text:
\section{Introduction}
Since the seminal work of Magalinski{\u\i} \cite{Mag59}, Feynman and Vernon \cite{
FV63}, Ullersma \cite{Ull66,*Ull66b,*Ull66c,*Ull66d}, and  Caldeira-Leggett
\cite{CL83}, the theory of open quantum systems has been source of great interest
and object of continuous development, refinement and application (cf. Refs.~\citenum{
Wei08,GZ10,BP02,Muk99,MK11} and references therein).
This theory has, for example,  provided a solid conceptual framework  to explain
fundamental phenomena such as the quantum-classical transition
\cite{Zur03}, violations of the second law of the thermodynamics \cite{EG&08}, the
survival of quantum features at high temperature \cite{GPZ10,PB11}, and has found
applications in several fields in physics and chemistry \cite{Wei08,GZ10,BP02,Muk99,MK11}.

Despite the beauty and power of this theory, a study of the dynamics of a particular
system can be very cumbersome due to the sheer complexity of correctly incorporating
the various time, energy and coupling scales.
In order to circumvent this problem various approximations, such as the
weak-coupling-to-the-bath, Markovian, high temperature,  or
the initial factorizing condition is usually invoked \cite{GSI88,Wei08,GZ10,BP02,Muk99,MK11}.
However, the development of modern experimental techniques for preparing
and manipulating physical and chemical systems has reached the regime where
such approximations are questionable.
This fact has encouraged the development  of techniques for more consistently  treating
and analyzing open quantum systems  (cf. Ref.~\citenum{PB12c}).

Despite  the refinement in  technique, no approach  is completely
approximation-free \cite{Wei08,GZ10,BP02,Muk99,MK11,PB12c}, and therefore some
physical features of the dissipative dynamics are often absent in these descriptions
\cite{Wei08,GZ10,BP02,PB12c,SB12}.

The breakdown of these common approximations is expected to occur in the low
temperature regime \cite{HI05} and/or in the presence of initial correlations between
the system and the environment \cite{GSI88} and, in particular, in the case of driven
non-equilibrium quantum systems \cite{ZH95,KK04,GPZ10,SN&11}.
The ubiquitous presence of this situation (an open quantum systems under the
presence of time dependent fields in, e.g., coherent control scenarios of chemical
systems \cite{SB11} and assorted  physical systems \cite{SN&11,PB13,PB12c}),
motivates a formal and detailed treatment of such dynamics.

In this paper, we derive formal \emph{exact} results for the non-Markovian
dynamics of a prototypical system, including the presence of initial correlations between
the system and the bath, and the possibility of arbitrary rapidly oscillating driving forces.
The method can be applied in the low temperature and strong coupling-to-the-bath regimes.
The particular system studied here is a charged quantum oscillator confined in a harmonic
trap that is  initially in thermodynamic equilibrium with its environment (non-factorized
initial conditions \cite{GSI88}).
For $t>0$, we start varying the frequency of the harmonic trap and couple the
oscillator, via the dipole, with an incoherent quantized blackbody
radiation field and with an arbitrary coherent laser field.
By means of the influence functional approach \cite{FV63,FH65,CL83,GSI88}, we
derive analytic closed expressions for the non-Markovian time evolution of the
covariance matrix elements of the quantum fluctuations valid at any temperature,
any system-environment coupling strength and incorporating the zero-point fluctuations
of the radiation.

This robust and general model allows us to address many different
physical problems in generic open quantum systems.
The results derived here can be used to study, for example:
\begin{enumerate}[i]
\item \emph{The incoherent excitation of open quantum systems}:
In particular, our results allow us to model situations when a molecular system
such as retinal or a photosynthetic light-harvesting complex, equilibrated with it surroundings, is excited
by either coherent (coherent laser pulses) or incoherent (sunlight or moonlight) light
sources, a subject of great significance in the chemical physics community \cite{JB91,BS11b,EC&07,HB11}.
One such application of this method is given in Ref.~\citenum{PB13}.

\item \emph{Environmentally-assisted one-photon phase control}:
That weak field
one-photon phase control is not possible for certain isolated quantum systems
is a known result (cf. Refs.~\citenum{SB11,SAB10,PYB13} and references therein).
Recently, it was suggested that the coupling to the environment could allow, in principle,
for the coherent manipulation of quantum systems in such cases \cite{SAB10,PYB13}.
However, it is still unclear what physical mechanisms are behind this process,
mainly because a formal study of this situation involves the presence of ultrafast field-induced
modulations in open quantum systems at low temperature, a situation where non-Markovian
processes cannot be disregarded and where approximations such as the rotating-wave
fail \cite{KK04}.
Moreover, in this case of environmentally-assisted control,
the presence of initial correlations between the
system and the bath is vital. A treatment of this problem using the method developed
herein is to be provided in Ref. \citenum{PB13c}.
\item \emph{Optimal-control-based cooling of quantum nano-resonators by means of
parametric driving}.
On the basis of a numerical protocol, it was recently shown that nano-mechanical
resonators can be cooled down by the delicate interplay of dissipative and driving
process \cite{SN&11}.
Being based on an optimal-control protocol, the possibility of parametrically driving the frequency
of the resonator with arbitrary rapidly oscillating fields, as we consider here, is a key element
in this cooling process.
As in the previous case, this scenario is well beyond standard approximation schemes, but
can be immediately cast as a particular case of our general model, with the great advantage
of having an analytic formulae for the system dynamics.

\item \emph{The establishment of a quantum limit on non-Markovian time scales}.
In thermodynamical equilibrium, quantum features survive in the limit
$\hbar \omega_0 / k_{\mathrm{B}}T > 1$,
where $\hbar \omega_0$ is a typical energy scale of the system and $T$ the temperature.
According to Ref.~\citenum{GPZ10}, quantum features can persist for higher temperatures under
non-equilibrium situations.
However, results in Ref.~\citenum{GPZ10} are based on the Markovian approximation, so the
derivation a quantum limit consistent with the presence of non-Markovian effects is desirable.
This problem can be addressed using our general approach and this study is currently in progress.
\item \emph{Non-Markovian thermodynamics}.
For quantum systems, it is possible to have very fast control of heat and entropy due to
anomalies induced by the non-Markovian character of the relaxation \cite{EG&08}.
Our general results, can be immediately applied to study, e.g., heat transport between
two non-Markovian reservoirs at the quantum regime.
\end{enumerate}

This provides a sample list of problems that can be
readily examined using the exact solution, derived below. It is the
variety of challenging problems that can be addressed that motivates this paper,
the derivation
of an all-in-one versatile model that can treat a host of problems and which
can be generalized to consider additional phenomena.

\section{Description of the System Hamiltonian
The Hamiltonian of a quantum oscillator confined in a harmonic trap and in contact
with a thermal bath comprises three parts:
the Hamiltonian of the quantum oscillator itself, $\hat{H}_{\mathrm{S}}$;
the Hamiltonian of the thermal bath, $\hat{H}_{\mathrm{TB}}$, here described
according to the Ullersma-Caldeira-Leggett model \cite{Ull66,CL83}, as
a collection of harmonic modes;
and an interaction term $\hat{H}_{\mathrm{S-TB}}$ between the two systems.
For typical bilinear coupling, the three contributions are
\begin{align}
\label{equ:HS}
\hat{H}_{\mathrm{S}} &=
\frac{1}{2m} \hat{p}^2  + \frac{1}{2}m\omega(t)^2 \hat{q}^2  \\
\hat{H}_{\mathrm{TB}} &=
\sum_{j}^{\infty}\left[\frac{\hat{p}_j^2}{2m_j} + \frac{m_j \omega_j^2}{2} \hat{q}_j^ 2\right]
\\
\label{equ:HSTB}
\hat{H}_{\mathrm{S-TB}} &=
-\hat{q} \sum_{j}^{\infty}c_j \hat{q}_j
+ \hat{q}^2\sum_{j}^{\infty} \frac{c_j^2}{2 m_j \omega_j^2}
\end{align}
with $\hat{p}$ and $\hat{q}$ the canonically conjugate momentum and position of the
oscillator (an analogous notation describes the bath modes),  $m$ the mass of the quantum oscillator and
$\omega^2(t) = \omega_0^2 + \omega_{\mathrm{P}}^2(t)$ the parametrically
modulated frequency. This frequency comprises two components: $\omega_0$,
a constant frequency and an arbitrary time-dependent-frequency $\omega_{\mathrm{P}}(t)$.
The magnitude of the interaction between the system and the bath is determined by the
coupling constants $c_j$.

In the presence of the blackbody radiation the Hamiltonian
$\hat{H}_0  = \hat{H}_{\mathrm{S}} +  \hat{H}_{\mathrm{S-TB}} +  \hat{H}_{\mathrm{TB}}$
needs to be augmented  to include the interaction with the field as well as the Hamiltonian
of the field modes,
\begin{equation}
\label{equ:HBB}
\hat{H}_{\mathrm{BB}} = \sum_{\mathbf{k},s}\hbar c k
   \left(\hat{a}_{\mathbf{k},s}^{\dagger} \hat{a}_{\mathbf{k},s}+\frac{1}{2}\right).
\end{equation}
Assuming that the charged oscillator interacts weakly with each mode in the field, we
can adopt a dipole-dipole type interaction, giving the overall Hamiltonian
\begin{linenomath}
\begin{equation}
\begin{split}
\label{equ:HamSysTBBB}
\hat{H} &= \frac{1}{2m}\left( \hat{p} - \frac{e}{c} \hat{A}\right)^2
   + \frac{1}{2}m\omega(t)^2 \hat{q}^2
\\&
+ \hat{H}_{\mathrm{S-TB}} +  \hat{H}_{\mathrm{TB}}
+\hat{H}_{\mathrm{BB}},
\end{split}
\end{equation}
\end{linenomath}
where $e/c$ is the coupling constant to the radiation, $\hat{a}_{\mathbf{k},s}$ and
$\hat{a}_{\mathbf{k},s}^{\dagger}$ are the annihilation and creation operators of
the  field mode of momentum $\mathbf{k}$ and polarization $s$. The
vector potential is given by
\begin{linenomath}
\begin{align}
\label{equ:AVPBBR}
 \hat{A} &= \sum_{\mathbf{k},s} \left(\frac{h c}{kV}\right)^{\frac{1}{2}}
 f_k \vec{\mathbf{e}}_{\mathbf{k},s}
 \left(
    \hat{a}_{\mathbf{k},s}+\hat{a}_{\mathbf{k},s}^{\dagger}\right),
\end{align}
\end{linenomath}
where $\hat{\mathbf{e}}$ is the polarization unit vector, $V$ is the volume of the
auxiliary cavity containing the field modes and $f_k$ is the electron form-factor
(Fourier transform of the charge distribution) that incorporates the electron
structure \cite{FLO85,FLO88}. We have assumed, with no loss of generality,
that the form factor and polarization vector are real.
Note that by virtue of Eq.~(\ref{equ:HBB}), Eq.~(\ref{equ:HamSysTBBB}) already
contains zero-point or vacuum fluctuations.

Equation (\ref{equ:HamSysTBBB}) can be generalized to include  an additional term
$- \hat{q} E_{\mathrm{L}}(t)$, which allows for the possible manipulation of
the charged oscillator, via dipole coupling, by means of the electric field
$E_{\mathrm{L}}(t)$ of a pulsed or continuous laser field.

