2
\frac{\ii}{2^{5}\pi^{\frac{d-1}{2}}\Gamma(\frac{d-1}{2})}
\nn\\
&\quad\times\Biggl\{
 e^{\ii\pi\,\frac{d-2}{2}}\,\Biggl[
  \frac{e^{\ii \pi\,\nu}}{\sin (\pi\,\nu)}
     \, _3\widehat{F}_2\Biggl(
         \afrac{\frac{d-3}{2},\,
                             \frac{d-3}{2}+\nu,\,
                             \frac{\frac{d-1}{2}+\nu-2-\ii\Delta E}{2}}
                            {1+\nu,\,
                             \frac{\frac{d+3}{2}+\nu-2-\ii\Delta E}{2}};\,
                             e^{-\ii0}\Biggr)
\nn\\
  &\qquad\qquad\qquad + \frac{e^{-\ii\pi\,\nu}}{\sin (-\pi\,\nu)}
     \, _3\widehat{F}_2\Biggl(
         \afrac{\frac{d-3}{2},\,
                             \frac{d-3}{2}-\nu,\,
                             \frac{\frac{d-1}{2}-\nu-2-\ii\Delta E}{2}}
                            {1-\nu,\,
                             \frac{\frac{d+3}{2}-\nu-2-\ii\Delta E}{2}};\,
                             e^{-\ii0}\Biggr) \Biggr]
\nn\\
 &\quad\quad\quad +e^{-\ii\pi\,\frac{d-2}{2}}\,\Biggl[
  \frac{e^{-\ii \pi\,\nu}}{\sin (\pi\,\nu)}
     \, _3\widehat{F}_2\Biggl(
         \afrac{\frac{d-3}{2},\,
                             \frac{d-3}{2}+\nu,\,
                             \frac{\frac{d-1}{2}+\nu-2-\ii\Delta E}{2}}
                            {1+\nu,\,
                             \frac{\frac{d+3}{2}+\nu-2-\ii\Delta E}{2}};\,
                             e^{\ii0}\Biggr)
\nn\\
  &\qquad\qquad\qquad + \frac{e^{\ii\pi\,\nu}}{\sin (-\pi\,\nu)}
     \, _3\widehat{F}_2\Biggl(
         \afrac{\frac{d-3}{2},\,
                             \frac{d-3}{2}-\nu,\,
                             \frac{\frac{d-1}{2}-\nu-2-\ii\Delta E}{2}}
                            {1-\nu,\,
                             \frac{\frac{d+3}{2}-\nu-2-\ii\Delta E}{2}};\,
                             e^{\ii0}\Biggr) \Biggr]
\Biggr\}
\nn\\
&\quad\quad\quad
+(\Delta E \to -\Delta E)\,.
\label{eq:Fdot2}
\end{align}

\subsection{Thermalization of a two-level detector in de Sitter space}
\label{sec:thermal}

As we discussed in the preceding subsections,
if we take the initial state for the scalar field
to be the instantaneous ground state at a finite past,
there exists a damping term in $\dot{\cF}$ with the relaxation time $\ell/2$.
This behavior of $\dot{\cF}$ is expected
to represent the relaxation of the surrounding medium.
In this subsection, as an analytically tractable example,
we consider the case where the detector is a two-level system
with energy eigenvalues $E_1$ and $E_2$ with $\Delta E \equiv E_2-E_1>0$\,,
and describe how the density distribution of the detector
approaches the Gibbs distribution with relaxation time $\ell/2$\,.

To proceed the analysis, we set the following assumptions:
\makeatletter
\renewcommand{\labelenumi}{(\roman{enumi})}
\renewcommand{\theenumi}{(\roman{enumi})}
\makeatother
\begin{enumerate}
\item
The initial distribution of the detector, $\rho(\tau_1)$,
is averaged over the initial proper time $\tau_1$
for the duration $\Delta\tau\gtrsim 1/\Delta E$\,.
This is for describing distributions at different energy levels to a good accuracy.
As a consequence, the off-diagonal elements of $\rho(\tau_1)$
can be effectively set to zero
because they are oscillatory during the time.
\label{enum:ave}
\item
The energy difference should be much larger than the natural energy scale
of de Sitter space, $\Delta E \gg \ell^{-1} = 1$\,,
in order for relaxation modes with relaxation times of order $\ell$
to be observed.
\label{enum:DeltaE}
\item The operator $\mu$ is off-diagonal with respect to
the basis $\ket{m}$ $(m=1,2)$,
\begin{align}
 \mu = \bigl( \bra{m}\mu\ket{n} \bigr)
  =\begin{pmatrix}
    0 & \mu_{12} \cr
    \mu_{12}^\ast & 0
   \end{pmatrix}\,.
\end{align}
This is simply for making the following analysis much easier.
\label{enum:mu}
\end{enumerate}
\makeatletter
\renewcommand{\labelenumi}{\arabic}
\renewcommand{\theenumi}{\arabic}
\makeatother

One can easily show that the master equation \eqref{dense} decomposes
into the diagonal and off-diagonal parts
under assumption \ref{enum:mu}.
The off-diagonal part is given by linear differential equations
of the form
\begin{align}
 \begin{pmatrix}
  \dot{\rho}_{12}(\tau) \cr \dot{\rho}_{21}(\tau)
 \end{pmatrix}
 &= \mathcal{M}(\tau,\tau_1)\,
   \begin{pmatrix}
    \rho_{12}(\tau) \cr \rho_{21}(\tau)
   \end{pmatrix}\,,
\\
 \mathcal{M}(\tau,\tau_1)
 &\equiv
 \begin{pmatrix}
  \ii\Delta E-\abs{\mu_{12}}^2\,\dot{\cF}(0;\,\tau,\,\tau_1;\,\eta_0) &
  \mu_{12}^2\,\dot{\cF}(0;\,\tau,\,\tau_1;\,\eta_0) \cr
  \mu_{12}^{\ast\,2}\,\dot{\cF}(0;\,\tau,\,\tau_1;\,\eta_0) &
  -\,\ii \Delta E- \abs{\mu_{12}}^2\,\dot{\cF}(0;\,\tau,\,\tau_1;\,\eta_0)
 \end{pmatrix}\,,
\end{align}
which can be integrated to
\begin{align}
 \begin{pmatrix}
  \rho_{12}(\tau) \cr \rho_{21}(\tau)
 \end{pmatrix}
 ={\rm T}\exp\Bigl(\int_{\tau_1}^\tau\!\!\rmd \tau'\,\mathcal{M}(\tau',\tau_1)\Bigr)
  \begin{pmatrix}
  \rho_{12}(\tau_1) \cr \rho_{21}(\tau_1)
 \end{pmatrix}.
\end{align}
Since we can effectively set $\rho_{12}(\tau_1)=\rho_{21}(\tau_1)=0$
due to assumption \ref{enum:ave},
we can also set $\rho_{12}(\tau)=\rho_{21}(\tau)=0$
[assuming that $\mathcal{M}(\tau',\tau_1)$ changes slowly
when averaging over $\tau_1$].