Since Eq. (\ref{equ:HamSysTBBB}) includes the diamagnetic term
$\hat{A}^2$, it is not suitable for a path integration calculation,
which is  why it is usually omitted \cite{CK87,BC91}.
However, the contribution of this term is relevant for the derivation
of the partition function of the oscillator in the presence of the blackbody
radiation (cf. the discussion in \cite{FLO88b,CK88}).
In our case, this term can be introduced by means of the Power-Zienau's
transformation \cite{PZ59} (see also Ref.~\citenum{HG36} for the original version and
Ref.~\citenum{YHJ81}
and references therein for a short historical review on this transformation),
$
\hat{T} = \exp\left\{\frac{\mathrm{i}}{\hbar}
\frac{e}{c}\hat{q} \cdot \hat{A}\right\},
$
which transforms
$\hat{p} \rightarrow \hat{p} + \frac{e}{c}\hat{A}$,
$\hat{q} \rightarrow \hat{q}$,
$\hat{p}_{\mathbf{k},s} \rightarrow \hat{p}_{\mathbf{k},s}
+ \frac{e}{c} m_{\mathbf{k}} \omega_{\mathbf{k}} \hat{q}$,
$\hat{q}_{\mathbf{k},s} \rightarrow \hat{q}_{\mathbf{k},s}$, where
we have defined \cite{FLO85,FLO88}
\begin{align}
\label{equ:AnhCreOpeField}
\hat{a}_{\mathbf{k},s} &=
(m_{\mathbf{k}} \omega_{\mathbf{k}} \hat{q}_{\mathbf{k},s}
+ \mathrm{i}\hat{p}_{\mathbf{k},s})/\sqrt{2m_{\mathbf{k}} \hbar \omega_{\mathbf{k}}},
\end{align}
with $m_{\mathbf{k}} = 4\pi e^2f_k^2/(\omega_{\mathbf{k}} V)$.
The corresponding total Hamiltonian reads
\begin{linenomath}
\begin{equation}
\begin{split}
\label{equ:HamSysTBBBb}
\hat{H} &= \frac{1}{2m} \hat{p}^2
 + \frac{1}{2}m\omega(t)^2 \hat{q}^2 - \hat{q} E_{\mathrm{L}}(t)
\\ & +  \sum_{j}\frac{\hat{p}_j^2}{2m_j}
+ \frac{m_j \omega_j^2}{2} \left(\hat{q}_j - \frac{c_j}{m_j \omega_j^2} \hat{q}\right)^2
\\ &
+ \sum_{\mathbf{k},s}
   \frac{1}{2 m_{\mathbf{k}}}\left(\hat{p}_{\mathbf{k},s} +
   m_{\mathbf{k}} \omega_{\mathbf{k}} \hat{q} \right)^2+
   \frac{1}{2}m_{\mathbf{k}}\omega_{\mathbf{k},s}^2 \hat{q}_{\mathbf{k},s}^2.
\end{split}
\end{equation}
\end{linenomath}
Here the oscillator is seen to be coupled to the momentum coordinate
$p_{\mathbf{k},s}$.

Recalling  that the electric field of the blackbody radiation is given by
\begin{linenomath}
\begin{equation}
\label{equ:DefQuaElecField}
\hat{E} = -\frac{\partial \hat{A}}{\partial t} =
 \textrm{i} \sum_{\mathbf{k},s} \left(\frac{h c^3}{V}\right)^{\frac{1}{2}}
 f_k \hat{\mathbf{e}}_{\mathbf{k},s}
 \left(
      \hat{a}_{\mathbf{k},s} - \hat{a}_{\mathbf{k},s}^{\dagger}\right),
\end{equation}
\end{linenomath}
and examining Eq.~(\ref{equ:AnhCreOpeField}), we see that in
Eq.~(\ref{equ:HamSysTBBBb}), the position of the oscillator is coupled
to the electric field of the blackbody radiation.
From an open-quantum-systems
perspective, this implies that any  statistical behavior induced by the blackbody
radiation (as seen elsewhere \cite{FLO85,FLO88,PB13})
is dictated by the stochastic fluctuations of the electric field.

\section{Initial Density Matrix
After characterizing the Hamiltonian of the total system, the next step in the description
of the dynamics is the determination of the initial state, taken to be the equilibrated state of S$+$TB.
By denoting the coordinates of S$+$TB as
$\mathcal{Q} = \{q,\mathbf{Q}\}$,
the matrix elements of the initial density operator of the system S plus the environment TB
can be calculated as \cite{GSI88}
\begin{equation}
\label{equ:initialStateSTB}
\bok{\bar{\mathcal{Q}}'' }{\hat{\rho}_{\beta}}{\bar{\mathcal{Q}}'} =
Z_{\beta_{\mathrm{TB}}}^{-1}
\int \limits_{\bar{\mathcal{Q}}'}^{\bar{\mathcal{Q}}''}\mathcal{D} \bar{\mathcal{Q}}
\exp\left(- \frac{1}{\hbar}S^{\mathrm{E}}[\bar{\mathcal{Q}}]\right),
\end{equation}
where the integral is over all paths $\bar{\mathcal{Q}}(\tau)$,
$0 \leq \tau \leq \hbar\beta_{\mathrm{TB}}$ with
$\bar{\mathcal{Q}}(0) = \bar{\mathcal{Q}}'$
and $\bar{\mathcal{Q}}(\hbar \beta_{\mathrm{TB}}) = \bar{\mathcal{Q}}''$.
The bar stands for the trajectories $\mathcal{Q}(t)$ in imaginary time
$t\rightarrow - \mathrm{i} \hbar \beta$, with $\beta_{\mathrm{TB}}=1/(k_{\mathrm{B}}T_{\mathrm{TB}})$.
In the imaginary-time path integral described by Eq.~(\ref{equ:initialStateSTB}),
$S^{\mathrm{E}}[\bar{\mathcal{Q}}]$ denotes the Euclidean action of
the system
$S^{\mathrm{E}}[\bar{\mathcal{Q}}] =
\int\limits_0^{\hbar \beta_{\mathrm{TB}}} \mathrm{d}\tau \mathcal{L}^{\mathrm{E}}(\bar{\mathcal{Q}},\dot{\bar{\mathcal{Q}}})
= S^{\mathrm{E}}_{\mathrm{S}}[\bar{q}] +
S^{\mathrm{E}}_{\mathrm{S-TB}}[\bar{q},\bar{\mathbf{Q}}]
+ S^{\mathrm{E}}_{\mathrm{TB}}[\bar{\mathbf{Q}}]$,
obtained by introducing a global minus sign in the
potential energy \cite{GSI88,Ing02}.
Since $\hat{\rho}_{\beta}$ denotes the density operator at $t=0$, we assume that
$\omega_{\mathrm{P}}(0)=0$ and $E_{\mathrm{L}}(0) = 0$ in Eq.~(\ref{equ:HamSysTBBBb}).
The matrix elements of the initial total density operator of interest are
\begin{equation}
\label{equ:initialState}
\begin{split}
\bok{\bar{q}_{\mathbf{k},s}'' \bar{\mathcal{Q}}'' }{\hat{\rho}(0)}{\bar{q}_{\mathbf{k},s}'\bar{\mathcal{Q}}'}&=
\bok{\bar{q}_{\mathbf{k},s}''}{\hat{\rho}_{\beta_{\mathrm{BB}}}}{\bar{q}_{\mathbf{k},s}'}
\bok{\bar{\mathcal{Q}}'' }{\hat{\rho}_{\beta}}{\bar{\mathcal{Q}}'},
\end{split}
\end{equation}
where $\hat{\rho}_{\beta_{\mathrm{BB}}}$ denotes the equilibrium density operator of
the radiation only, at temperature $T_{\mathrm{BB}}$ and defined as in Eq.~(\ref{equ:initialStateSTB})
using the Euclidean action $S^{\mathrm{E}}_{\mathrm{BB}}[\bar{q}_{\mathbf{k},s}]$ of the
bare radiation, $\hat{\rho}_\beta$ is the thermal
density operator of system and bath introduced in Eq.~(\ref{equ:initialStateSTB}),
and the blackbody-radiation-mode-coordinates $q_{\mathbf{k},s}$ are defined
in Eq.~(\ref{equ:AnhCreOpeField}).
When a system S is in contact with two thermal baths, what is usual in the literature
(cf. Refs.~\citenum{PB13,PB13c}) is that the initial state of the total system is assumed to be
factorized, in our case this is equivalent to take
$\hat{\rho}(0) = \hat{\rho}_{\beta_{\mathrm{BB}}} \otimes \hat{\rho}_{\beta_{\mathrm{TB}}}
\otimes \hat{\rho}_{\mathrm{S}}$, being $ \hat{\rho}_{\beta_{\mathrm{TB}}}$
and $ \hat{\rho}_{\mathrm{S}}$ the density operator of the thermal bath TB and
the system S, respectively. Here we deal with a more complex situation because
we take into account the initial correlations between the system S and the thermal bath
TB.

In general, one would also like to study the system S prepared
in a state other than  the
equilibrium state, e.g., in a coherent state or in a squeezed state.
According to Refs.~\citenum{SG86,GSI88}, one could prepare a different
initial state by allowing the operators $\hat A_n, \hat A_n'$ acting only in the system
Hilbert space of S to generate an initial non-equilibrium density operator of system
and bath
$
\hat{\rho}_{\beta}^{\lambda} = \sum_n (\hat{A}_n \otimes \hat{1}_{\mathrm{TB}})
\hat{\rho}_\beta (\hat{1}_{\mathrm{TB}} \otimes \hat{A}_n')\,,
$
where $\hat{1}_{\mathrm{TB}}$ denotes the unit operator in the Hilbert space of the
bath alone.
In the position representation, the matrix elements of $\hat{\rho}_{\beta}^{\lambda}$
are given by
\begin{equation}
\label{equ:initialStateGen}
\begin{split}
\bok{\mathcal{Q}_+'}{\rho_{\beta}^{\lambda}}{\mathcal{Q}_-'}&=
\int \mathrm{d}\bar{\mathcal{Q}}''\mathrm{d}\bar{\mathcal{Q}}'
\lambda(q_{+}',{\bar q}'',q_{-}',{\bar q}')
\\&\times
\delta(\mathbf{Q}_+'-\bar{\mathbf{Q}}'') \delta(\mathbf{Q}_-'-\bar{\mathbf{Q}}')
\bok{\bar{\mathcal{Q}}'' }{\hat{\rho}_{\beta}}{\bar{\mathcal{Q}}'} ,
\end{split}
\end{equation}
where the propagating function
$
\lambda(q_{+}',\bar q'',q_{-}',\bar q')
=\sum_n \sbok{q_+'}{\hat{A}_n}{\bar{q}''} \sbok{\bar{q}'}{\hat{A}_n'}{q_{-}'}
$
characterizes the action of these operators.
The delta functions  indicate that the imaginary-time paths for the bath degrees
of freedom are continuously connected to the real-time paths describing the time
evolution of the initial state \cite{SG86,GSI88}.
The thermal initial state in Eq.~(\ref{equ:initialStateSTB}) can be recovered
by setting the auxiliary operators $\hat{A}_n$ and $\hat{A}_n'$ to $\hat{1}_{\mathrm{S}}$,
which yields $\lambda(q_{+}',{\bar q}'',q_{-}',{\bar q}') = \delta(q_{+}'-{\bar q}'')
\delta(q_{-}'-{\bar q}')$.
As distinct from $\bar{\mathcal{Q}}'$ and $\bar{\mathcal{Q}}''$, which are the endpoints
of a single imaginary-time-trajectory, $\mathcal{Q}_{+}'$ and $\mathcal{Q}_{-}'$ denote
the initial condition for two real-time-trajectories ``$+$'' and ``$-$''
(see below). For
the case of unitary evolution these can be identified as the forward and backward trajectories
associate to the unitary time-evolution-operator and its adjoint, respectively.
Additional details about different initial preparations can be found in
Refs.~\citenum{SG86,GSI88}.

\section{Dynamics of the System}
In this section we solve for the time evolution of the initial density matrix
[Eq.~(\ref{equ:initialState})] under the action of the Hamiltonian
[Eq.~(\ref{equ:HamSysTBBBb})], using the Feynman and Vernon
influence functional approach \cite{FV63,FH65}. For this problem we require  a
mixture of the influence functional for factorizing initial conditions
\cite{FV63,FH65,CL83,ZH95}  in order to consider the effect of the radiation, and for
non-factorizing initial conditions \cite{SG86,GSI88} in order to correctly characterize
the equilibrium state between the oscillator and the thermal bath at $t=0$.

It is worth mentioning that the standard path integral calculations are performed for position-position
couplings \cite{FV63,FH65,FV63,FH65,CL83,ZH95,SG86,GSI88,Wei08} [cf. the term
$\hat{q}\, \hat{q}_j$ in Eq.~(\ref{equ:HamSysTBBBb})], and for momentum-momentum
coupling \cite{CL83}.
The oscillator-radiation coupling is of position-momentum type,
$\hat{q}\,\hat{p}_{\mathbf{k},s}$ [cf. Eq.~(\ref{equ:HamSysTBBBb})].
By means of a set of unitary transformations \cite{FLO88}, one could invert the
role of the momentum and position operators of the field [see
Eq.~\ref{equ:AnhCreOpeField}], with the caveat that this generates an additional
term proportional to the initial position of the oscillator (see below).
However, since the system described in Eq.~(\ref{equ:HamSysTBBBb}) is still linear,
the path integral calculation can  also be carried out analytically for the
position-momentum coupling.  This is the approach  followed below.

\subsection{Derivation of the propagating function and the influence functional}
The time evolution of the system S is described by the reduced density operator
$\hat{\rho}_{\mathrm{S}}(t) = \mathrm{tr}_{\mathrm{TB},\mathrm{BB}}\hat{\rho(t)}$.
Following Ref.~\citenum{GSI88}, we obtain that the matrix elements of $\hat{\rho}_{\mathrm{S}}(t)$
are given by
\begin{equation}
\label{equ:ForTimeEvo}
\begin{split}
\bok{q_+''}{\hat{\rho}_{\mathrm{S}}(t)}{q_-''} &=
\int  \mathrm{d}q_+' \mathrm{d}q_-' \mathrm{d}\bar{q}'' \mathrm{d}\bar{q}'
\lambda(q_{+}',{\bar q}'',q_{-}',{\bar q}')
\\&\times
J(q_+'',q_-'',t;q_+',q_-',0;\bar{q}'',\bar{q}') ,
\end{split}
\end{equation}
where $J(q_+'',q_-'',t;q_+',q_-',0;\bar{q}'',\bar{q}')$ is the propagating function of the
system density matrix which can be expressed in terms of the functional
phase $\Sigma[q_+,q_-,\bar{q}]$ by means of the three-fold path integral
expression
\begin{equation}
\label{equ:GenPropFunct}
\begin{split}
J(q_+''&,q_-'',t;q_+',q_-',0;  \bar{q}'',\bar{q}')  =
\\&
\frac{1}{Z} \int\limits_{q_+'}^{q_+''} \mathcal{D}q_+
\int\limits_{q_-'}^{q_-''} \mathcal{D}q_-
\int\limits_{\bar{q}'}^{\bar{q}''}  \mathcal{D}\bar{q}
\exp\left(\frac{\mathrm{i}}{\hbar} \Sigma[q_+,q_-,\bar{q}]\right),
\end{split}
\end{equation}
where $Z$ normalizes $J(q_+'',q_-'',t;q_+',q_-',0;  \bar{q}'',\bar{q}')$ to
$\delta(q_+''-q_+')\delta(q_-''-q_-') \sbok{\bar{q}''}{\hat{\rho}_{\beta,\mathrm{S}}}{\bar{q}'}$
at $t=0$, being $\hat{\rho}_{\beta,\mathrm{S}} = \mathrm{tr}_{\mathrm{TB}}(\hat{\rho}_{\beta})$.
The real time path-integrals over $q_+$ and $q_-$ are carried out subject
to the endpoints $q_+(0) = q_+'$, $q_+(t) = q_+''$, $q_-(0) = q_-'$ and
$q_-(t) = q_-''$, while the imaginary time path integral are over
$\bar{q}(0)=\bar{q}'$ and $\bar{q}(\hbar\beta_{\mathrm{TB}})=\bar{q}''$.
Recall that the imaginary time path-integral allows for the calculation of the equilibrated
density operator of S$+$TB and the influence of their initial correlations in the
subsequence time evolution.

After tracing over the degree of freedom of TB and BB, and after defining
$q_{+} = (r + x)/2$ and $q_{-} = r - x$, we have that the functional phase
$\Sigma [x,r,\bar{q}]$ is given by
\begin{widetext}
\begin{equation}
\label{equ:GenSigma}
\begin{split}
&\Sigma [x,r,\bar{q}] =
\\
& \mathrm{i} \int\limits_0^{\hbar \beta_{\mathrm{TB}}} \mathrm{d}\tau
\left[
\frac{m}{2} \dot{\bar{q}}^2 + \frac{1}{2}m\omega_0^2 \bar{q}^2 + \frac{1}{2}
\int\limits_0^{\hbar \beta_{\mathrm{TB}}}\mathrm{d}\sigma
k_{\mathrm{TB}}(\tau - \sigma)\bar{q}(\tau) \bar{q}(\sigma)
\right]
+
\int\limits_0^{\hbar \beta_{\mathrm{TB}}}\mathrm{d}\tau \int\limits_0^{t}\mathrm{d} s
K_{\mathrm{TB}}^*(s - \mathrm{i} \tau)\bar{q}(\tau) x(s)
\\
& + \int\limits_0^{t} \mathrm{d}s \left\{m \dot{x}(s) \dot{r}(s) - m \omega_0^2 r(s) x(s)
- m \omega_{\mathrm{P}}(t)^2 r(s) x(s) + E_{\mathrm{L}}(t) x(s)
- r'\eta_{\mathrm{TB}}(s)x(s)\right\}
\\
\phantom{\Sigma [x,r,\bar{q}] =}
&- \int\limits_0^{t}\mathrm{d}s
\left\{
\int\limits_0^{t}\mathrm{d} u
\left[ \eta_{\mathrm{TB}}(s) + \eta_{\mathrm{BB}}(s)\right] x(s) \dot{r}(u)
- \frac{\mathrm{i}}{2} \int\limits_0^{t}\mathrm{d} u
\left[ K^{\mathrm{re}}_{\mathrm{TB}}(s-u) + K_{\mathrm{BB}}(s-u)\right] x(s) x(u)
\right\}.
\end{split}
\end{equation}
\end{widetext}
Note that $r' = q_+' + \frac{1}{2}q_-'$, $r'' = q_+'' + \frac{1}{2}q_-''$ and
analogously  for $x''$ and $x'$.
The various kernels entering into Eq. (\ref{equ:GenSigma}) are defined
in the next section.

The first term in the first line of Eq. (\ref{equ:GenSigma}) accounts  for the
equilibrium density operator of S in the presence of the thermal bath TB while
the second term containing $\bar{q}(\tau)x(s)$ is responsible of the the effect
of initial correlations between the environment and the system on the
subsequent time evolution.
The first three terms in the second line are responsible of the evolution under
 the parametric harmonic potential, while the fourth term is
responsible for the evolution induced by the laser field $E_{\mathrm{L}}$.
As previously noted, we take $\omega_{\mathrm{P}}(0) =0$ and $E_{\mathrm{L}}(0) =0$.
The last term in the second line of Eq.~(\ref{equ:GenSigma}) arises  from
the incoherent excitation induced by the position-position coupling to the
thermal bath TB.
Since the coupling to the blackbody radiation field BB is of a different nature,
position-momentum coupling, this transient term proportional to the initial
position $r'$ is not present.  However, if one changes the role of position
and momentum, as discussed above, this transient term enters implicitly.
The terms in the third line constitute the exponent of the influence functional
of the Feynman-Vernon theory under the action of the thermal
bath TB and  blackbody radiation BB.

The additional time integration in the last line of Eq.~(\ref{equ:GenSigma}) over $u$
accounts for the non-local time (non-Markovian) evolution of the density operator.
Although the temporal non-locality is determined by the various kernels in a
cumbersome way, we can identify two kinds of non-Markovian contributions:
 one from the dissipative part and determined by the non-local
character of $\eta_{\mathrm{TB}}$ and $\eta_{\mathrm{BB}}$, and
 a second  determined by the thermal fluctuations  described (see below)
by the kernels $K^{\mathrm{re}}_{\mathrm{TB}}$ and $K_{\mathrm{BB}}$.
The presence of the latter is not determined by the
presence of the former, i.e., in the limit of local dissipative Ohmic kernels,
$\eta_{\mathrm{TB}}(s) \sim \delta(s)$, the non-local character of the thermal
fluctuations is still present; it only vanishes in the high temperature regime
\cite{Wei08,Pac10}.
\subsection{Kernels in the functional action}
The quantities introduced in the effective action \( \Sigma [x,r,\bar{q}_x]\)
are defined in terms of the bath spectral density $J_{\mathrm{TB}}$
and the blackbody-radiation spectral density $J_{\mathrm{BB}}$.
These spectral densities are determined  \cite{CL83,GSI88,FLO85,FLO88} from the parameters of the bath
modes and the coupling constants by means of
\begin{align}
\label{equ:DefJwTB}
J_{\mathrm{TB}}(\omega)&= \frac{\pi}{2} \sum_{j=1}^{\infty} \frac{c_j^2}{m_j\omega_j}
\delta(\omega - \omega_j),
\\
\label{equ:DefJwBB}
J_{\mathrm{BB}}(\omega)&= \frac{\pi}{2} \sum_{\mathbf{k},s}^{\infty}
m_{\mathbf{k}}\omega_{\mathbf{k}}^3
\delta(\omega - \omega_{\mathbf{k}}).
\end{align}
Assuming that the thermal bath is dense in the frequency of the modes
\cite{CL83}, it is customary to describe the spectral density in Eq.~(\ref{equ:DefJwTB})
by, e.g., assuming the Ohmic model
\begin{equation}
\label{equ:JwTB}
J_{\mathrm{TB}}(\omega) = m \gamma_{\mathrm{TB}} \omega
\Omega_{\mathrm{TB}}^2/(\Omega_{\mathrm{TB}}^2 + \omega^2 ),
\end{equation}
where $\gamma_{\mathrm{TB}}$ is the coupling constant to the bath TB and
$\Omega_\mathrm{TB}$ is a cutoff parameter related to the inverse of
the bath memory time.
In contrast with the thermal bath case, no assumption on the functional
form of the spectral density of the blackbody is needed in the continuous
limit \cite{FLO85,FLO88,FLO87,BC91}.

The transversality condition implies  that only two of the three components
of $\mathbf{k}$ contribute to the coupling \cite{FLO88}, giving a global
factor of two-thirds for the spectral density in Eq.~(\ref{equ:DefJwBB}).
In the continuous limit,
$\sum_{\mathbf{k}}\rightarrow \frac{V}{(2\pi)^3}\int\mathrm{d}\mathbf{k}$,
the spectral density for the blackbody radiation is
\begin{equation}
\label{equ:JwBB}
J_{\mathrm{BB}}(\omega) = M \tau_{\mathrm{BB}} \, \omega^3 \Omega_{\mathrm{BB}}^2/
\left(\Omega_{\mathrm{BB}}^2 + \omega^2\right),
\end{equation}
where $M = m + M \tau_{\mathrm{BB}} \Omega_{\mathrm{BB}}$ is the renormalized mass,
$\tau_{\mathrm{BB}} = 2 e^2 /3 M c^3$ and $\Omega_{\mathrm{BB}}$ is a frequency cutoff.
This path-integral-based expression coincides completely with
the seminal results in Refs.~\citenum{FLO85,FLO88,FLO87} using the quantum
Langevin formalism.
It also coincides with the result derived in Ref.~\citenum{BC91} using the standard path integral
approach.
However, we need to note that in Ref.~\citenum{BC91}, the system, an electron, is interacting
with its own radiation; here, by difference, we consider the system as being irradiated
by an external blackbody radiation such as sunlight or moonlight for $t>0$.
This is precisely what allows us to separate the initial density operator of the system and the radiation.
This natural emerging functional form of $J_{\mathrm{BB}}(\omega)$ reveals, from a
statistical viewpoint, the intrinsic non-Markovian character of the radiation \cite{FLO85,FLO88,FLO87,BC91}.
This fact implies that
the two point correlation function, $\langle \hat{E}(t'') \hat{E}(t'') \rangle_{\mathrm{BB}}$,
of the electric field in Eq.~(\ref{equ:DefQuaElecField}) is not delta correlated.
From an optics point of view \cite{MW64a,*MW64b,*MW67} this means that
the blackbody radiation is coherent, although the coherence time is very short,
$\sim$1.3~fs at $T_{\mathrm{BB}} = 5900$~K (cf. Ref.~\citenum{PB13} and Chap.~13 in Ref.~\citenum{MW95}).

Once we have condensed the relevant information for the thermal bath and
the radiation field in the spectral densities Eqs.~(\ref{equ:JwTB}) and
(\ref{equ:JwBB}), we are in the position to define the various functions
entering into the functional actions $\Sigma[x,r,\bar{q}]$.