The diagonal part can then be obtained from \eqref{master} and \eqref{wmk} as
\begin{align}
 \begin{pmatrix}
  \dot{\rho}_{11}(\tau) \cr \dot{\rho}_{22}(\tau)
 \end{pmatrix}
 =
 \begin{pmatrix}
  -w_{21}(\tau,\tau_1;\eta_0) & w_{12}(\tau,\tau_1;\eta_0) \cr
  w_{21}(\tau,\tau_1;\eta_0) & -w_{12}(\tau,\tau_1;\eta_0)
 \end{pmatrix}
 \begin{pmatrix}
  \rho_{11}(\tau) \cr \rho_{22}(\tau)
 \end{pmatrix}
\end{align}
with
\begin{align}
 w_{m k}(\tau,\tau_1;\eta_0)=\lambda^2\,\abs{\mu_{mk}}^2\,
 \dot\cF(E_m-E_k; \tau,\tau_1; \eta_0)\,.
\label{wmk_reduced}
\end{align}
Using Eqs.~\eqref{eq:cF-expansion}, \eqref{eq:cF-0-tau1} and \eqref{eq:cF-2-tau1}
together with assumptions \ref{enum:ave}
and \ref{enum:DeltaE},
one can easily show that $\dot{\cF}$ comes to take the following form
after averaging over $\tau_1$\,:
\begin{align}
 \dot{\cF}_{\rm av}(\Delta E;\tau;\eta_0)
 &\equiv \frac{1}{\Delta\tau}\int^{\tau_1+\Delta\tau}_{\tau_1}\!\!\rmd \tau^{\prime}_1\,
 \dot{\cF}(\Delta E;\tau,\tau^{\prime}_1;\eta_0)
\nn\\
 &=\dot{\cF}^{\eq}(\Delta E) +e^{-2(\tau-\tau_0)}\,\Delta\dot{\cF}^{(0)}(\Delta E)
 + \mathcal{O}(e^{-4(\tau-\tau_0)})\,,
\label{eq:average}
\end{align}
where $\dot{\cF}^{\eq}(\Delta E)$ and $\Delta\dot{\cF}^{(0)}(\Delta E)$ are
given by \eqref{eq:Fdot-eq} and \eqref{eq:Fdot2}.
Note that the uninteresting $\tau_1$-dependent terms
proportional to $e^{-(\frac{d-1}{2}\pm\nu\pm\ii\Delta E)\,(\tau-\tau_1)}$
have totally disappeared from $\dot{\cF}$ due to the averaging procedure
[see \eqref{eq:cF-0-tau1} and \eqref{eq:cF-2-tau1}].
Replacing $\dot\cF(\Delta E; \tau,\tau_1;\eta_0)$ in \eqref{wmk_reduced}
by $\dot\cF_{\rm av}(\Delta E; \tau;\eta_0)$,
we obtain the master equation for the diagonal elements of the form
\begin{align}
\begin{pmatrix}
 \dot{\rho}_{11}(\tau) \cr
 \dot{\rho}_{22}(\tau)
 \end{pmatrix}
&=
\begin{pmatrix}
 -w_{+}(\tau;\tau_0) &w_{-}(\tau;\tau_0)\cr
 w_{+}(\tau;\tau_0) &-w_{-}(\tau;\tau_0)
\end{pmatrix}
\begin{pmatrix}
 \rho_{11}(\tau) \cr
 \rho_{22}(\tau)
\end{pmatrix}
\label{2level-master}
\end{align}
with
\begin{align}
 w_{\pm}(\tau;\tau_0) &\equiv
 \lambda^2\,\abs{\mu_{12}}^2\,\bigl[
 \dot{\cF}^{\eq}(\pm\Delta E) +e^{-2(\tau-\tau_0)}\,\Delta\dot{\cF}^{(0)}(\pm\Delta E)
 +{\cal O}\bigl(e^{-4(\tau-\tau_0)}\bigr)
 \bigr]\,.
\end{align}