The kernels $K_{\mathrm{TB}}(s-\mathrm{i}\tau)$ and $k_{\mathrm{TB}}(\tau)$
are given by \cite{GSI88}
\begin{linenomath}
\begin{align}
\label{equ:KTB}
K_{\mathrm{TB}}(s-\mathrm{i}\tau) &= K^{\mathrm{re}}_{\mathrm{TB}}(s-\mathrm{i}\tau)
+ \mathrm{i} K^{\mathrm{im}}_{\mathrm{TB}}(s-\mathrm{i}\tau),
\\
\label{equ:kkTB}
k_{\mathrm{TB}}(\tau) &= \frac{m}{\hbar \beta_{\mathrm{TB}}}
\sum_{n=-\infty}^{\infty}\zeta_n(0) \exp(\mathrm{i} \nu_n \tau),
\end{align}
\end{linenomath}
with the Matsubara frequencies $\nu_n = 2\pi n/\hbar\beta_{\mathrm{TB}}$.
The kernel $K_{\mathrm{TB}}(s-\mathrm{i}\tau)$ contains the information of the
thermal fluctuations due to TB and its influence in the lost of coherence as well
as the in the decay of correlations between the system and the bath.
The kernel $k_{\mathrm{TB}}(\tau)$ contains the influence of the bath on the
thermal equilibrium state of the system S.
The real and imaginary parts of $K_{\mathrm{TB}}(s-\mathrm{i}\tau) $ are given
by
\begin{align}
 K^{\mathrm{re}}_{\mathrm{TB}}(s-\mathrm{i}\tau) &=
 \int\limits_{0}^{\infty} \frac{\mathrm{d} \omega}{\pi}
 J_{\mathrm{TB}}(\omega)\frac{\cosh[\omega(\frac{1}{2}\hbar\beta_{\mathrm{TB}})-\tau]}
                 {\sinh(\frac{1}{2}\omega\hbar\beta_{\mathrm{TB}})} \cos(\omega s)
\nonumber\\&
=\frac{m}{\hbar\beta_{\mathrm{TB}}}\sum_{n=-\infty}^{\infty}g_n(s)\exp(\mathrm{i}\nu_n \tau),
\\
 K^{\mathrm{im}}_{\mathrm{TB}}(s-\mathrm{i}\tau) &=
 -\int\limits_{0}^{\infty} \frac{\mathrm{d} \omega}{\pi}
 J_{\mathrm{TB}}(\omega)\frac{\sinh[\omega(\frac{1}{2}\hbar\beta_{\mathrm{TB}})-\tau]}
                 {\sinh(\frac{1}{2}\omega\hbar\beta_{\mathrm{TB}})} \sin(\omega s)
\nonumber \\&=\frac{m}{\hbar\beta_{\mathrm{TB}}}\sum_{n=-\infty}^{\infty}\mathrm{i}
f_n(s)\exp(\mathrm{i}\nu_n \tau).
\end{align}
while
$ \zeta_{n}(s) =\frac{1}{m}\int\limits_{0}^{\infty} \frac{\mathrm{d} \omega}{\pi}
               \frac{J_{\mathrm{TB}}(\omega)}{\omega}\
               \frac{2\nu_n^2}{\omega^2 + \nu_n^2}\cos(\omega s)$
or in terms of the damping kernel $\gamma_{\mathrm{TB}}(s)$ \cite{GSI88},
$ \zeta_{n}(s) = \frac{1}{2} |\nu_n|\int\limits_0^{\infty} \mathrm{d}u\,
\gamma_{\mathrm{TB}}(u) [\exp(-|\nu_s(s+u)|) + \exp(-|\nu_s(s-u)|) ]$.
For the case when system-bath interactions are neglected, i.e., when there is no mixing
between the imaginary time trajectory $\bar{q}(\tau)$ and the real time trajectory $x(s)$ in
Eq.~(\ref{equ:GenSigma}), only the real part of $K_{\mathrm{TB}}(s)$ contributes to
the dynamics, which is consistent with calculations derived under the factorizing initial
condition assumption.

The functions $g_n(s)$ and $f_n(s)$ can be expressed in terms of the damping
kernels
\begin{equation}
\gamma_{\mathrm{TB},\mathrm{BB}}(s)
=
\frac{2}{m} \int\limits_{0}^{\infty} \frac{\mathrm{d} \omega}{\pi}
               \frac{J_{\mathrm{TB},\mathrm{BB}}(\omega)}{\omega}\cos(\omega s),
\end{equation}
and $\zeta_n(s)$ as
$g_n(s) = \gamma_{\mathrm{TB}}(s)-\zeta_n(s)$ and
$f_n = -\frac{1}{\nu_n}\frac{\mathrm{d}}{\mathrm{d}s}\zeta_n(s)$.
The spectral density in Eq.~(\ref{equ:JwTB}) generates the damping
kernel
$\gamma(s) = \gamma_{\mathrm{TB}}
\Omega_{\mathrm{TB}} \exp(-\Omega_{\mathrm{TB}}|s|)$.
In the limit when the cutoff frequency $\Omega_{\mathrm{TB}}$ tends to infinity,
$\gamma(s) \rightarrow 2 \gamma_{\mathrm{TB}} \delta(s)$, which corresponds
to Markovian Ohmic dissipation. The spectral density in Eq.~(\ref{equ:JwBB})
generates
$
\gamma_{\mathrm{BB}}(s) =\tau_{\mathrm{BB}} \Omega_{\mathrm{BB}}^2
\left[2 \delta(s) - \Omega_{\mathrm{BB}} \exp(-\Omega_{\mathrm{BB}}|s|)\right].
$
Note that there is a fundamental limitation to the use of Eq. (\ref{equ:JwBB}).
That is, in the limit $\Omega_{\mathrm{BB}} \rightarrow\infty$, we get the
surprising result that
$\gamma_{\mathrm{BB}}(s) = 0$, i.e. no relaxation \cite{FO91,FO98}.
This corresponds to  the point-electron limit
[$f_k = \Omega_{\mathrm{BB}}^2/(\Omega_{\mathrm{BB}}^2 + \omega_k^2) = 1$
in Eq.~(\ref{equ:AVPBBR})] and is unphysical  because even for the electron,
$\Omega_{\mathrm{BB}}$ remains finite, although large.
According to Refs.~\citenum{FO91,FO98}, there is a natural upper value given by
$\Omega_{\mathrm{BB}} = {\tau_{\mathrm{BB}}}^{-1}$, which corresponds to
two-thirds of the time for a photon to traverse the classical electron radius
($r_\mathrm{cl}^{\mathrm{e}} = 2.818\times10^{-15}$m).
Beyond this natural limit, causality is violated \cite{FO91} and the bare mass $m$
takes negative values \cite{FO91}.

Finally, the kernel $K_{\mathrm{BB}}(s)$ is given by
\begin{equation}
\label{equ:KBB}
 K_{\mathrm{BB}}(s) = \int\limits_{0}^{\infty} \frac{\mathrm{d} \omega}{\pi}
               J_{\mathrm{BB}}(\omega)
               \coth\left(\frac{\omega \hbar\beta_{\mathrm{BB}}}{2}\right)\cos(\omega s).
\end{equation}
This kernel is responsible for the decoherence due to thermal fluctuations induced
by the blackbody radiation, while the kernels
$\eta_{\mathrm{TB}}(s)=m \gamma_{\mathrm{TB}}(s)$ and
$\eta_{\mathrm{BB}}(s)=  m \gamma_{\mathrm{BB}}(s)$ in Eq.~(\ref{equ:GenSigma})
induce the relaxation process.

\subsection{Explicit calculation of the propagating function}
The explicit calculation of the propagating function demands  evaluating  the path integral  in
Eq.~(\ref{equ:GenPropFunct}).
Since, the system is linear, the path integral can be performed by evaluating
the action in Eq.~(\ref{equ:GenSigma}) along its stationary trajectories and
condensing the effect of the fluctuations in a global time dependent factor
\cite{GSI88,Wei08}.
The extremum of the action for imaginary time is given by
\begin{equation}
m \ddot{\bar{q}} - m \omega_0^2 \bar{q} -
\int\limits_0^{\hbar \beta_{\mathrm{TB}}}
\mathrm{d} \sigma k_{\mathrm{TB}}(\tau -\sigma) \bar{q}(\sigma)
= -\mathrm{i}\int\limits_0^t \mathrm{d} s K_{\mathrm{TB}}^*(s-\mathrm{i}\tau)x(s),
\end{equation}
where we can see how the dynamics in real time, represented by $x(s)$, drives
the system-bath correlations, by driving the imaginary time path $q(\tau)$ in a
non-local way.
For real time, the  action is stationary along
\begin{align}
\begin{split}
&\hspace{-0.5cm}m \ddot{r} + m \omega(t)^2 r -  E_{\mathrm{L}}(t)
+\frac{\mathrm{d}}{\mathrm{d} s}\int\limits_0^s \mathrm{d} u \,\eta(s-u)r(u)
\\
&\hspace{-0.5cm}= r'\eta_{\mathrm{BB}}(s) + \mathrm{i}\int\limits_0^t \mathrm{d} u K(s-u)x(u)
+ \hspace{-0.25cm}\int\limits_0^{\hbar \beta_{\mathrm{TB}}}  \mathrm{d} \hspace{-0.0125cm} \tau
K_{\mathrm{TB}}^*(s - \mathrm{i} \tau)\bar{q}(\tau),
\end{split}
\\
\begin{split}
m \ddot{x} &+ m \omega(t)^2 x
- \frac{\mathrm{d}}{\mathrm{d} s}\int\limits_s^t \mathrm{d} u \,\eta(s-u)x(u)=0,
\end{split}
\end{align}
where we have defined
\begin{equation*}
\eta(s) = \eta_{\mathrm{TB}}(s) + \eta_{\mathrm{BB}}(s), \quad
K(s) = K^{\mathrm{re}}_{\mathrm{TB}}(s) + K_{\mathrm{BB}}(s).
\end{equation*}
The term $ r'\eta_{\mathrm{BB}}(s)$ appears here as a consequence of the sudden
turn-on of the blackbody radiation. Since, we assume that the parametric driving,
as well as the laser field and the blackbody radiation act after $t=0$, the equilibrium
state of our system coincides with the one derived in Ref.~\citenum{GSI88}, so we
 need to focus only on the evaluation of the real part of the action.

The real part of the action is stationary along the solution to the
equation of motion
\begin{linenomath}
\begin{align}
\begin{split}
\label{equ:FinalEOMr}
m \ddot{r} &+ m \frac{\mathrm{d}}{\mathrm{d} s}\int\limits_0^s \mathrm{d} u \,\gamma(s-u)r(u)
+ m \omega(t)^2 r
\\
&=\bar{E}_{\mathrm{L}}(s) + \mathrm{i} m \int\limits_0^t \mathrm{d} u R(s-u)x(u),
\end{split}
\\
\label{equ:FinalEOMx}
m \ddot{x}  &- m \frac{\mathrm{d}}{\mathrm{d} s}\int\limits_s^t \mathrm{d} u \,\gamma(s-u)x(u)
+ m \omega(t)^2 x = 0,
\end{align}
\end{linenomath}
where we have defined
$\bar{E}_{\mathrm{L}}(s) =
E_{\mathrm{L}}(s) + r'\eta_{\mathrm{BB}}(s) + m [\bar{x} C_1(s) -
\mathrm{i} \bar{x} C_2(s)]$ with $\bar{r} = (\bar{q}+\bar{q}')/2$ and
$\bar{x} = \bar{q}-\bar{q}'$. Additionally, we have defined
$\eta(s) = m \gamma(s)$ and
\begin{equation*}
R(s,u) = R_{\mathrm{TB}}(s,u) +  K(s-u)/m
\end{equation*}
 with
\begin{align}
\begin{split}
 R_{\mathrm{TB}}&(s,u) = -\Lambda_{\mathrm{TB}} C_{1}(s)C_{1}(u)
\\&
+ \frac{1}{\hbar\beta_{\mathrm{TB}}}
\sum_{n=-\infty}^{\infty} u_{n}[g_{n}(s) g_{n}(u) - f_{n}(s) f_{n}(u)],
\end{split}
\\
 C_{1}(s) &= \frac{1}{\hbar\beta_{\mathrm{TB}}\Lambda_{\mathrm{TB}}}
 \sum_{n=-\infty}^{\infty} u_{n} g_{n}(s),
\\
 C_{2}(s) &= \frac{1}{\hbar\beta_{\mathrm{TB}}}\sum_{n=-\infty}^{\infty} u_{n} \nu_{n} f_{n}(s),
\end{align}
where $ \Lambda_{\mathrm{TB}} =\frac{1}{\hbar\beta_{\mathrm{TB}}}
\sum_{n=-\infty}^{\infty} u_{n}$ can be related to the second moment of the position
of the system at equilibrium with TB,
$\langle q^2 \rangle_{\mathrm{equil.}} = (\hbar/m)\Lambda_{\mathrm{TB}}$, and
$u_{n} = (\omega_0^2 + \nu_n^2 + \zeta_n)^{-1}$.

Since, for a harmonic potential, the functional action $\Sigma [x,r,\bar{q}_x]$
can be evaluated using only the real part of the trajectories $r(s)$ and $x(s)$
\cite{HA85,SG86,GSI88,PID10}, we  need to solve only for the real part of Eq.
(\ref{equ:FinalEOMr}) and (\ref{equ:FinalEOMx}).
Due to the linear character of (\ref{equ:FinalEOMr}), the solution to the
homogeneous part can be written as
\begin{equation}
\begin{split}
r(s) &= r'' \frac{\phi_1(s)}{\phi_1(t)} + r' \left(\phi_2(s) -
\frac{\phi_2(t)}{\phi_1(t)} \phi_1(s) \right),
\end{split}
\end{equation}
where $\phi_1(s)$ is the fundamental solution for $r(0)=0$ and $\dot{r}(0)=1$,
while $\phi_2(s)$ is the fundamental solution for $r(0)=1$ and $\dot{r}(0)=0$.
Thus, for Eqs. (\ref{equ:FinalEOMr}) and (\ref{equ:FinalEOMx}) we have
\begin{align}
\label{equ:Finar}
\begin{split}
 r^{\mathrm{re}}(s) &= r'' \frac{\phi_1(s)}{\phi_1(t)} + r'\left(\phi_2(s) - \frac{\phi_2(t)}{\phi_1(t)} \phi_1(s) \right)
\\       &+ \frac{1}{m}\int_0^s \mathrm{d}u \phi_1(s-u) \bar{F}'(u)
           - \frac{1}{m}\frac{\phi_1(s)}{\phi_1(t)}\int_0^t \mathrm{d}u \phi_1(t-u) \bar{F}'(u).
\end{split}
\end{align}
Since $\bar{F}'(s)$ contains the term induced by the sudden coupling to the radiation,
$r'\eta_{\mathrm{BB}}(s)$, we can see that $r(s)$ is driven by this sudden turn on.

Following a similar procedure for $x(s)$, we get
\begin{align}
\label{equ:Finax}
\begin{split}
 x(s) &= x'' \frac{\varphi_1(s)}{\varphi_1(t)} + x' \left(\varphi_2(s) -
 \frac{\varphi_2(t)}{\varphi_1(t)} \varphi_1(s) \right),
\end{split}
\end{align}
where $\varphi_1(s)$ is the fundamental solution for $x(0)=0$ and $\dot{x}(0)=1$,
while $\varphi_2(s)$ is the fundamental solution for $x(0)=1$ and $\dot{x}(0)=0$.

For the particular case of no frequency modulation, $\phi_1(s)$, $\phi_2(s)$,
$\varphi_1(s)$ and $\varphi_2(s)$ can be derived from standard Laplace techniques
\cite{GSI88}.
For Markovian dissipation, $\Omega_{\mathrm{TB}}\rightarrow \infty$ in Eq.~(\ref{equ:JwTB}),
and harmonic modulation of the frequency, $\phi_1(s)$, $\phi_2(s)$, $\varphi_1(s)$ and $\varphi_2(s)$
are related to the Mathieu functions \cite{ZH95,GPZ10}.
For more general cases, these functions must be calculated numerically.
However, the functional form of Eqs. (\ref{equ:Finar}) and (\ref{equ:Finax})
is very convenient of the subsequent analytical calculations.
For further convenience we define
\begin{align}
\label{equ:v1v2}
v_1(t,s) &= \varphi_2(s) -  \frac{\varphi_2(t)}{\varphi_1(t)} \varphi_1(s),
\quad
v_2(t,s) =\frac{\varphi_1(s)}{\varphi_1(t)} ,
\\
\label{equ:u1u2}
u_1(t,s) &= \phi_2(s) -  \frac{\phi_2(t)}{\phi_1(t)} \phi_1(s),
\quad
u_2(t,s) =\frac{\phi_1(s)}{\phi_1(t)}.
\end{align}

The influence functional in Eq.~(\ref{equ:GenPropFunct}) can now be
rewritten as
\begin{equation}
\label{equ:GenPropFunctFinal}
\begin{split}
J(r'',x'',t;r',x',0;\bar{r},\bar{x})  =
\frac{1}{N(t)}
\exp\left(\frac{\mathrm{i}}{\hbar} \Sigma[r,x,\bar{r},\bar{x}]\right),
\end{split}
\end{equation}
where $N(t)$ is a normalization factor given by
$N(t) = 2\pi \hbar \frac{1}{m} |\dot{u}_2(t,0)
|\left(2\pi \hbar \frac{1}{m}\Lambda_{\mathrm{TB}}\right)^{1/2}$.
After evaluating Eq. (\ref{equ:GenSigma}) along $r^{\mathrm{re}}(s)$ and $x(s)$, we get
\begin{widetext}
\begin{equation}
\begin{split}
\label{equ:FinalSigma}
\Sigma& [r'',x'',r',x',\bar{r},\bar{x}]=
\\
& \mathrm{i}m \left( \frac{1}{2\Lambda_{\mathrm{TB}}}\bar{r}^2
+ \frac{\Omega_{\mathrm{TB}}}{2}\bar{x}^2\right)
+ m \left[x'' r'' \dot{u}_2(t,t) + x' r' \dot{u}_1(t,0) \right]
-m\left[x' r'' \dot{u}_2(t,0) - x'' r' \dot{u}_1(t,t) \right]
\\
& + m\int\limits_0^{t} \mathrm{d}s
\left[x' v_1(t,s)+ x'' v_2(t,s) \right]
\left[\frac{1}{m}E_{\mathrm{L}}(s) + r' \gamma_{\mathrm{BB}}(s) +
\bar{r} C_1(s) - \mathrm{i}\bar{x}C_2(s)\right]
\\
\phantom{\Sigma [r'',x'',r',x',\bar{r},\bar{x}]}\hspace{-0.75cm}
&+\frac{\mathrm{i}}{2}m {x'}^2 \int_0^t \mathrm{d}s  \int_0^t \mathrm{d}u
R(s,u)  v_1(t,s) v_1(t,u)
+\frac{\mathrm{i}}{2}m x' x'' \int_0^t \mathrm{d}s  \int_0^t \mathrm{d}u
R(s,u) v_1(t,s) v_2(t,u)
\\
&+\frac{\mathrm{i}}{2}m x'' x'\int_0^t \mathrm{d}s  \int_0^t \mathrm{d}u
R(s,u) v_2(t,s) v_1(t,u)
+\frac{\mathrm{i}}{2}m {x''}^2 \int_0^t \mathrm{d}s  \int_0^t \mathrm{d}u
R(s,u)  v_2(t,s) v_2(t,u),
\end{split}
\end{equation}
\end{widetext}
where $\Omega_{\mathrm{TB}} =\frac{1}{\hbar\beta_{\mathrm{TB}}}
\sum_{n=-\infty}^{\infty} u_{n}(\omega_0^2 + \zeta_n)$ can be related
to the second moment of the momentum,
$\langle p^2 \rangle_{\mathrm{equil.}} = \hbar m\Omega_{\mathrm{TB}}$,
at equilibrium with TB.
The first term in Eq.~(\ref{equ:FinalSigma}) containing $\Lambda_{\mathrm{TB}}$
and $\Omega_{\mathrm{TB}}$ can be associated to the thermal equilibrium state
influenced by the presence of the thermal bath TB.
These results provide the general expression for the influence functional.
\subsection{Limiting cases}
The general result in Eq.  (\ref{equ:GenPropFunctFinal})
includes, and agrees with, several limiting cases. These include:
\begin{enumerate}[i]
\item \emph{In absence of blackbody radiation and with no parametric
modulation of the frequency}, Eq.~(\ref{equ:GenPropFunctFinal}) reduces to the result in
Ref.~\citenum{GSI88}.

\item \emph{In absence of the thermal bath and for no laser field and
no parametric modulation of the frequency}, Eq.~(\ref{equ:GenPropFunctFinal}) is the
formal path integral equivalent of Refs.~\citenum{FLO85,FLO88} (based on the quantum
Langevin equation formalism).

\item \emph{In absence of the thermal bath and with no parametric
modulation of the frequency}, Eq.~(\ref{equ:GenPropFunctFinal}) is the formal path integral
equivalent of Refs.~\citenum{FLO85,FLO88} to the result in Refs.~\citenum{LFO90,LFO90b}.

\item \emph{In absence of blackbody radiation, for no laser field,
harmonic modulation of the frequency and factorized initial conditions},
Eq.~(\ref{equ:GenPropFunctFinal}) reduces to Ref.~\citenum{ZH95} (see also Ref.~\citenum{GPZ10}).

\item \emph{In absence of  blackbody radiation, for no laser field,
no modulation of the frequency and for factorized initial conditions},
Eq.~(\ref{equ:GenPropFunctFinal}) reduces to Refs.~\citenum{CL83,GWT84} (see also Ref.~\citenum{PID10}
for a description in terms of the Wigner function and Ref.~\citenum{HPZ92} for a master equation
approach).
\end{enumerate}

\subsection{Explicit form of the propagating function}
For an initial thermal state, i.e.  $\lambda(q_{+}',{\bar q}'',q_{-}',{\bar q}') = \delta(q_{+}'-{\bar q}'')
\delta(q_{-}'-{\bar q}')$ in Eq.~(\ref{equ:initialStateGen}) and correspondingly $\bar{r}=r'$ and
$\bar{x} = x'$, the influence functional in Eq.~(\ref{equ:GenPropFunctFinal}) with the function
phase given in Eq.~(\ref{equ:FinalSigma}) can be written in the very compact form
\begin{align}
\label{equ:InfFuncMatForm}
\begin{split}
J&(r'',x'',t;,r',x',0;r',x') = \frac{1}{N(t)}
\\&\times
\exp\left\{-\frac{1}{2} \mathbf{x}^{\mathrm{T}} \mathsf{A}(t) \mathbf{x}
+ E_{\mathrm{L}}^+(s) x' +  E_{\mathrm{L}}^-(s) x'' \right\}
\end{split}
\end{align}
where
$ \mathbf{x}^{\mathrm{T}} \mathsf{A}(t) \mathbf{x}=
 \mathbf{x}_f^{\mathrm{T}} \mathsf{A}_f(t) \mathbf{x}_f+
\mathbf{x}_i^{\mathrm{T}} \mathsf{A}_i(t) \mathbf{x}_i
+2\mathbf{B} \mathbf{x}_i$
where $\mathbf{x} = (x'', r'',x',r')$, $\mathbf{x}_{i} = (x', r')$,
 $\mathbf{x}_f = (x'', r'')$,
 $\mathbf{B} = (\mathsf{A}_{13} x'' + \mathsf{A}_{23} r'', \mathsf{A}_{14} x'')$
with the time dependent matrix $\mathsf{A}(t)$ given by
\begin{widetext}
\begin{equation}
\mathsf{A}(t) = \frac{m}{\hbar}\left(
\begin{array}{cccc}
R_{22}(t) & -\mathrm{i} \dot{u}_2(t,t)& -C_2^-(t) + R_{12}(t)&
-\mathrm{i} \dot{u}_1(t,t) - \mathrm{i} \tilde{C}_1^-(t)
\\
-\mathrm{i}\dot{u}_2(t,t)& 0 & \mathrm{i} \dot{u}_2(t,0) & 0
\\
-C_2^-(t) + R_{12}(t) & \mathrm{i} \dot{u}_2(t,0)  &
\Omega_{\mathrm{TB}} - 2 C_2^+(t) + R_{11}(t)&
-\mathrm{i} \dot{u}_1(t,0) - \mathrm{i} \tilde{C}_1^+(t)
\\
-\mathrm{i} \dot{u}_1(t,t) - \mathrm{i} \tilde{C}_1^-(t)& 0 &
-\mathrm{i} \dot{u}_1(t,0) - \mathrm{i} \tilde{C}_1^+(t)&
1/\Lambda_{\mathrm{TB}}
\end{array}
\right),
\end{equation}
\end{widetext}
being
\begin{linenomath}
\begin{equation}
\label{equ:Cjpm}
C_j^+(t) =
\int\limits_0^t  \mathrm{d}s  C_j(s) v_1(t,s),
\quad
C_j^-(t) =
\int\limits_0^t \mathrm{d}s C_j(s) v_2(t,s).
\end{equation}
The $C_j^{\pm}$s functions account for the influence of the initial correlations between
the system and the bath on the system dynamics. $\tilde{C}_1^{\pm}(t)$ is obtained by replacing
$C_1(t) \rightarrow C_1(t) +  \gamma_{\mathrm{BB}}(t) $ in the definition of
$C_1^{\pm}(t)$ in Eq.~(\ref{equ:Cjpm}) and contains the effect of the turn on of the
interaction with the blackbody radiation.
The effects of the laser field $E_{\mathrm{L}}$ on the dynamics are encoded in
\begin{equation}
\label{equ:Epm}
E_{\mathrm{L}}^+(t) =
\int\limits_0^t  \mathrm{d}s  E_{\mathrm{L}}(s) v_1(t,s),
\quad
E_{\mathrm{L}}^-(t) =
\int\limits_0^t \mathrm{d}s E_{\mathrm{L}}(s)v_2(t,s).
\end{equation}
The decoherence dynamics is governed by the $R_{ij}(t)$ functions given by
\begin{equation}
\label{equ:Rij}
R_{ij}(t) =
\int_0^t \mathrm{d}s  \int_0^t \mathrm{d}u R(s,u)
v_i(t,s) v_j(t,s).
\end{equation}
\end{linenomath}

Once we have completely characterized the functional form of the propagating
function, we proceed in the next sections to derive explicit expressions for the
covariance matrix elements of the quantum fluctuations and the time evolution
of the reduced density-operator.