From a general argument following \eqref{eq:Fdot-eq},
we have already seen that, as $\tau$ becomes large,
the density distribution $\rho_{mm}(\tau)$ approaches the Gibbs distribution
at temperature $1/2\pi\ell$\,,
\begin{align}
 \rho^{\eq}_{mm}
 = \frac{e^{-2\pi\ell E_m}}{Z} \quad
 \bigl(Z=e^{-2\pi\ell E_1}+e^{-2\pi\ell E_2}\bigr)\,.
\end{align}
In order to investigate how the detector relaxes to this equilibrium,
we expand $\rho_{mm}(\tau)$ as
$\rho_{mm}(\tau)\equiv \rho^{\eq}_{mm}+\Delta\rho_{mm}(\tau)$
and keep small quantities to the first order,
assuming that $\Delta\rho_{mm}/\rho^{\eq}_{mm}\ll 1$
and $e^{-2(\tau-\tau_0)}\ll 1$\,.
Then \eqref{2level-master} becomes
\begin{align}
 \frac{\rmd}{\rmd\tau}
 \Delta\rho_{11}(\tau)
 &=-\lambda^2\,\abs{\mu_{12}}^2\,\bigl[\,
 e^{-2(\tau-\tau_0)}\,\Delta\dot{\cF}^{(0)}(\Delta E)\,\tanh(\pi\Delta E)
\nn\\
 &\qquad\qquad\qquad  +\bigl(\dot{\cF}^{\eq}(\Delta E)+\dot{\cF}^{\eq}(-\Delta E)\bigr)\,\Delta\rho_{11}(\tau)\,
\bigr]\,.
\label{master11}
\end{align}
We used the fact that
$\Delta\dot{\cF}^{(0)}(\Delta E)$ is an even function of $\Delta E$
[see \eqref{eq:Fdot2}]
and $\Delta\rho_{11}=-\Delta\rho_{22}$\,.
This equation can be solved easily,
and we find that the relaxation behavior of the density distribution
is given by
\begin{align}
 &\Delta\rho_{11}(\tau)
\nn\\
 &=e^{-\lambda^2\,\abs{\mu_{12}}^2\,A(\Delta E)\,(\tau-\tau_2)}
 \Delta\rho_{11}(\tau_2)
\nn\\
 &\quad-\frac{\lambda^2\,\abs{\mu_{12}}^2\,\Delta\dot{\cF}^{(0)}(\Delta E)}
         {\lambda^2\,\abs{\mu_{12}}^2\,A(\Delta E)-2}
 \,\tanh(\pi\Delta E)
 \bigl( e^{-2(\tau-\tau_2)}
 -e^{-\lambda^2\,\abs{\mu_{12}}^2\,A(\Delta E)\,(\tau-\tau_2)-2(\tau_2-\tau_0)}
 \bigr) \,.
\label{relax}
\end{align}
Here,
\begin{align}
 A(\Delta E)\equiv \dot{\cF}^{\rm eq}(\Delta E)+\dot{\cF}^{\rm eq}(-\Delta E) \,,
\label{eq:A-def}
\end{align}
and $\tau_2$ is an arbitrary proper time after $\tau_1$ $(\tau_2>\tau_1)$,
where the linear approximation is well justified.

We note that the function $A(\Delta E)$
has the following asymptotic form for large $\abs{\Delta E}$\,:
\begin{align}
 A(\Delta E) \sim \frac{\abs{\Delta E}^{d-3}}
                       {2^{d-2}\,\pi^{\frac{d-3}{2}}\,\Gamma\bigl(\frac{d-1}{2}\bigr)}
 \qquad \bigl(\,\abs{\Delta E}\to \infty\bigr)\,,
\label{eq:A-DeltaE}
\end{align}
as can be easily shown
by using the asymptotic form of the Gamma function (8.328-1 of \cite{GR}),
\begin{align}
 \lim_{\abs{y}\to \infty} \abs{\Gamma(x+\ii y)}
 \sim \sqrt{2\pi}\,e^{-\frac{\pi}{2}\,\abs{y}}\,
 \abs{y}^{x-\frac{1}{2}}\quad (x,\,y\in\mathbb{R})\,.
\end{align}
We now consider a detector that satisfies the inequality
\begin{align}
 \lambda^2\,\abs{\mu_{12}}^2\,(\Delta E)^{d-3}\,\ell\gg 1 \,.
\label{eq:ideal-assumption}
\end{align}
Then, by using the asymptotic form \eqref{eq:A-DeltaE},
$A(\Delta E)\sim \abs{\Delta E}^{d-3}$\,,
and the inequality \eqref{eq:ideal-assumption},
the damping terms in \eqref{relax}
which are proportional to
$e^{-\lambda^2\,\abs{\mu_{12}}^2\,A(\Delta E)\,(\tau-\tau_2)}$
rapidly disappear from the expression,
leaving only the term proportional to $e^{{}-2(\tau-\tau_2)}$\,,
\begin{align}
 \Delta\rho_{11}(\tau) \sim {}
 -\frac{\Delta\dot{\cF}^{(0)}(\Delta E)}
         {A(\Delta E)}
 \,\tanh(\pi\ell\Delta E)\,e^{-2(\tau-\tau_2)/\ell}\,,
\label{eq:universal}
\end{align}
where we have restored the curvature radius $\ell$\,.
Note that the coefficient $\Delta\dot{\cF}^{(0)}(\Delta E)/A(\Delta E)$
does not depend on details of the detector
(such as $\lambda \,\mu_{12}$).
The remaining damping term in \eqref{eq:universal}
corresponds to the desired relaxation mode
with the relaxation time $\ell/2$\,.
Since only those rapidly disappearing terms depend on details of detector,
one may say that such detector under consideration is an {\em ideal detector},
in the sense that
it quickly loses its own nonequilibrium properties
and gets adjusted to its environment almost instantaneously.

\section{Conclusion and discussions}
\label{sec:conclusion}

In this paper, we have considered an Unruh-DeWitt detector
staying in the Poincar\'e patch of de Sitter space.
The main difference of our setup from those in the literature
is in that the scalar field
(before interacting with the detector)
is not in the Bunch-Davies vacuum (nor in the $\alpha$-vacuum).
Then the Unruh-DeWitt detector behaves
as if it is in a nonequilibrium environment.
In order to deal with such situations,
we first derived the master equation
which describes a finite time evolution of
the density matrix of an Unruh-DeWitt detector in arbitrary geometry.

We then applied the framework to de Sitter space.
We showed that there exists a damping term in $\dot{\cF}$
with a relaxation time of the form $\ell/\alpha$
if the initial state of scalar field
is chosen such that the Wightman function
takes the form \eqref{Wightman-decomposition} with \eqref{eq:DeltaG+} and \eqref{f_ab-alpha}.
In particular,
if we take the initial state to be the instantaneous ground state at a finite past,
the relaxation time is always given by $\ell/2$\,.
We further gave an explicit description of the relaxation process
for a two-level detector.

We here should stress again that we are not considering just
the thermalization process of a detector dipped in a thermal bath.
In fact, since the initial state is chosen
to be different from the Bunch-Davies vacuum,
the detector should initially behave as if it is in a medium
which is not in thermodynamic equilibrium.
As time goes on, the detector comes to behave as if it is in a thermal bath,
since the difference of the initial state from the Bunch-Davies vacuum
becomes irrelevant at later times
(i.e., $\Delta\dot{\cF}/\dot{\cF}_{\rm BD} \ll 1$ at later times).

In reality, there can be many relaxation processes for such a detector,
some of which are simply the processes
where the detector gets adjusted to its environment.
However, these processes usually depend on details of the detector,
and thus can be neglected
by considering an ideal detector
which quickly responds to changes in its environment.
The terms proportional to
$e^{-\lambda^2\,\abs{\mu_{12}}^2\,A(\Delta E)\,\tau}$
in \eqref{relax} actually represent such processes.
On the other hand, there are terms including the factors
$e^{-(\frac{d-1}{2}\pm{\rm Re}\,\nu)\,\tau}$ in $\dot{\cF}$.
This kind of terms always exist
even when the initial state is the Bunch-Davies vacuum,
and thus do not have relevance to the relaxation of the nonequilibrium medium
depicted in Fig.~\ref{fig:medium} (b).
For the case of two-level detector considered in section \ref{sec:thermal},
such damping terms disappear from the expression
after taking an average over the start-up time $\tau_1$\,.
There is also a damping term proportional to $e^{-\alpha\,\tau/\ell}$
[see \eqref{delta_fdot_asympt}],
which appears only when the initial state is different from the Bunch-Davies vacuum
and thus describes the nonequilibrium dynamics of the surrounding medium.
In particular, the relaxation time takes a universal value $\ell/2$
(i.e.\ $\alpha=2$)
for the instantaneous ground states at finite pasts.
We expect that the relaxation time $\ell/2$
gives a quantity representing the nonequilibrium thermodynamic character
intrinsic to de Sitter space,
just as the temperature of the final Gibbs distribution
represents the Gibbons-Hawking temperature intrinsic to de Sitter space.

In the previous studies,
thermal properties of de Sitter space
have been investigated mainly in the context of equilibrium thermodynamics.
The relaxation processes discussed in this paper
may serve as examples related to the nonequilibrium dynamics
of de Sitter space.

\section*{Acknowledgments}

We thank Y.~Hamada, H.~Ishimori, T.~Kameyama, H.~Kawai,
K.~Murase, K.~Oda and A.~Ogasahara
for useful discussions on the Unruh-DeWitt detector.
This work was supported by the Grant-in-Aid for the Global COE program
``The Next Generation of Physics, Spun from Universality and
Emergence" from the Ministry of Education, Culture, Sports,
Science and Technology (MEXT) of Japan.
This work was also supported by MEXT (Grant No.\,23540304).

redpajama_set_name: RedPajamaArXiv


text:
Home entertainment New Netflix series for Benedict Cumberbatch
New Netflix series for Benedict Cumberbatch
Good news for all Benedict Cumberbatch fans. "Sherlock" Star has a new, action-packed thriller series for us: "39 Steps".
Finally there is a new thriller series with Benedict Cumberbatch. Because the "Sherlock" star is now daring to try another classic: "39 Steps". After the inclusion of Sherlock Holmes in the Netflix series of the same name since 2010, the actor is now embarking on a new project based on a book. Again, this is about investigation and an action-packed thriller series, as the series is considered 'loud'tvline.comTo be a remake of "39 Steps". John Buchan's book was first filmed in 1935 by none other than Alfred Hitchcock. Now it should shine on the screen again in the modern version. Benedict Cumberbatch will serve as the executive producer of the mini-series. It is currently unknown whether the actor will also appear in front of the camera. This will not be the first time, however, that the "Sherlock" actor takes a role in front of and behind the camera at the same time.
"39 Steps": This is about Benedict Cumberbatch's new series
Until now, there is not a lot of information about the new minaret "39 Steps" starring Benedict Cumberbatch. Hitchcock is likely to be based in Scotland based on the feature film and the original book. Here we meet Hane, who is accused of killing a secret agent. He wants to prove his innocence and during his investigation a young woman arrives who wants to uncover a secret spy ring with him. Will they be able to clear his name? Mark L. Smith knows that, because the author of "The Revenant" also wrote the screenplay for "39 Steps". Benedict Cumberbatch has also brought in a famous "Patrick Melrose" director, Edward Berger, to direct. There is no more information about the artists. Nor which streaming service gets the rights and promising series are allowed to air. However, one great contender is Netflix. We are already very excited and look forward to Benedict Cumberbatch's "The 39 Steps".
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text:
Cory Booker Says Joe Biden 'Took A Step Towards Healing' By Apologizing For Remarks About Segregationists
Drew Angerer / Getty Images
Democrat Cory Booker praised on Saturday fellow presidential candidate Joe Biden for apologizing about remarks he had made about pro-segregation senators, Mediaite reports.
"I'm frustrated that it took so long," Booker said, proceeding to thank Biden for apologizing.
"I'm grateful for him doing it — I mean, we can't have a culture where — we can't have a leader that can't stand up and say 'I've been imperfect and made mistakes and I apologize,'" the New Jersey Democrat said.
"He took a step towards healing today," he added.
"We need to extend grace to each other. I'm never going to not accept somebody I respect and admire that has come to terms with this and apologized."
As Mediaite notes, Booker was one of the first Democrats to condemn Biden's comments about pro-segregation senators.
As previously reported by The Inquisitr, former Vice President Joe Biden has come under scrutiny for touting his ability to work with segregationist Republicans during what he claims was an era of "civility."
During a June donor dinner, Biden touted his friendships with segregationist senators James O. Eastland of Mississippi and Herman Talmadge of Georgia.
This is not out of the ordinary for the Delaware Democrat, it seems, given that a recently unearthed video shows him praising notorious segregationist Strom Thurmond as "one of my closest friends."
Biden's past opposition to desegregation busing and complimentary remarks about pro-segregation senators served as the basis of the attack California Senator Kamala Harris launched at him during the first Democratic primary debate. Harris' attack not only caused her to surge in the polls, but prompted others to criticize the former vice president.
Biden apologized on Saturday for his remarks about pro-segregation senators.
"I'm sorry for the pain and misconception," he said.
The Delaware Democrat has issued a slew of apologies since officially announcing his candidacy, including an attempted apology to Anita Hill.
Joe Biden apologizes for praising segregationist senators, invokes Obama in defending his civil rights recordhttps://t.co/hvyICF9gSA
— Newsweek (@Newsweek) July 7, 2019
Although his lead has dropped, Biden is still the Democratic front-runner, leading in virtually every single poll.
According to The Washington Examiner, on the campaign trail, Biden is doing damage control and relying on his relationship with former President Barack Obama to counter the attacks from other Democrats in the race.
During a campaign stop in South Carolina, Biden argued that other presidential candidates are focusing on the early days of his political career in order to avoid talking about his time as Obama's vice president.
As HuffPost notes, no Democrat has directly attacked the Obama-Biden legacy, which perhaps comes as no surprise given how popular Obama still is among Democratic voters.
Andrew Yang Jumps From Zero To 4.9 Percent In New Hampshire Poll July 16, 2019
Loren Lott Reacts To Inclusion In New 'The Young And The Restless' Opening July 16, 2019