\section{Time evolution of the covariance matrix elements
Due to the linear nature of the system under consideration, the dynamics as well as the statistical
properties can be characterized in terms of the first and second moments \cite{GWT84},
defined as
$\langle f(q(t)) \rangle = \int \mathrm{d} r''
f(r'') \rho_{\mathrm{S}}(r'',x''=0,t)$, or in terms of the propagating function in
Eq.~(\ref{equ:InfFuncMatForm}) by,
\begin{equation}
\langle f(q(t)) \rangle = \int \mathrm{d} r'' \mathrm{d} r' \mathrm{d} x'
f(r'') J(r'',x''=0,r',x',r',x').
\end{equation}
So, we can now make use of the explicit form of the propagating
function in Eq.~(\ref{equ:InfFuncMatForm}).
Note that the propagating function in Eq.~(\ref{equ:InfFuncMatForm})
also contains  the initial thermal state $\hat{\rho}_{\beta,\mathrm{S}}$.
These moments define the variances or dispersion relations
\begin{align}
\label{equ:sigmaqq}
\sigma_{qq}(t) &= \langle q^2(t) \rangle - \langle q(t) \rangle^2,
\\
\label{equ:sigmaqp}
\sigma_{qp}(t) &= \frac{1}{2}\langle q(t)p(t) + p(t)q(t)\rangle
- \langle q(t) \rangle\langle p(t) \rangle,
\\
\label{equ:sigmapp}
\sigma_{pp}(t) &= \langle p^2(t) \rangle - \langle p(t) \rangle^2,
\end{align}
which will be used in order to express explicitly  the time evolution of the
density matrix in Eq.~(\ref{equ:DenMatSFinal}).

The first moments are determined by
\begin{align}
\label{equ:q}
\langle q(t) \rangle &= \frac{1}{m \dot{u}_2(t,0)}\int_0^t \textrm{d}s\, v_1(t,s)E_{\mathrm{L}}(s)
\\
\label{equ:p}
\langle p(t) \rangle &=
\int_0^t \textrm{d}s \left[v_1(t,s) \frac{\dot{u}_2(t,t)}{\dot{u}_2(t,0)}
+ v_2(t,s)\right] E_{\mathrm{L}}(s)
\end{align}
where the overdot denotes the derivative with respect to $s$, i.e.,
$\dot{u}_2(t,0) = \partial u_2(t,s)/\partial s|_{s=0}$.

The second moments are given by
\begin{align}
\begin{split}
\label{equ:qq}
\langle q^2(t) \rangle &= \langle q(t) \rangle^2
+
\frac{\hbar^2}{m^2 \dot{u}^2_2(t,0)}
\left[M_{11}(t) -\frac{\hbar}{m}{M_{12}(t)}^2\right] ,
\end{split}
\\
\begin{split}
\label{equ:pq}
\frac{1}{2}\langle pq + qp \rangle &= m \langle q^2(t) \rangle \dot{u}_2(t,t)
+ \frac{\hbar}{\dot{u}_2(t,0)} \left[R_{12}(t) - C_2^-(t) \right]
\\ &+ \mathrm{i} \frac{\hbar^2\Lambda_{\mathrm{TB}}}{m \dot{u}_2(t,0)}
\left[\tilde{C}_1^-(t) + \dot{u}_1(t,t)\right]M_{12}(t)
\\
&+  \langle q(t) \rangle \int_0^t \mathrm{d}s v_2(t,s)E_{\mathrm{L}}(s),
\end{split}
\\
\begin{split}
\label{equ:pp}
\langle p^2(t) \rangle &= \hbar m R_{22}(t)
+ \hbar m \Lambda_{\mathrm{TB}} \left[\tilde{C}_1^-(t) +\dot{u}_1(t,t)\right]^2
\\ &
- m^2 \dot{u}^2_2(t,t)\langle q^2(t) \rangle
+m\dot{u}_2(t,0)\langle pq + qp \rangle
\\
&+ \left[\int_0^t \hspace{-0.1cm}\mathrm{d}s v_2(t,s)E_{\mathrm{L}}(s)\right]^2,
\end{split}
\end{align}
where
\begin{equation}
\mathsf{M} = \frac{m}{\hbar}
\left(
\begin{tabular}{cc}
$\Omega_{\mathrm{TB}} - 2C_2^+(t) + R_{11}$ &
$-\mathrm{i}\left[\dot{u}_1(t,0) + \tilde{C}_1^+(t)\right]$
\\
$-\mathrm{i}\left[\dot{u}_1(t,0) + \tilde{C}_1^+(t)\right]$ &
$1/\Lambda_{\mathrm{TB}}$
\end{tabular}
\right).
\end{equation}

In the absence of the parametric driving, the blackbody radiation and the laser field,
Eqs.~(\ref{equ:qq})-(\ref{equ:pp}) are time-independent and coincide with the
expressions (6.62)-(6.64) in Ref.~\citenum{GSI88}, i.e.,
$\langle q^2(t) \rangle = (\hbar/m)\Lambda_{\mathrm{TB}}$,
$\langle pq + qp \rangle = 0$ and
$\langle p^2(t) \rangle = \hbar m \Omega_{\mathrm{TB}} $.

\section{Time evolution of the reduced density-operator
If one is interested in the reduced-density operator itself, it can be written in terms of the second moments
as
\begin{equation}
\label{equ:DenMatSFinal}
\begin{split}
\bok{r''}{\hat{\rho}_{\mathrm{S}}(t)}{x''} &= \left(2\pi\sigma_{qq}(t)\right)^{-1/2}
\exp\left[-\frac{1}{2\sigma_{qq}(t)}\left[r'' - \langle q(t) \rangle\right]^2
\right. \\
&- \left. \frac{1}{2\hbar^2} \left(\sigma_{pp}(t)
- \frac{\sigma_{pq}(t)^2}{\sigma_{qq}(t)} \right) {x}''^2
\right.
\\
&+ \left.\frac{{\rm i}}{\hbar}\left\{ \langle p(t) \rangle +
 \frac{\sigma_{pq}(t)^2}{\sigma_{qq}(t)}
 \left(r'' - \langle q(t) \rangle \right)\right\}x''\right].
\end{split}
\end{equation}
In summary, to obtain the time evolution of the reduced density operator, we proceed
as follows:
\begin{enumerate}[i]
\item One first specifies the spectral density $J_{\mathrm{TB}}(\omega)$
to obtain the function
describing the modulating force in Eq.~(\ref{equ:HamSysTBBBb}).
This permits us to obtain
the fundamental solutions $\phi_{1,2}(s)$ and $\varphi_{1,2}(s)$
in Eq.~(\ref{equ:Finar})
and  Eq.~(\ref{equ:Finax}), respectively.
\item One then calculates
the kernels $K_{\mathrm{TB}}(s)$, $k(\tau)$ and $K_{\mathrm{BB}}(s)$
defined in Eqs.~(\ref{equ:KTB}), (\ref{equ:kkTB}) and (\ref{equ:KBB}).
\item With the fundamental solutions  obtained, and all the kernels calculated,
we calculate the auxiliary functions $v_{1,2}(t,s)$ and $u_{1,2}(t,s)$ given in Eqs.~(\ref{equ:v1v2})
and (\ref{equ:v1v2}), and subsequently the functions $C_j^{\pm}(t)$, $E^{\pm}(t)$ and $R_{ij}(t)$
defined by Eqs.~(\ref{equ:Cjpm}), (\ref{equ:Epm}) and (\ref{equ:Rij}), respectively.
\item One then calculates the first and second moments given in
Eqs.~(\ref{equ:q}-\ref{equ:pp}), and subsequently the dispersion relations in
Eqs.~(\ref{equ:sigmaqq}-\ref{equ:sigmapp}) and system dynamics via Eq. (\ref{equ:DenMatSFinal}).
\end{enumerate}

This brief prescription concludes our completely formal and approximation-free
treatment.
Note that we have successfully applied the method to a number of cases,
some of which are reported elsewhere \cite{PB13,PB13c}.
\section{Concluding Remarks
As discussed in the introduction, the results derived here can be used to study a wide variety of
problems, e.g., the incoherent \cite{PB13} or coherent \cite{PB13c} excitation of open quantum
systems in order to provide physical insight into the role of coherences detected
in photosynthetic light-harvesting complexes (for a review in the subject see Ref.~\citenum{PB12c}).
In doing so, we need to translate the propagating function in Eq.~(\ref{equ:InfFuncMatForm})
into the energy basis in order to identify the incoherent/coherent nature of the
excitation.

Our results allow us to directly study the possibility of environmentally assisted one-photon phase control
\cite{SAB10,PYB13} provided by the fact that the initial equilibrium density matrix deviates from the canonical
distribution. In this respect, we
eliminate the incoherent radiation and the frequency modulation
contributions and
focus on how the phase information encoded in $E_{\mathrm{L}}(t)$ can be used to manipulate the populations
of the oscillator \cite{PB13c}.

Additionally, the analytic closed expression could be useful in understanding the
delicate balance between dissipation and driving under non-Markovian evolution
that has been pointed out in Ref.~\citenum{SN&11} in the context of optimal control theory
and cooling of nano-mechanical resonators.
In particular, the optimal cooling protocol addressed in Ref.~\citenum{SN&11} by means of numerical
techniques, can be analyzed in great detail from the second moments derived in
Eqs.~(\ref{equ:qq})--(\ref{equ:pp}) and the theory of variational calculus.

In Ref. ~\citenum{GPZ10}, it was established that the usual quantum limit, $\hbar \omega / k_{\mathrm{B}}T >1$,
needs to be reformulated for out-of-equilibrium systems. However, in that work a Markovian Ohmic
spectral density, $\Omega_{\mathrm{TB}}\rightarrow \infty$ in Eq.~(\ref{equ:JwTB}), was used, disregarding
in this way the dynamics during non-Markovian time scales.
The physical system considered in Ref.~\citenum{GPZ10} consisted of two identical harmonic oscillators
with time-depend coupling, in the normal mode description we get two independent parametric
oscillators. So, under the same circumstances considered in Ref.~\citenum{GPZ10}, the results derived here
allow us to explore the limit for the presence of quantum features in
non-Markovian-driven-open-quantum systems, which is of great importance in, e.g., quantum
statistical mechanics or control theory.

\begin{acknowledgements}
LAP acknowledges discussions with Gert-Ludwig Ingold with pleasure.
This work was supported by the US Air Force Office of Scientific Research under contract
number FA9550-10-1-0260, by \emph{Comit\'e para el Desarrollo de la Investigaci\'on}
--CODI-- of Universidad de Antioquia, Colombia, under contract number E01651 and by
the \textit{Colombian Institute for the Science and Technology Development} --COLCIENCIAS--
under the contract number 111556934912
\end{acknowledgements}

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text:
Den Stærkere sejrer er en amerikansk stumfilm fra 1920 af Cecil B. DeMille.