If you're gonna use a red so bold it looks like fire, your product better deliver. The brand new for 2010 Dainese Red Line Racing Suit absolutely does. This suit is across-the-board quality and, in our opinion, primo at its $1499 price. Hit the track!
Bursting out like a hot ball of lead, the 2010 Dainese Red Line is packed with features. The video above digs really deep into what's got us going about the suit, so I'll just scratch the surface here.
This is in Dainese's upper echelon of motorcycle racing suits, which means no expense was spared. First off, D-Skin leather, Dainese's proprietary leather, is a massive safety upgrade over traditional cowhide. Using special silicone and wax treatments, the leather has a much softer and more supple feel. More important is the protection: 60% greater tensile strength (lateral tear) and 80% greater abrasion-resistance (road-sliding) than regular cowhide.
That's a pretty serious upgrade.
The next major feature is the 3D Bubble Liner, which creates air pockets between the three dimensional extrusions in the liner. This in turn creates air pockets resulting in microclimate at your core. The body's desired temperature will then be maintained, insulating you from outside hot or cold variables.
For ease of getting in and out, Dainese utilized a W-frame construction, with double zips at the chest, offering added comfort and flexibility. On the same note, S1 Stretch Panels can be found throughout the suit for on-bike mobility. Also, you'll have co-injected titanium sliders on the shoulders with Dainese's knee sliders atop the standard CE rated internal armor. I almost forgot, it also has a water bag built into the speed hump. Like I said, across-the-board quality.
We're super-high on this suit based on all the features and functionality we see at this price point.
If you want to compare their line-up, check out all the Dainese Racing Suits at RevZilla. We'll be rolling out reviews of them shortly.