Medvirkende
 Elliott Dexter som David Markely
 Gloria Swanson som Ruth Anderson
 Monte Blue som Jim Dirk
 Theodore Roberts som Luke Anderson
 Claire McDowell
 Michael D. Moore som Bobby
 Julia Faye
 Jim Mason
 Togo Yamamoto
 Theodore Kosloff
 William Boyd

Referencer

Eksterne henvisninger

Stumfilm fra USA
Amerikanske film fra 1920
Dramafilm fra USA

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I love love love making bread. Love it. I love the way the yeast smells as it ferments. I love the way the dough feels when I'm shaping a loaf. I love the smell of fresh bread right out of the oven. And most of all I love the taste and texture of good homemade bread.
I've been experimenting lately with different breads. I'm not going to lie, it can be hard, time consuming, and tiring. But these experiments are well-worth it in my opinion. So, I wanted to put together a few tips and tricks I've learned along the way. Below are links to a few of my stand-by, basic (read: easy/cheats guide) bread recipes as well.
The inspiration for this recipe simply came from some leftover pineapple. I had bought pre-packaged, pre-sliced pineapple and had about a cup of it left — just sitting there in its own juices. So I figured I may as well make something out it instead of letting it go bad.
4 ingredients and a little gelatin, and you've got what contestants on MasterChef struggle with almost every episode — a beautifully light panna cotta. Don't get me wrong, this seemingly simple dessert isn't without its complications, but with a little bit of practice and experimentation (and isn't that what cooking is all about anyway?), you'll wow your family and friends with this recipe.
You were looking at my Turkey Burger recipe and wondering about the bun, weren't you? No, I didn't buy them at the supermarket — I have this thing about fresh baked bread (love love love it), so I can't resist every opportunity to make it. Burger night is no exception.


text:
Home › 19th century Art › American Art › Sculptor
William Wetmore Story | Neoclassical sculptor
William Wetmore Story - Semiramis, 1873
William Wetmore Story (February 12, 1819 - October 7, 1895) was an American sculptor, art critic, poet and editor.
William Wetmore Story was the son of jurist Joseph Story and Sarah Waldo (Wetmore) Story. He graduated at Harvard College in 1838 and at the Harvard Law School in 1840, continued his law studies under his father, was admitted to the Massachusetts bar, and prepared two legal treatises of value - Treatise on the Law of Contracts not under Seal (2 vols., 1844) and Treatise on the Law of Sales of Personal Property, 1847.
William Wetmore Story - Cleopatra, 1858
William Wetmore Story - Delilah, 1877
William Wetmore Story - John Marshall in the Supreme Court
Abandoning the law, he devoted himself to sculpture, and after 1850 lived in Rome, whither he had first gone in 1848, and where he was intimate with the Brownings and with Walter Savage Landor. In 1856, he received a commission for a bust of his late father, which resides in the Memorial Hall/Lowell Hall. Story's apartment, in Palazzo Barberini, became a central location for Americans in Rome. During the American Civil War his letters to the Daily News in December 1861 (afterwards published as a pamphlet, "The American Question", i.e. of neutrality), and his articles in Blackwood's, had considerable influence on English opinion.
One of his most famous works, Cleopatra, (1858) was described and admired in Nathaniel Hawthorne's romance, The Marble Faun, and is on display at the Virginia Museum of Fine Arts in Richmond, Virginia.
Another work, the Angel of Grief, has been replicated near the Stanford Mausoleum at Stanford University. Among his other life-size statues he completed were those of Saul; Sapho: Electra; Semiramide; Delilah; Judith; Medea; Jerusalem Desolate; Sardanapolis; Solomon, Orestes; Canidia and Shakespeare. His Sibyl and Cleopatra were exhibited at the 1863 Universal Exposition in London.
Story submitted a design for the Washington Monument, then under construction. Although the Washington National Monument Society concluded that his design seemed "vastly superior in artistic taste and beauty" to the obelisk already under construction, the obelisk continued to be built, and is what we see today as the monument. In addition, Story sculpted a bronze statue of Joseph Henry on the Mall in Washington, D.C., the scientist who served as the Smithsonian Institution's first Secretary. His works Libyan Sibyl, Medea and Cleopatra are on display at the High Museum of Art in Atlanta, GA.
Story died at Vallombrosa Abbey, Italy; a place to which he had a sentimental attachment, and which he chronicled in an informal travel journal, Vallombrosa in 1881. He is buried with his wife, Emelyn Story, in the Protestant Cemetery, Rome, under a statue of his own design (Angel of Grief). A posthumous biography of Wetmore (and his circle), entitled William Wetmore Story and His Friends, was penned by Henry James.
His children also pursued artistic careers: Thomas Waldo Story (1855-1915) became a sculptor, Julian Russell Story (1857-1919) was a successful portrait painter, and Edith Marion (1844-1907), the marchesa Peruzzi de' Medici, became a writer.
William Wetmore Story - Medea, 1865
William Wetmore Story - Statue of John Marshall, 1883
William Wetmore Story - The Libyan Sibyl, 1867
William Wetmore Story - Angel of Grief / L'Angelo del Dolore, 1894
William Wetmore Story - Beethoven bronze
William Wetmore Story - George Peabody statue
William Wetmore Story - Jerusalem in Her Desolation, 1879
William Wetmore Story - Joseph Henry Statue
William Wetmore Story - Sappho, 1863
William Wetmore Story (Salem, 12 febbraio 1819 - Vallombrosa, 7 ottobre 1895) è stato uno scultore, critico d'arte, poeta ed editore Statunitense.
Nacque da Sarah Waldo Wetmore e dal marito, il giurista Joseph Story. Si laureò al Collegio Harvard nel 1838, e alla Scuola di legge di Harvard nel 1840, proseguendo gli studi grazie al padre, venne ammesso alla barra del Massachusetts, preparando due trattati giuridici di valore - Treatise on the Law of Contracts not under Seal (2 voll., 1844) e Treatise on the Law of Sales of Personal Property, 1847.
Abbandonando la professione in legge, si dedicò all'attività artistica per la scultura. Nel 1850 si trasferì definitivamente a Roma, dopo essere stato in un primo momento nel 1848, dove venne a contatto con Browning e Walter Savage Landor. Nel 1856, ricevette la commissione per realizzare un busto del suo defunto padre, situato nella Memorial Hall/Lowell Hall.
L'appartamento di Story, a Palazzo Barberini, divenne una centro di contatti culturali per la comunità statunitense nella capitale italiana. Durante la guerra civile americana le sue lettere inviate al Daily News, pubblicate nel dicembre 1861, ed i suoi articoli su Blackwood's, ebbero una considerevole influenza sull'opinione pubblica inglese.

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text:
About Lions International
About Lions New Zealand
History of Lions Clubs
LEO Clubs (12 - 30 Years)
Lions Clubs International Foundation
Lloyd Morgan Lions Clubs Charitable Trust
Child Mobility Foundation
Lions Mints
Heads Up For Kids
Diabetes NZ
Kidney Kids
NZ Liver Transplant Service
World Festival of Magic
Lions Christmas Cakes
Global Action Teams
My Lion
One Community at a Time
Across the globe, Lions are rolling up their sleeves and taking action. With over 1.4 million members, we're the largest service organization in the world. And we're as dedicated to helping those in need today as we were over 100 years ago when we first began. Visit the Lions Clubs International website.
Each Lion and every club is changing lives in ways both great and small. Service is a personal journey that impacts the people serving and the people we serve.
Lions serve. It's that simple, and it has been since we first began in 1917. Our clubs are places where individuals join together to give their valuable time and effort to improving their communities, and the world.
To be the global leader in community and humanitarian service
Where There's a Need, There's a Lion
Our history is a long one. And a proud one. For over 100 years, we've served with uncommon kindness, putting the needs of our neighbours, our communities and our world first. We look back at our past with pride, but we know it's only the beginning. Every day, we're building on our legacy with the goal of making an even greater impact in the future.
Every journey begins with a single step. One act of service, one encouraging word, one gift of generosity is often all it takes to bring hope where it's needed most.
Over the last 100 years, the kindness of Lions and Leos has multiplied across borders, oceans, and continents. With over 1.4 million members, we now have an opportunity to truly change our world.
Lions Clubs International Fact Sheet
We are 10,000 men and women serving in clubs across New Zealand, Tonga, Samoa, American Samoa and Fiji.
Complete the form and we'll help to match you up with your local club...
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text:
Novogrygorivka (en ) est un village situé dans le raïon de Beryslav, lui même situé dans l'oblast de Kherson, en Ukraine.

Historique
Lors de l'invasion de l'Ukraine par la Russie, le , Volodymyr Zelensky annonce que Novogrygorivka a été repris par les forces armées de l'Ukraine à l'armée russe, alors mises en difficulté par la contre-offensive de Kherson.

Notes et références

Ville dans l'oblast de Kherson


text:
Home COVID
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COVID Fred Jones - January 13, 2022
www.bloomberg.com By Kanoko Matsuyama No, Smoking Marijuana Does Not Protect You From Covid-19 Cannabis compounds prevented the virus that causes Covid-19 from penetrating healthy human cells, according to a laboratory study published in the Journal of Nature Products. The two compounds commonly found in hemp...
Rep. Alexandra Ocasio-Cortez Announces She Contracted A Breakthrough Case Of COVID-19
www.bet.com By Jamila Bey Her office says the New York representative had gotten a booster shot last fall. New York Rep. Alexandria Ocasio-Cortez announced that she tested positive for COVID-19. She was the fourth member of congress over the weekend to announce...
COVID Fred Jones - December 30, 2021
www.kjrh.com By: Brady Halbleib As the Omicron variant surges across the country and flu season underway, many are wondering what those sniffles and sneezes might actually mean? So far, Oklahoma is on track to have a typical flu season based on state...
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www.news9.com By: Erica Rankin COVID-19 cases across the United States have continued on an upward trend. Cases have surpassed what Oklahomans saw during the surge at the end of August. "If it is that much more transmissible and that many more people...
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COVID Fred Jones - December 8, 2021
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www.newson6.com A person in California became the first in the U.S. to have an identified case of the COVID-19 omicron variant, the White House announced Wednesday as scientists continue to study the risks posed by the new strain of the...
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www.fox23.co Oklahoma's Attorney General John O'Connor just tweeted that the Federal Court has granted his request to stop the requirement of COVID-19 vaccines on healthcare workers. O'Connor calls this a "major win for Oklahoma's healthcare heroes who have fearlessly braved the...
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COVID Fred Jones - November 24, 2021
www.aframnews.com Earlier this month, the U.S. Food and Drug Administration authorized the use of the Pfizer vaccine for children ages 5-11. The move has garnered criticism and speculation about the vaccine's effectiveness and health risks for children. Many parents are still...
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COVID Fred Jones - November 4, 2021
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This entry was posted on Saturday, October 8th, 2011 at 2:49 pm and is filed under Development, Tutorials. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.
This is very good Carl.
I wonder whether you might know how easy (or otherwise) it is to traverse through folders that are part of the list read based upon the uniqueid and/or filref that is returned?
I'm new to SharePoint and I used the username and password I use to log in to SharePoint server.
Sorry about the slow response. Would you be able to raise an issue over on https://github.com/thybag/PHP-SharePoint-Lists-API/issues with some more details about the problem your having. By the sounds of the copy of sharepoint your using may have authentication using basic auth disabled (potentially using something like NTLM instead?).


Owning a pet is a responsibility. And this responsibility incurs both monthly and annual costs for the expected and unexpected needs of the pet. Bearded Dragons are not the most expensive nor difficult of reptile pets. But they do require TLC to ensure their health and happiness.
They are a good pet for a beginner. But caring for your bearded dragon will require forethought and a certain amount of resources for the set up. These resources can be costly. But it's a noble expense to ensure the health and happiness of your beloved bearded dragon. Read through our article on how to properly care for a bearded dragon if you need more information.
If you are purchasing a very young juvenile and don't want to fork over a ton of money for an adult sized habitat, then you will be looking at approximately $125 for a 20 gallon horizontal tank. They are easy to come by, fairly light weight, and relatively easy to clean.
But remember, this tank is ideal for a juvenile bearded dragon. When your BD matures he or she is going to need a habitat at least twice that size. Then you will need an entirely new enclosure. And larger bearded dragon habitats can average about $325.
The smaller tank is fine, especially if you think you'll be owning more juvenile reptiles in the future. But if you are going to be a 'one reptile' owner, then it's likely you'll be spending money on a juvenile tank that you may never use again.
If you first buy the small enclosure and then replace it with a bigger one after a couple of years, you will have spent at least $450 in cage costs. Over the lifetime of your pet (about 10 years or so) you will have spent $45 per year in containment costs, as opposed to $30 annually if you purchased the larger size to begin with. Some reptiles (like corn snakes) need a small enclosure when young or they become insecure. But beardies like all the room they can get right from the beginning, so that is something to consider.
If you do decide on an adult sized enclosure, there are some really nice ones available from a number of providers who ship if you do not find what you need at a local pet shop. Sometimes a nice 40-60 gallon can be acquired for about $270 and free shipping if you shop carefully. If you do decide to begin with a smaller habitat, Exo Terra has a number of sizes, from a 20 gallon equivalent all the way up to their 'large wide' size for later in your pet's life, with a custom back wall and handy front opening doors with a lock ($350).
Once you have selected your BDs home size and design, he or she is going to need correct light and heat. Lighting that provides adequate UVB exposure will run about $30 - $50 for bulb and holder. Some holders clip to the side, some are designed to rest right on top of the enclosure's screen. Those that clip and swivel will cost $15 to $20, with the other kind running around $10 at the most. You will want to add a timer to this arrangement and that will run about $15-$20. This is especially important for busy working pet parents or young reptile keepers who may be a bit forgetful.
An undertank heating pad to keep your friend toasty at night will cost around $30. Reptile rocks are cheaper, but they have a bad reputation these days for a number of reasons and are not used as commonly as they used to be. If you don't have the ability to adjust the heat throughout the day, you may wish to invest in a second timer that turns the pad off for several hours in the middle of the day when the light is on and turns it back on at about dusk. Between the lamp and the heating pad, you will need an electrical strip with surge protection, which will be about another $15. The lamp bulb will need to be replaced yearly, so count on another $15-$20 annually for that expense.
Temperature and humidity need to be monitored, so add $20 for a digital laser thermometer and $25 for a side of the tank hygrometer. Although I don't particularly like them, especially for a large enclosure, you can get a combo thermometer/hygrometer that mounts to the inside of the tank for around $17.
You will also need a reptile fogger or mister system if you are away from home a lot. These start at $50 for the cheap ones and go up to $130 for models with better reviews.
Complete this ensemble with some plants ($10), a hammock ($10) a hide ($14) and some dishwasher safe faux branches ($30), a water dish ($5) and you are nearly ready. Last but not least, a beardie safe substrate such as that which Zoo Med produces ($7.00/bag) is needed. One bag will cover a small enclosure, two will be needed for a large one.
The price tag for all of this for an adult enclosure?
Approximately $550 before you even purchase your pet! And this doesn't include the cost of the actual bearded dragon.
This is something to think about and plan for, which is what all good animal parents should do anyway. Never, ever, buy a pet spontaneously just because they are so fluffy or appealing or interesting that you simply have to have it that very day. You will be wasting your money on the purchase and condemning your new pet to an early grave. That said, let's assume that you have crafted a wonderful and sustainable habitat for your new beardie. What is going to be the monthly cost of maintaining your friend?
Your BD will need to eat, of course, so that means purchasing food as a rule (unless you are raising your own). Buying food items from commercial suppliers is highly recommended. Because even though you spend money buying creatures that you feel you could go out into the yard and catch yourself, the food that you purchase will be clean and free of diseases and most importantly, parasites.
Little dragons eat fewer bugs at once, but they eat more often, so the price is relatively the same for raising a juvenile. It is for this reason that many keepers raise their own food items. Crickets chirp, and if they get loose you may find yourself with a very outdoorsy sound inside your home. That's why many keepers prefer to raise roaches. Dubia roaches are more expensive initially than crickets (you generally have to buy a colony from a breeder) and take up to 3 months to start producing enough young to keep your dragon happy, but the long term savings can be well worth the extra bother.
No matter the age of the bearded dragon, you'll need to factor in an adequate supply of vegetables. endive, kale, mustard greens, and chicory are just a few of the acceptable greens you can feed your beardie. But it's just as important to know what you cannot feed them, such as spinach, rhubarb, and avacados. Knowing what you cannot feed them can be beneficial to your budget. Because if your bearded dragon is fed the wrong type of food, then you'll have to factor in cost for a trip to the vet.
Veterinary Care - these are hard to predict, but start saving $20 per week. This will cover routine check ups or emergencies. (or if you are not disciplined enough then consider pet insurance).
Electrical – depending on the lamp used, the electricity needed to keep your pet lighted and toasty can run $10-$15 per month.
This is one of the most unpredictable aspects of owning any pet…professional health care and the expense of providing it. More and more pet owners are choosing to invest in pet insurance. The monthly cost for such insurance for a reputable company should be no more than $10/month. Yes, that's $120 per year that you may never use, but consider these figures.
If you take your BD to the vet annually, your cost could be $100 at a minimum (including office visit charge). If your beardie actually gets sick, you will be spending a great deal more. And then there's your gas expense for driving back and forth to the vet. This may be a little in an urban area, or a lot in a rural area. Pet insurance will not reimburse this, but wouldn't it be nice if all of the veterinary expenses were covered!
Consider this… if your beardie is an expensive morph directly from a breeder for around $400, and your set up is about that price as well, then you have plopped down a sizable chunk of change, not to mention the monthly expenses. So protecting your investment with insurance really makes good financial sense in the long run. The peace of mind of knowing your pet can be permitted to receive the best vet care available is also not a bad thing for the conscientious pet owner!


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Does History Link to Geography?
historyinpolitics	Location, Theory	 7th Sep 2020 10th Sep 2020 3 Minutes
Although separate disciplines, history and geography are tightly intertwined. While history attempts to examine human society, culture and experience through a temporal lense, geography does so through a spatial one.
A map of the world, surrounded by classical imagery. (Credit: Marzolino, via iStock.)
Physical geography and natural phenomena have undoubtedly influenced the course of history: the 1815 eruption of Mount Tambora, which caused massive global cooling, attests to this. It is human geography, however, which is more closely linked to history. History is, for example, an incredibly important factor in the development (or decline) of cities. The relative economic power of London, owing to it's strategic location and history as the Roman capital of Britannia, encouraged the Hanseatic League to set up it's main English trading post there. At this time the government was located in Westminster, but the relative autonomy afforded to the Corporation of London made it the main commercial centre on the island. Having its own government and liberal economic laws, and a notable lack of interference from the national government, made it attractive to traders. The wealth this generated caused an explosion in the urban population and forced the city to abandon the grid-iron street patterns favoured by the Romans to accommodate the growth. This led to an incredibly dense settlement with narrow winding streets, which in turn necessitated the development of public mass transit come industrialisation, now an essential component of every major city in the world.
In stark contrast to the wealth historical factors have helped to produce, an inability to develop regions of Southern Africa can, to a large extent, be traced to apartheid era debt afflicting the region. Not only must South Africans repay debts of a regime that oppressed them, but people in countries such as Lesotho, Malawi, and Mozambique have been saddled with enormous debts fighting South Africa's apartheid wars. These debts have made Southern Africa the poorest region in the world. As poverty is closely linked to high birth and mortality rates, the region's history has resulted in modern demographics unseen in other parts of the world, with incredibly young populations and rapid growth. This has resulted in dense urban developments and rapid land degradation as a result of overgrazing and the necessity of intensive agriculture.
Migration, another incredibly topical theme in modern geography, has also been affected by historical European colonialism. Push and pull factors are a foundation of modern studies of migration, and yet colonial ties have only been appreciated as a significant pull factor in recent history. While much credence was traditionally given to economic and social factors, historical factors have recently been afforded more attention. The Caribbean is a region characterised by a diverse group of people – forced migrants (slaves) from Africa as well as indentured and voluntary migrants from Europe and Asia. In the twentieth century the migration flow reversed and citizens emigrated in large numbers – overwhelmingly to the colonial motherland. While Britain was notable in its extensive attempts to restrict the volume of migrants, France and the Netherlands welcomed and indeed encouraged migration from the French Antilles and Suriname respectively. This is responsible for rapid demographic and social changes in those nations in the late 1900s which have shaped their population pyramids and urbanisation to this day.
History is also an oft-cited reason for territorial claims, such as those in the South China Sea. While geographical factors are the predominant reason why these islands are desirable, – with an abundance of oil and militarily strategic locations – a lack of clarity as to historical ownership has been used to justify claims. The 1951 Treaty of San Francisco failed to specify the status of the islands which left them open to competing territorial claims. China, following a border dispute with the Soviet Union, laid claim to all of the islands, not wanting Soviet ally North Vietnam to exert control over its coastline. After a minor naval skirmish with North Vietnam, China occupied Johnson reef and used this as a base to further its influence in the region. Today no less than seven nations compete for control over this trade passage worth $3.37 trillion.
Although it is obvious from these examples and countless more that history and geography are almost inseparable, historical geography remains a relatively new field with innumerable unexplored areas of research. Ultimately, it is only through an interdisciplinary approach that we can truly appreciate and understand that which we study. Hopefully, geographers and historians can move beyond their differences and appreciate this in future.
Michael Hendöe
Published 7th Sep 2020 10th Sep 2020
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One thought on "Does History Link to Geography?"
davidatqcm says:
interesting Mike, interesting


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"Happy Little Pill" is a single by singer and songwriter Troye Sivan. The song is the first single from TRXYE, his first extended play (EP) released by a major label. It was released digitally on 25 July 2014. The song was written by Troye Sivan, Brandon Rogers, and Tat Tong. "Happy Little Pill" peaked at number 10 on the Australian ARIA Singles Chart (aided by the sales of his extended play TRXYE that were not eligible to chart on the Australian Albums Chart) and was certified gold by the Australian Recording Industry Association for sales exceeding 35,000 copies. The single received critical acclaim for its mature sound.
Troye Sivan Mellet (born 5 June 1995 in Johannesburg, South Africa), better known as Troye Sivan, is an Australian actor, YouTuber and singer-songwriter. His YouTube channel has over 250 million views and more than 4,1 million subscribers. He is the third most subscribed Australian YouTube channel. In July of 2014 he released his first single titled "Happy Little Pill" from the TRXYE EP that was released on August 15 2014. As an actor, he played young James Howlett in the 2009 X-Men film X-Men Origins: Wolverine Show more ...
In the crowd, alone
And every second passing, reminds me I'm not home
Bright lights and city sounds, are ringin' like a drone
Unknown, Unknown
Oh, glazed eyes, empty hearts
Buyin' happy from shopping carts
Nothin' but time to kill
Sippin' life from bottles
Tight skin, bodyguards
Gucci down the boulevard
Cocaine, dollar bills, and
My happy little pill, take me away
Dry my eyes, bring colour to my skies
My sweet little pill, take my hunger
But within, numb my skin
Like a rock, afloat
Sweatin' conversations, seep into my bones
Four walls are not enough, I'll take a dip into
The unknown, unknown
Sippin' love from bottles
Blazed eyes, empty hearts
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1) Youth
2) My My My!
3) Wild
4) Dance to This (feat. Ariana Grande)
5) FOOLS
6) Happy Little Pill
7) Bloom
8) TALK ME DOWN
9) The Good Side
10) Bite
11) Lucky Strike
12) Animal
13) Seventeen
14) THE QUIET
15) Strawberries & Cigarettes
16) For Him.
17) EASE
18) SUBURBIA
19) LOST BOY
20) Plum
21) Touch
22) HEAVEN
23) COOL
24) What a Heavenly Way to Die
25) Blue
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2) BROODS - Bridges
3) Taylor Swift - Wonderland
4) Lorde - Bravado
5) Natalia Kills - Saturday Night
6) Lorde - Ribs
7) Sia - Burn the Pages
8) Charli XCX - Doing It (feat. Rita Ora)
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10) MØ - XXX 88 (feat. Diplo)
11) Betty Who - Somebody Loves You
12) Ellie Goulding - Anything Could Happen
13) Marina & the Diamonds - Lies
14) Natalia Kills - Devils Don't Fly
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19) MØ - Slow Love
20) Troye Sivan - Touch
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A 2010 laureate of the prestigious Fondation Marcel Bleustein-Blanchet in France, violinist Etienne Gara was invited to participate in the Perlman Music Program in New York in 2009 and 2010. Later he joined violinist Itzhak Perlman for the winter residency of the festival in Sarasota, Florida and for a tour in Israel. Mr. Gara has performed throughout Europe and North America in prominent halls such as Jordan Hall in Boston, the Marigny Theatre in Paris, the Wiener Saal and the Marmor Saal of the Mirabell Palace in Salzburg. Among several prizes, Mr. Gara won the Music Live Now competition, created by Sir Yehudi Menuhin. Born in Paris to a French mother and Hungarian father, Étienne Gara studied at the Conservatoire National de Région de Boulogne-Billancourt, continued his studies at the Mozarteum University of Salzburg, and completed his Performer Diploma at Indiana University. He earned his Master of Music at New England Conservatory, under renowned pedagogue and violinist Donald Weilerstein. Mr. Gara is currently under the tutelage of celebrated violinist Midori, at the University of Southern California.
Presenter / Producer: Classical Crossroads - Interludes.


The year is 1971. With Cold War tensions at their peak, the U.S.S. Farnley has been selected to take part in Operation Marathon, the brainchild of ambitious and unscrupulous one-star Admiral Eickhoff, who sees the dangerous experiment as his stepping stone to greater power and national prestige. The World War Two-era destroyer--with her outdated technology, low crew morale, and mentally unstable captain--would be an ideal component of the Admiral's experiment due to her chief characteristic: her expendability.
Failure, however, is not an option for the new Engineering Officer Lee. His enthusiastic faith in the viability of the aged vessel and his dedication to duty with honor puts him at immediate odds with soon-to-retire Chief Petty Officer Ross, while ultimately inspiring his fellow crewmen to help return the Farnley to fighting form. But their dedicated efforts are creating a powerful and unforgiving enemy--and when unexpected catastrophe befalls the Farnley, those manning the old destroyer will suddenly be forced to choose between duty and disobedience, between life and death.
Larry Laswell's The Marathon Watch is a gripping, authentically detailed naval adventure that harkens back to the heyday of the late, great storyteller Tom Clancy. Based in part on the author's actual military experiences, it provides a fascinating inside look at the shadowy intricacies of U.S. Navy politics wrapped around an enthralling story of honor, determination, and grave peril during the dark days of America's Cold War.
Featuring a richly developed cast of officers and crewmen, Laswell's thrilling chronicle transports the reader aboard the aging World War Two-era destroyer, U.S.S. Farnley, into the perilous heart of an ill-conceived experiment. There the frailties of human nature, the ambitions of unscrupulous men, and the terrible vicissitudes of nature will combine to create a disaster at sea that only courage, ingenuity, and a devotion to duty can possibly overcome.