text:
Remaining Aligned on the Challenges Facing Taiwan
Michael D. Swaine, Ryo Sahashi
Alliance Policy Coordination Brief
Summary: Washington and Tokyo should proactively keep common ground amid rising tensions between mainland China and Taiwan.
This brief is part of the China Risk and China Opportunity for the U.S.-Japan Alliance project.
Issue Background
Taiwan is the most sensitive and dangerous security issue influencing the relationship between China and the U.S.-Japan alliance. The continued stability of the bilateral Sino-U.S. relationship, as well as China's productive relations with Japan and other U.S. allies, has largely depended on the maintenance of an explicit understanding on Taiwan reached when Beijing and Washington normalized diplomatic ties in 1979. At its core was a specific quid pro quo in which the United States, Japan, and others recognized the People's Republic of China (PRC) as the sole legitimate government of China, while acknowledging or respecting the Chinese position that Taiwan is a part of China. In return, Beijing dropped its long-standing stance of liberating Taiwan by force in favor of a primary stress on peaceful unification, regarding force as a last resort.
For this understanding to remain a basis for stability will require the United States, Japan, and other nations to continue limiting their interactions with and support for Taiwan to the unofficial, nondiplomatic arena and for these countries to remain open to peaceful reunification. As a statement of policy, Washington refuses to support any unilateral, non-agreed-upon efforts by either Beijing or Taipei to alter the status quo regarding Taiwan toward either formal independence or unification. Alongside this understanding with China, the United States, as part of the Taiwan Relations Act (TRA), is required to maintain its capacity to assist the island if coerced or attacked by China and to sustain Taiwan's self-defense capabilities. The United States is not obligated, however, to protect Taiwan from Chinese military attack. That said, under the TRA, any U.S. administration is obliged to consult with Congress on the appropriate response if it is deemed that Taiwan faces a serious security threat.
For its part, Beijing is expected to continue to seek a peaceful resolution to the Taiwan issue through negotiations and to avoid any actions that would coerce or seriously threaten Taiwan. The Chinese government passed an anti-secession law in 2005 that provides conditions under which China might employ non-peaceful means to prevent the permanent separation of Taiwan from mainland China. Article 8 of the law states:
Michael D. Swaine
Swaine was a senior fellow at the Carnegie Endowment for International Peace and one of the most prominent American analysts in Chinese security studies.
@Dalzell60
In the event that the "Taiwan independence" secessionist forces should act under any name or by any means to cause the fact of Taiwan's secession from China, or that major incidents entailing Taiwan's secession from China should occur, or that possibilities for a peaceful reunification should be completely exhausted, the state shall employ non-peaceful means and other necessary measures to protect China's sovereignty and territorial integrity.
Nonetheless, Beijing has also repeatedly reaffirmed its commitment to peaceful unification as a priority, without relinquishing its sovereign right to employ force if necessary.
The Stakes
These understandings and commitments pose risks for the alliance relationship largely because of the high stakes involved on both sides and the clear possibility for miscalculation and overreaction. For the United States, the credibility of its commitments to a friendly (albeit not allied) power is clearly at stake. Tokyo and others would likely view the U.S. abandonment of Taiwan as a weakening of the U.S. commitment to its allies and an indication of Washington's willingness to concede the dominant power position in Asia to Beijing.
Ryo Sahashi
Ryo Sahashi is an associate professor of international politics at the Institute for Advanced Studies on Asia at the University of Tokyo.
As for Tokyo, although the Japanese Diet has not passed any laws equivalent to the TRA to specify the country's commitment or interests on Taiwan, Japanese leaders have repeatedly stressed the need for the issue to be handled peacefully. In addition, the peaceful management of the Taiwan issue certainly involves Japan's security, given Tokyo's almost certain involvement in any future armed crisis or conflict over the island in support of U.S. forces based on or deployed from the Japanese islands. The United States could not defend Taiwan without the consent of the Japanese government to use U.S. bases in Japan and without logistical and medical (rear-area) support from the Japanese Self-Defense Forces.
For China, the nationalist credentials of the PRC government and hence the legitimacy of the regime in the eyes of the Chinese people are dependent on the leadership's ability to prevent the permanent separation of Taiwan from the mainland and to achieve eventual unification. And, of course, all the relevant powers have a high stake in avoiding a conflict that could undermine regional peace and prosperity.
The dangers of miscalculation and overreaction derive from several factors, including:
heightened Sino-U.S. distrust: the high and arguably growing level of overall distrust between Beijing and Washington today, which increases all sides' tendency to interpret others' motives and behaviors in the worst possible light;
the specter of Chinese coercion: China's growing military and economic abilities to pressure or threaten Taiwan;
Taiwan's unsettled domestic politics: the increasingly unpredictable, fragmented nature of Taiwan's domestic politics, which adds uncertainty over Taipei's commitment to avoiding provocative behavior; and
hardening allied views of China: the emergence of views among some U.S. and Japanese defense analysts that appear inconsistent with the long-standing One China policy approach of both governments.
Any of these factors, or a combination thereof, could cause Beijing, Washington, or Tokyo to take actions viewed by others as unacceptably provocative, or to generate crisis-inducing overreactions to real or imagined threats.
In recent years, and especially since Chinese President Xi Jinping and U.S. President Donald Trump came to power, several developments have enhanced the saliency of the aforementioned negative or potentially destabilizing factors, thereby increasing the dangers and complexities that the Taiwan issue presents for the U.S.-Japan alliance and overall relations between China, Japan, and the United States. These developments include:
The first factor is the political weakening of the pro–One China Kuomintang (KMT) and the commensurate strengthening of the pro-independence Democratic Progressive Party (DPP) in Taiwan; alongside this trend, Beijing's economic leverage over Taipei is deepening, public support in Taiwan for unification is declining, and popular resentment toward mainland China on the island is arguably growing, partly as a result of increasing cross-strait contacts.
The second consideration is an appreciable hardening of Beijing's stance toward Taipei, largely as a result of the election of DPP stalwart Tsai Ing-wen to the presidency and the emergence of a more assertive Chinese leader (Xi). This hardening is reflected in several developments, including decreased cross-strait political contacts (outside of the KMT), harsh Chinese criticism of Tsai's government, Chinese efforts to compel foreign airlines to drop references to Taiwan as a country, unprecedented Chinese military displays near Taiwan, and Beijing's successful efforts to poach a few of Taipei's remaining diplomatic allies.
Third, there are growing concerns in U.S. and Japanese defense circles over the declining capacity of both countries to deter or defeat in a timely fashion any future possible Chinese military action against Taiwan. Over the past two decades, Beijing has acquired the capacity to place at high risk any U.S. or Japanese forces that might attempt to intervene in a cross-strait military conflict.
Fourth, the United States has engaged in some rash, unpredictable moves that call into question continued U.S. support for the One China policy, including remarks and actions by Trump questioning the continued suitability of the policy, and various congressional statements and actions, including the Taiwan Travel Act and the Asia Reassurance Initiative Act, that call for greatly enhanced U.S.-Taiwan contacts that Beijing could interpret as violations of the understanding between the two countries regarding Taiwan.
Finally (and closely related to the previous two factors), there is an overall deepening hostility toward China within some U.S. and Japanese political circles, including a tendency to view Beijing as a supposedly existential threat committed to splitting the alliance, driving the United States out of Asia, and dominating the region. This mentality arguably increases the tendency of some U.S. and Japanese analysts to regard Taiwan as a strategic asset to be kept separate from China in the struggle to counter and contain Beijing's growing regional power and influence. Such a viewpoint, if supported by U.S. (and possibly some Japanese) policy moves, could lead Beijing to conclude that Washington (and Tokyo) has abandoned the One China policy.
In recent years, partly to cope with the resulting potential dangers of these developments, and to enhance the overall capabilities of the alliance, Tokyo and Washington have worked individually and in concert to improve U.S.-Japan interoperability, enhance maritime and air capabilities, and strengthen their electronic warfare systems and their command, control, communications, computer, intelligence, surveillance, and reconnaissance systems, among other actions. The two partners have also worked to create a more certain level and scope of Japanese support for U.S. forces in a potential Taiwan conflict.
Beyond this, Washington has continued to offer defense systems to Taiwan, maintain or enhance military-to-military consultations with Taipei, and at times reiterate the basic tenets of U.S. policy toward Taiwan, including both the One China policy and the obligations set forth in the TRA.
Unfortunately, as indicated above, on the political and diplomatic front, the Trump administration has sent confusing messages to both Taipei and Beijing. For instance, the administration at times has seemingly questioned the meaning and continued relevance of the One China policy and apparently endorsed ill-advised congressional actions, such as the Taiwan Travel Act. While Beijing and Taipei certainly engage in risky behavior, these U.S. moves have unnerved some Japanese observers since they are not confident about the end state Washington is trying to achieve. Moreover, recent U.S. policy statements describing a largely zero-sum relationship with China and labeling Beijing as a "revisionist" power dedicated to undermining the United States and driving it out of Asia add to the above-mentioned fear that Washington might view Taiwan as a strategic asset that Beijing should be denied.
Several highly dangerous risks emerge as a result of the above factors, such as the possibility that China would discard its long-standing peaceful stance on resolving the Taiwan issue by opting for more forceful measures, or that Beijing would apply highly destabilizing economic and other pressures on Taipei, or that Washington would reach an understanding with Beijing on Taiwan's unification that in some way compromises Japanese security, or alternatively, shift to support de facto Taiwanese independence, thus triggering a conflict with China. The challenge is to assess the likelihood that such risks actually materialize under current or future conditions.
The state of the Sino-U.S. bilateral relationship: The most important factor influencing the likelihood of these risks materializing is the character of the overall political and strategic relationship between the United States and China. Although a source of concern on their own, other considerations in and of themselves such as adverse changes in Taiwan's domestic political environment, the further development of Chinese military capabilities of relevance to Taiwan, or even the general state of cross-strait relations will not trigger the above extreme risks. But one or more such developments could produce such risks if they were to occur amid a fundamental shift in Sino-U.S. relations toward confrontation and hostility.
Such a shift, by creating enormous bilateral distrust and leading both sides to completely accept a zero-sum calculus on all strategic issues, would call into question the longtime basic understanding on Taiwan that has existed since—and, indeed, made possible—the normalization of Sino-U.S. relations. This, in turn, would cause both Beijing and Washington to employ worst-case assessments of any of the above adverse changes. And that would enormously raise the likelihood of miscalculation and overreaction toward Taiwan, drastically increasing the likelihood of conflict. Unfortunately, under current conditions, the chance of such an ugly dynamic is arguably increasing.
A new quid pro quo on Taiwan: The risk that the United States and China might reach a new understanding on Taiwan that compromises Japanese security is not part of the current dynamic surrounding Sino-U.S. ties. Such an outcome could conceivably result from one of three developments: 1) a major improvement in Sino-U.S. relations that virtually ends strategic distrust and produces a new understanding on Taiwan that results in coercive pressure on the island unacceptable to Tokyo; 2) a major decline in U.S. influence across the Western Pacific, leading Washington to make major concessions to Beijing on Taiwan; or 3) an increase in China's attractiveness to Taiwan as a result of China's hypothetical democratization, resulting in a form of unification. As things stand now, however, none of these three developments is likely to occur for the foreseeable future.
Potential Opportunities
The Taiwan issue also presents several potential opportunities for the U.S.-Japan alliance, including the possibility that Beijing could revise its policy stance to accept an autonomous Taiwan, discard the option of using force to achieve unification, or permit Taiwan to expand its diplomatic and political associations. Even in the absence of such developments, Washington might reach an understanding with both Beijing and Tokyo that limits the further militarization of the Taiwan situation, making things more stable.
Under current conditions, none of these opportunities is likely to occur in the near to medium term, although the second possibility is arguably more likely than the first. Again, the larger Sino-U.S. dynamic is the most critical factor, and current trends are moving in decidedly adverse directions that reduce the likelihood of such opportunities emerging.
Areas for Further Study
Further research and reflection are required to define more clearly the possible differences between the United States and Japan over the meaning and likelihood of the aforementioned risks and opportunities. It will also be necessary to assess more precisely how the two allies can work together to reduce risks while increasing the possibility that opportunities emerge in the future.
Michael D. Swaine is a senior fellow at the Carnegie Endowment for International Peace and one of the most prominent American analysts in Chinese security studies.
The China Risk and China Opportunity for the U.S.-Japan Alliance project examines different perspectives between the alliance members and discusses ways in which Washington and Tokyo can effectively respond to China's rise. The project is led by the Japan Forum on International Relations (JFIR) and the Carnegie Endowment for International Peace. This piece is part of an accompanying series of policy briefs that explore various China-related risks and opportunities for the U.S.-Japan alliance in the areas of regional and international order, trade and technology, security, and foreign relations.
Policy Brief Series
Managing Risks and Opportunities for the U.S.-Japan Alliance Through Coordinated China Policy
Shin Kawashima, Matake Kamiya, and James L. Schoff
Responding to China's Complicated Views on International Order
Mira Rapp-Hooper, Michael S. Chase, Matake Kamiya, Shin Kawashima, and Yuichi Hosoya
Working Toward a Free and Open Indo-Pacific
Nicholas Szechenyi and Yuichi Hosoya
Bridging the Belt and Road Divide
Carla P. Freeman and Mie Ōba
Avoiding Friendly Fire Amid Economic Tensions With China
James L. Schoff and Asei Ito
Competing With China on Technology and Innovation
Bolstering the Alliance Amid China's Military Resurgence
Michael S. Chase, Satoru Mori, and Masafumi Iida
Staying in Sync on North Korea's Nuclear Program
Evans J.R. Revere and Matake Kamiya
Michael Swaine and Ryo Sahashi
The U.S.-Japan Alliance: Past Is Prologue
Why Little Lithuania Is Taking On Mighty China
Denis Kishinevsky
At Biden and Xi's Virtual Summit, What Can Be Accomplished?
Paul Haenle
Nathaniel Sher
South Korea Beyond Northeast Asia: How Seoul Is Deepening Ties With India and ASEAN
Kathryn Botto


text:
It's a sunny Big West season, so far, for…
It's a sunny Big West season, so far, for Anosike and Cal State Fullerton
On his third school, the muscular Anosike has led the Titans to a 4-0 conference record
Cal State Fullerton forward E.J. Anosike looks for an opening in a Big West Conference game against UCI at the Bren Center in Irvine on Thursday, January 20, 2022. (Photo by Paul Rodriguez, Contributing Photographer)
By Mark Whicker | MWhicker@scng.com | Daily News
PUBLISHED: January 21, 2022 at 1:19 p.m. | UPDATED: January 21, 2022 at 2:57 p.m.
Cal State Fullerton played at Northern Arizona in December, on one of those days when Flagstaff seemed like the North Pole.
E.J. Anosike shivered and looked at the sky. Snow was imminent. Anosike remembered how nice it looks until it starts invading your socks. He gave the side-eye to CSF coach Dedrique Taylor.
"I didn't come out here for this," Anosike told him.
He came "out here" from Tennessee because he wanted a bonus year of college basketball, wanted to wave goodbye in a meaningful way, and without gloves.
He is the leading scorer in the Big West Conference and the main driver of Cal State Fullerton's 4-0 league record. The Titans won, 65-63, at UC Irvine on Thursday night.
Anosike played three years at Sacred Heart in Fairfield, Conn., not terribly far from his home of East Orange, N.J. He went to Tennessee last year. Along the way, he got a bachelor's degree and an MBA.
Vincent Lee, Anosike's partner in the post, came from Nevada. Tray Maddox Jr. came from Oakland University, near Detroit. Damari Milstead came from San Francisco. There are freshmen, too, but this is what college basketball is now. With no mandatory sit-out year, players are flying off the shelves.
A school like Cal State Fullerton and a league like the Big West wasn't supposed to thrive this way. The case of Elijah Harkless, who went from Cal State Northridge to Oklahoma, was far more likely.
But players have their own motivations. They aren't just names on a greaseboard.
Anosike knew the Titans because he knew Kyle Allman, the leader of the 2018 Big West Tournament championship team. And he knew that people often wear shorts on campus in January.
As Lute Olson said when he came to Long Beach State from Iowa, "I don't have to scrape any of that 'fair and warmer' off my windshield."
"I didn't know anything about E.J. but when he became available we dug in our heels and looked at him," Taylor said. "Location is important. We sell it. You can go an hour and a half one way and be in the snow if you want to, or you can go a half-hour the other way and be in the sand. Very few places can say that.
"The transfers bring maturity and professionalism. You see E.J. and he's the same every day. He works. Our whole team is now emulating him."
The Titans have the biceps and the composure of grown men. Anosike, at 6-foot-7 and 236 pounds, averages 18.7 points and 7.9 rebounds. As they got acquainted during nonconference season, they realized their wins would happen in the lane. They average about seven more foul shots than their opponents. At UCI they shot the first 16 free throws of the game.
But then Anosike had more than 1,000 points and 1,000 rebounds at Sacred Heart, which plays in the Northeast Conference, a rough equivalent of the Big West. He also graduated in three years with a 3.57 GPA.
Tennessee was a natural next stop. Anosike's sister Nicky was a teammate of Candace Parker's in Knoxville and went to three Final Fours, winning two, while graduating with a triple major.
E.J., 12 years younger, was a ballboy for the Lady Vols. Nicky became a WNBA All-Star at Minnesota and was an L.A. Spark in 2012, and also played on national teams.
But E.J.'s year at Tennessee wasn't as eventful. COVID-19 barred the fans, and Anosike averaged 8.6 minutes and 1.7 points.
"It was a great experience being around Nicky's teams," E.J. said. "I got to see what (Coach) Pat Summitt was like. My mom saw a lot of herself in Pat, and they were close. I got to see what a woman's empowerment can look like in a male-dominated industry."
Nicky was a high school coach in Anderson County, Tenn., but she resigned after a dangerous and difficult pregnancy.
"She's the one who put the basketball in my hand," E.J. said. "We'd go out to the park every Saturday. I finally sneaked a win against her when I was 14. After that, she didn't want to play me anymore."
But how disorienting are three different programs in different locales with different pressures and coaches?
"I'm just grateful to get to play five years," Anosike said. "You find a family wherever you go.
"I'm not looking for the flashy, big-name stuff. You tune all of that out and you just focus on the actual playing, getting better on a daily basis. It's really not that different. Just a different level."
Wherever you go, there's a trophy to win. Anosike wants sunshine to reflect off his.
Cal State Fullerton sports
Mark Whicker
Mark Whicker | Sports Columnist
Mark Whicker has worked for SCNG newspapers since 1987. He has been a sports columnist for newspapers since 1980 and a professional sportswriter since 1974. He was selected for induction to the U.S. Basketball Writers Hall of Fame in 2020. Mark can be reached at mwhicker@scng.com or mwhicker03@gmail.com.
MWhicker@scng.com
Follow Mark Whicker @MWhicker03
Pac-12 financial mess: On the fate of the networks, an emergency recourse and the legal framework


"Poppa's Got a Brand New Badge"
SpringShield was a security agency in Springfield set up by Homer Simpson.
After Homer caused an electrical blackout in Springfield, all of the citizens began looting businesses. Chief Wiggum was incompetent in restoring order, so Homer decided to take the law into his own hands and set up SpringShield. Although Homer, Lenny, and Carl were the only employees, they had a lot of business due to a television commercial which Marge described as "very effective". They also turned out to be much more efficient at fighting crime than the Springfield Police Department. Because of this, Mayor Quimby fired Chief Wiggum and proclaimed SpringShield to be Springfield's new police force. As a symbol of the transfer of authority, Quimby took the Chief's badge away from Wiggum and gave it to Homer.
SpringShield enjoyed a brief period of success, until Homer busted Fat Tony selling weasels with cotton balls stuck to them as "Toy Poodles". Fat Tony vowed that he and his associates would gun down Homer if he didn't leave Springfield by noon the following day. Homer resolved to stay, and at noon the following day, Fat Tony and his gang showed up at the Simpson home as promised. They were about to shoot Homer when Maggie, surreptitiously, used a rifle to shoot (albeit only injure) all of the gang.
Homer then decided to give the badge to the next person he saw, who, coincidentally, turned out to be Wiggum, marking the end of SpringShield.
Modified on January 3, 2015, at 15:20.


For sale is a 2011 Chevrolet Malibu LS. This vehicle has undergone a complete inspection and received all necessary repairs by our Certified Technicians. Repairs include, NEW brake pads, brake rotors, tie rod ends, wiper blades, hub bearings, calipers, battery, and fresh lube, oil, and filter. It comes equipped with an automatic transmission, front wheel drive, and a 4-cylinder engine with 135k miles. A few nice options include, automatic headlights, chrome wheels, air conditioning, climate control, cruise control, power locks and windows, power mirrors, power and tilt steering, Satellite Radio, CD Player, MP3 Enabled, alarm system, child proof locks, running lights, driver and passenger airbags, side airbags, OnStar Service, front and rear defrost, tinted glass, keyless entry, and cloth seats with lumbar support. If interested in this vehicle, you may contact us anytime or just stop in for a test drive. Thank You and have a Great Day!